Results of Parabolic Cylinder Functions: Difference between revisions

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Notation: 12.1 Special Notation
Properties: 12.2 Differential Equations
12.3 Graphics
12.4 Power-Series Expansions
12.5 Integral Representations
12.6 Continued Fraction
12.7 Relations to Other Functions
12.8 Recurrence Relations and Derivatives
12.9 Asymptotic Expansions for Large Variable
12.10 Uniform Asymptotic Expansions for Large Parameter
12.11 Zeros
12.12 Integrals
12.13 Sums
12.14 The Function ${\displaystyle{\displaystyle W\left(a,x\right)}}$
12.15 Generalized Parabolic Cylinder Functions
Applications: 12.16 Mathematical Applications
12.17 Physical Applications
Computation: 12.18 Methods of Computation
12.19 Tables
12.20 Approximations
12.21 Software

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+<div style="-moz-column-count:2; column-count:2;">
-{| class="wikitable sortable" style="margin: 0;"
+; Notation : [[12.1|12.1 Special Notation]]<br>
-|-
+; Properties : [[12.2|12.2 Differential Equations]]<br>[[12.3|12.3 Graphics]]<br>[[12.4|12.4 Power-Series Expansions]]<br>[[12.5|12.5 Integral Representations]]<br>[[12.6|12.6 Continued Fraction]]<br>[[12.7|12.7 Relations to Other Functions]]<br>[[12.8|12.8 Recurrence Relations and Derivatives]]<br>[[12.9|12.9 Asymptotic Expansions for Large Variable]]<br>[[12.10|12.10 Uniform Asymptotic Expansions for Large Parameter]]<br>[[12.11|12.11 Zeros]]<br>[[12.12|12.12 Integrals]]<br>[[12.13|12.13 Sums]]<br>[[12.14|12.14 The Function <math>\paraW@{a}{x}</math>]]<br>[[12.15|12.15 Generalized Parabolic Cylinder Functions]]<br>
-! scope="col" style="position: sticky; top: 0;" | DLMF
+; Applications : [[12.16|12.16 Mathematical Applications]]<br>[[12.17|12.17 Physical Applications]]<br>
-! scope="col" style="position: sticky; top: 0;" | Formula
+; Computation : [[12.18|12.18 Methods of Computation]]<br>[[12.19|12.19 Tables]]<br>[[12.20|12.20 Approximations]]<br>[[12.21|12.21 Software]]<br>
-! scope="col" style="position: sticky; top: 0;" | Constraints
-! scope="col" style="position: sticky; top: 0;" | Maple
-! scope="col" style="position: sticky; top: 0;" | Mathematica
-! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
-! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
-! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
-! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
-|-
-| [https://dlmf.nist.gov/12.2.E2 12.2.E2] || [[Item:Q4087|<math>\deriv[2]{w}{z}-\left(\tfrac{1}{4}z^{2}+a\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}-\left(\tfrac{1}{4}z^{2}+a\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])-((1)/(4)*(z)^(2)+ a)*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]-(Divide[1,4]*(z)^(2)+ a)*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.299038106+.4999999999*I
-Test Values: {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.299038106+1.000000000*I
-Test Values: {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.2990381056766582, 0.4999999999999999]
-Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.299038105676658, 0.9999999999999999]
-Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E3 12.2.E3] || [[Item:Q4088|<math>\deriv[2]{w}{z}+\left(\tfrac{1}{4}z^{2}-a\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}+\left(\tfrac{1}{4}z^{2}-a\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])+((1)/(4)*(z)^(2)- a)*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]+(Divide[1,4]*(z)^(2)- a)*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.299038106+1.000000000*I
-Test Values: {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.299038106+.4999999999*I
-Test Values: {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.299038105676658, 0.9999999999999999]
-Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.2990381056766582, 0.4999999999999999]
-Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E4 12.2.E4] || [[Item:Q4089|<math>\deriv[2]{w}{z}+\left(\nu+\tfrac{1}{2}-\tfrac{1}{4}z^{2}\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}+\left(\nu+\tfrac{1}{2}-\tfrac{1}{4}z^{2}\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])+(nu +(1)/(2)-(1)/(4)*(z)^(2))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]+(\[Nu]+Divide[1,2]-Divide[1,4]*(z)^(2))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9330127024+.8660254039*I
-Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9330127024+1.366025404*I
-Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [296 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.9330127018922196, 0.8660254037844386]
-Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.4330127018922191, 0.5000000000000001]
-Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E5 12.2.E5] || [[Item:Q4090|<math>\WhittakerparaD{\nu}@{z} = \paraU@{-\tfrac{1}{2}-\nu}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerparaD{\nu}@{z} = \paraU@{-\tfrac{1}{2}-\nu}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderD(nu, z) = CylinderU(-(1)/(2)- nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[\[Nu], z] == ParabolicCylinderD[- 1/2 -(-Divide[1,2]- \[Nu]), z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
-|-
-| [https://dlmf.nist.gov/12.2.E6 12.2.E6] || [[Item:Q4091|<math>\paraU@{a}{0} = \frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{0} = \frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}</syntaxhighlight> || <math>\realpart@@{(\frac{3}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, 0) = (sqrt(Pi))/((2)^((1)/(2)*a +(1)/(4))* GAMMA((3)/(4)+(1)/(2)*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), 0] == Divide[Sqrt[Pi],(2)^(Divide[1,2]*a +Divide[1,4])* Gamma[Divide[3,4]+Divide[1,2]*a]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 4]
-|-
-| [https://dlmf.nist.gov/12.2.E7 12.2.E7] || [[Item:Q4092|<math>\paraU'@{a}{0} = -\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU'@{a}{0} = -\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>subs( temp=0, diff( CylinderU(a, temp), temp$(1) ) ) = -(sqrt(Pi))/((2)^((1)/(2)*a -(1)/(4))* GAMMA((1)/(4)+(1)/(2)*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[ParabolicCylinderD[- 1/2 -(a), temp], {temp, 1}]/.temp-> 0) == -Divide[Sqrt[Pi],(2)^(Divide[1,2]*a -Divide[1,4])* Gamma[Divide[1,4]+Divide[1,2]*a]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/12.2.E8 12.2.E8] || [[Item:Q4093|<math>\paraV@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{1}{4}}}{\left(\EulerGamma@{\frac{3}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{1}{4}}}{\left(\EulerGamma@{\frac{3}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}</syntaxhighlight> || <math>\realpart@@{(\frac{3}{4}-\frac{1}{2}a)} > 0, \realpart@@{(\frac{1}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, 0) = (Pi*(2)^((1)/(2)*a +(1)/(4)))/((GAMMA((3)/(4)-(1)/(2)*a))^(2)* GAMMA((1)/(4)+(1)/(2)*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, 0] + ParabolicCylinderD[-(a) - 1/2, -(0)]) == Divide[Pi*(2)^(Divide[1,2]*a +Divide[1,4]),(Gamma[Divide[3,4]-Divide[1,2]*a])^(2)* Gamma[Divide[1,4]+Divide[1,2]*a]]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.7978845608028653, Times[0.7978845608028655, GAMMA[1.0]]]
-Test Values: {Rule[a, 0.5]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E9 12.2.E9] || [[Item:Q4094|<math>\paraV'@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{3}{4}}}{\left(\EulerGamma@{\frac{1}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV'@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{3}{4}}}{\left(\EulerGamma@{\frac{1}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{4}-\frac{1}{2}a)} > 0, \realpart@@{(\frac{3}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>subs( temp=0, diff( CylinderV(a, temp), temp$(1) ) ) = (Pi*(2)^((1)/(2)*a +(3)/(4)))/((GAMMA((1)/(4)-(1)/(2)*a))^(2)* GAMMA((3)/(4)+(1)/(2)*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, temp] + ParabolicCylinderD[-(a) - 1/2, -(temp)]), {temp, 1}]/.temp-> 0) == Divide[Pi*(2)^(Divide[1,2]*a +Divide[3,4]),(Gamma[Divide[1,4]-Divide[1,2]*a])^(2)* Gamma[Divide[3,4]+Divide[1,2]*a]]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -0.7978845608028653
-Test Values: {Rule[a, -0.5]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E10 12.2.E10] || [[Item:Q4095|<math>\Wronskian@{\paraU@{a}{z},\paraV@{a}{z}} = \sqrt{2/\pi}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraU@{a}{z},\paraV@{a}{z}} = \sqrt{2/\pi}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(CylinderU(a, z))*diff(CylinderV(a, z), z)-diff(CylinderU(a, z), z)*(CylinderV(a, z)) = sqrt(2/Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])}, z] == Sqrt[2/Pi]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .708254234e-1-.722805450e-2*I
-Test Values: {a = 3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .4257865765+.241883787*I
-Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.7978845608028654, Times[Complex[-3.533949646070574*^-17, -3.533949646070574*^-17], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.7978845608028654, Times[Complex[0.0, -2.1203697876423444*^-16], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E11 12.2.E11] || [[Item:Q4096|<math>\Wronskian@{\paraU@{a}{z},\paraU@{a}{-z}} = \frac{\sqrt{2\pi}}{\EulerGamma@{\frac{1}{2}+a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraU@{a}{z},\paraU@{a}{-z}} = \frac{\sqrt{2\pi}}{\EulerGamma@{\frac{1}{2}+a}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>(CylinderU(a, z))*diff(CylinderU(a, - z), z)-diff(CylinderU(a, z), z)*(CylinderU(a, - z)) = (sqrt(2*Pi))/(GAMMA((1)/(2)+ a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(a), - z]}, z] == Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.2.E12 12.2.E12] || [[Item:Q4097|<math>\Wronskian@{\paraU@{a}{z},\paraU@{-a}{+ iz}} = - ie^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraU@{a}{z},\paraU@{-a}{+ iz}} = - ie^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(CylinderU(a, z))*diff(CylinderU(- a, + I*z), z)-diff(CylinderU(a, z), z)*(CylinderU(- a, + I*z)) = - I*exp(+ I*Pi*((1)/(2)*a +(1)/(4)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(- a), + I*z]}, z] == - I*Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]</syntaxhighlight> || Failure || Failure || Successful [Tested: 42] || Successful [Tested: 42]
-|-
-| [https://dlmf.nist.gov/12.2.E12 12.2.E12] || [[Item:Q4097|<math>\Wronskian@{\paraU@{a}{z},\paraU@{-a}{- iz}} = + ie^{- i\pi(\frac{1}{2}a+\frac{1}{4})}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraU@{a}{z},\paraU@{-a}{- iz}} = + ie^{- i\pi(\frac{1}{2}a+\frac{1}{4})}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(CylinderU(a, z))*diff(CylinderU(- a, - I*z), z)-diff(CylinderU(a, z), z)*(CylinderU(- a, - I*z)) = + I*exp(- I*Pi*((1)/(2)*a +(1)/(4)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(- a), - I*z]}, z] == + I*Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]</syntaxhighlight> || Failure || Failure || Successful [Tested: 42] || Successful [Tested: 42]
-|-
-| [https://dlmf.nist.gov/12.2.E13 12.2.E13] || [[Item:Q4098|<math>\paraU@{-n-\tfrac{1}{2}}{-z} = (-1)^{n}\paraU@{-n-\tfrac{1}{2}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-n-\tfrac{1}{2}}{-z} = (-1)^{n}\paraU@{-n-\tfrac{1}{2}}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(- n -(1)/(2), - z) = (- 1)^(n)* CylinderU(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), - z] == (- 1)^(n)* ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.2.E14 12.2.E14] || [[Item:Q4099|<math>\paraV@{n+\tfrac{1}{2}}{-z} = (-1)^{n}\paraV@{n+\tfrac{1}{2}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{n+\tfrac{1}{2}}{-z} = (-1)^{n}\paraV@{n+\tfrac{1}{2}}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(n +(1)/(2), - z) = (- 1)^(n)* CylinderV(n +(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + n +Divide[1,2]], Pi]*(Sin[Pi*(n +Divide[1,2])] * ParabolicCylinderD[-(n +Divide[1,2]) - 1/2, - z] + ParabolicCylinderD[-(n +Divide[1,2]) - 1/2, -(- z)]) == (- 1)^(n)* Divide[GAMMA[1/2 + n +Divide[1,2]], Pi]*(Sin[Pi*(n +Divide[1,2])] * ParabolicCylinderD[-(n +Divide[1,2]) - 1/2, z] + ParabolicCylinderD[-(n +Divide[1,2]) - 1/2, -(z)])</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.2.E15 12.2.E15] || [[Item:Q4100|<math>\paraU@{a}{-z} = -\sin@{\pi a}\paraU@{a}{z}+\frac{\pi}{\EulerGamma@{\frac{1}{2}+a}}\paraV@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{-z} = -\sin@{\pi a}\paraU@{a}{z}+\frac{\pi}{\EulerGamma@{\frac{1}{2}+a}}\paraV@{a}{z}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, - z) = - sin(Pi*a)*CylinderU(a, z)+(Pi)/(GAMMA((1)/(2)+ a))*CylinderV(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), - z] == - Sin[Pi*a]*ParabolicCylinderD[- 1/2 -(a), z]+Divide[Pi,Gamma[Divide[1,2]+ a]]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[2.097331412545913, 1.9154557103012664], Times[Complex[-2.097331412545913, -1.9154557103012664], GAMMA[2.0]]]
-Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.668689589092481, 2.108602350101492], Times[Complex[0.668689589092481, -2.108602350101492], GAMMA[2.0]]]
-Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E16 12.2.E16] || [[Item:Q4101|<math>\paraV@{a}{-z} = \frac{\cos@{\pi a}}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sin@{\pi a}\paraV@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{-z} = \frac{\cos@{\pi a}}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sin@{\pi a}\paraV@{a}{z}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, - z) = (cos(Pi*a))/(GAMMA((1)/(2)- a))*CylinderU(a, z)+ sin(Pi*a)*CylinderV(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, - z] + ParabolicCylinderD[-(a) - 1/2, -(- z)]) == Divide[Cos[Pi*a],Gamma[Divide[1,2]- a]]*ParabolicCylinderD[- 1/2 -(a), z]+ Sin[Pi*a]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.3494376482945125, -0.44804866867585064], Times[Complex[0.1478618109503913, 0.18958829384201614], GAMMA[-1.5]]]
-Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.1936070900897449, -0.06991225535058408], Times[Complex[-0.5050655153080368, 0.029582824673347826], GAMMA[-1.5]]]
-Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E17 12.2.E17] || [[Item:Q4102|<math>\sqrt{2\pi}\paraU@{-a}{+ iz} = \EulerGamma@{\tfrac{1}{2}+a}\left(e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{z}+e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{-z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2\pi}\paraU@{-a}{+ iz} = \EulerGamma@{\tfrac{1}{2}+a}\left(e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{z}+e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{-z}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2*Pi)*CylinderU(- a, + I*z) = GAMMA((1)/(2)+ a)*(exp(- I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(a, z)+ exp(+ I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(a, - z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2*Pi]*ParabolicCylinderD[- 1/2 -(- a), + I*z] == Gamma[Divide[1,2]+ a]*(Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(a), z]+ Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(a), - z])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.2.E17 12.2.E17] || [[Item:Q4102|<math>\sqrt{2\pi}\paraU@{-a}{- iz} = \EulerGamma@{\tfrac{1}{2}+a}\left(e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{z}+e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{-z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2\pi}\paraU@{-a}{- iz} = \EulerGamma@{\tfrac{1}{2}+a}\left(e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{z}+e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{-z}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2*Pi)*CylinderU(- a, - I*z) = GAMMA((1)/(2)+ a)*(exp(+ I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(a, z)+ exp(- I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(a, - z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2*Pi]*ParabolicCylinderD[- 1/2 -(- a), - I*z] == Gamma[Divide[1,2]+ a]*(Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(a), z]+ Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(a), - z])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.2.E18 12.2.E18] || [[Item:Q4103|<math>\sqrt{2\pi}\paraU@{a}{z} = \EulerGamma@{\tfrac{1}{2}-a}\left(e^{- i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{+ iz}+e^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{- iz}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2\pi}\paraU@{a}{z} = \EulerGamma@{\tfrac{1}{2}-a}\left(e^{- i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{+ iz}+e^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{- iz}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2*Pi)*CylinderU(a, z) = GAMMA((1)/(2)- a)*(exp(- I*Pi*((1)/(2)*a +(1)/(4)))*CylinderU(- a, + I*z)+ exp(+ I*Pi*((1)/(2)*a +(1)/(4)))*CylinderU(- a, - I*z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2*Pi]*ParabolicCylinderD[- 1/2 -(a), z] == Gamma[Divide[1,2]- a]*(Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), + I*z]+ Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), - I*z])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.2.E18 12.2.E18] || [[Item:Q4103|<math>\sqrt{2\pi}\paraU@{a}{z} = \EulerGamma@{\tfrac{1}{2}-a}\left(e^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{- iz}+e^{- i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{+ iz}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2\pi}\paraU@{a}{z} = \EulerGamma@{\tfrac{1}{2}-a}\left(e^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{- iz}+e^{- i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{+ iz}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2*Pi)*CylinderU(a, z) = GAMMA((1)/(2)- a)*(exp(+ I*Pi*((1)/(2)*a +(1)/(4)))*CylinderU(- a, - I*z)+ exp(- I*Pi*((1)/(2)*a +(1)/(4)))*CylinderU(- a, + I*z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2*Pi]*ParabolicCylinderD[- 1/2 -(a), z] == Gamma[Divide[1,2]- a]*(Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), - I*z]+ Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), + I*z])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.2.E19 12.2.E19] || [[Item:Q4104|<math>\paraU@{a}{z} = + ie^{+ i\pi a}\paraU@{a}{-z}+\frac{\sqrt{2\pi}}{\EulerGamma@{\tfrac{1}{2}+a}}e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{+ iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = + ie^{+ i\pi a}\paraU@{a}{-z}+\frac{\sqrt{2\pi}}{\EulerGamma@{\tfrac{1}{2}+a}}e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{+ iz}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = + I*exp(+ I*Pi*a)*CylinderU(a, - z)+(sqrt(2*Pi))/(GAMMA((1)/(2)+ a))*exp(+ I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(- a, + I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == + I*Exp[+ I*Pi*a]*ParabolicCylinderD[- 1/2 -(a), - z]+Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]*Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), + I*z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.2.E19 12.2.E19] || [[Item:Q4104|<math>\paraU@{a}{z} = - ie^{- i\pi a}\paraU@{a}{-z}+\frac{\sqrt{2\pi}}{\EulerGamma@{\tfrac{1}{2}+a}}e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{- iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = - ie^{- i\pi a}\paraU@{a}{-z}+\frac{\sqrt{2\pi}}{\EulerGamma@{\tfrac{1}{2}+a}}e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{- iz}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = - I*exp(- I*Pi*a)*CylinderU(a, - z)+(sqrt(2*Pi))/(GAMMA((1)/(2)+ a))*exp(- I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(- a, - I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == - I*Exp[- I*Pi*a]*ParabolicCylinderD[- 1/2 -(a), - z]+Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]*Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), - I*z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.2.E20 12.2.E20] || [[Item:Q4105|<math>\paraV@{a}{z} = \frac{- i}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sqrt{\frac{2}{\pi}}e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{+ iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{z} = \frac{- i}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sqrt{\frac{2}{\pi}}e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{+ iz}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, z) = (- I)/(GAMMA((1)/(2)- a))*CylinderU(a, z)+sqrt((2)/(Pi))*exp(- I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(- a, + I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Divide[- I,Gamma[Divide[1,2]- a]]*ParabolicCylinderD[- 1/2 -(a), z]+Sqrt[Divide[2,Pi]]*Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), + I*z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.4621744673825597, -0.43960813814518984], Times[Complex[3.533949646070574*^-17, 0.0], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.0415095884926804, 0.5968092652227893], Times[Complex[1.0601848938211722*^-16, 3.533949646070574*^-17], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E20 12.2.E20] || [[Item:Q4105|<math>\paraV@{a}{z} = \frac{+ i}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sqrt{\frac{2}{\pi}}e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{- iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{z} = \frac{+ i}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sqrt{\frac{2}{\pi}}e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{- iz}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, z) = (+ I)/(GAMMA((1)/(2)- a))*CylinderU(a, z)+sqrt((2)/(Pi))*exp(+ I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(- a, - I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Divide[+ I,Gamma[Divide[1,2]- a]]*ParabolicCylinderD[- 1/2 -(a), z]+Sqrt[Divide[2,Pi]]*Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), - I*z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.4621744673825599, -0.4396081381451897], Times[Complex[3.533949646070574*^-17, 0.0], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.0415095884926797, 0.5968092652227891], Times[Complex[1.0601848938211722*^-16, 3.533949646070574*^-17], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.4.E1 12.4.E1] || [[Item:Q4109|<math>\paraU@{a}{z} = \paraU@{a}{0}u_{1}(a,z)+\paraU'@{a}{0}u_{2}(a,z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \paraU@{a}{0}u_{1}(a,z)+\paraU'@{a}{0}u_{2}(a,z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = CylinderU(a, 0)*(exp(-(1)/(4)*(z)^(2))*(1 +(a +(1)/(2))*((z)^(2))/(factorial(2))+(a +(1)/(2))*(a +(5)/(2))*((z)^(4))/(factorial(4))+ ..))+ subs( temp=0, diff( CylinderU(a, temp), temp$(1) ) )*(exp(-(1)/(4)*(z)^(2))*(z +(a +(3)/(2))*((z)^(3))/(factorial(3))+(a +(3)/(2))*(a +(7)/(2))*((z)^(5))/(factorial(5))+ ..))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == ParabolicCylinderD[- 1/2 -(a), 0]*(Exp[-Divide[1,4]*(z)^(2)]*(1 +(a +Divide[1,2])*Divide[(z)^(2),(2)!]+(a +Divide[1,2])*(a +Divide[5,2])*Divide[(z)^(4),(4)!]+ \[Ellipsis]))+ (D[ParabolicCylinderD[- 1/2 -(a), temp], {temp, 1}]/.temp-> 0)*(Exp[-Divide[1,4]*(z)^(2)]*(z +(a +Divide[3,2])*Divide[(z)^(3),(3)!]+(a +Divide[3,2])*(a +Divide[7,2])*Divide[(z)^(5),(5)!]+ \[Ellipsis]))</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.8412106300093095, 0.2667685495532514], Times[Complex[-0.8618940502981999, 0.18957697416081104], Plus[Complex[0.8660254037844387, 0.49999999999999994], …]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.7641562690755331, 0.8367141764786244], Times[Complex[-1.1066938670748312, -0.24342165324123666], Plus[Complex[-0.4999999999999998, 0.8660254037844387], …]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.4.E2 12.4.E2] || [[Item:Q4110|<math>\paraV@{a}{z} = \paraV@{a}{0}u_{1}(a,z)+\paraV'@{a}{0}u_{2}(a,z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{z} = \paraV@{a}{0}u_{1}(a,z)+\paraV'@{a}{0}u_{2}(a,z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(a, z) = CylinderV(a, 0)*(exp(-(1)/(4)*(z)^(2))*(1 +(a +(1)/(2))*((z)^(2))/(factorial(2))+(a +(1)/(2))*(a +(5)/(2))*((z)^(4))/(factorial(4))+ ..))+ subs( temp=0, diff( CylinderV(a, temp), temp$(1) ) )*(exp(-(1)/(4)*(z)^(2))*(z +(a +(3)/(2))*((z)^(3))/(factorial(3))+(a +(3)/(2))*(a +(7)/(2))*((z)^(5))/(factorial(5))+ ..))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, 0] + ParabolicCylinderD[-(a) - 1/2, -(0)])*(Exp[-Divide[1,4]*(z)^(2)]*(1 +(a +Divide[1,2])*Divide[(z)^(2),(2)!]+(a +Divide[1,2])*(a +Divide[5,2])*Divide[(z)^(4),(4)!]+ \[Ellipsis]))+ (D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, temp] + ParabolicCylinderD[-(a) - 1/2, -(temp)]), {temp, 1}]/.temp-> 0)*(Exp[-Divide[1,4]*(z)^(2)]*(z +(a +Divide[3,2])*Divide[(z)^(3),(3)!]+(a +Divide[3,2])*(a +Divide[7,2])*Divide[(z)^(5),(5)!]+ \[Ellipsis]))</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [40 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.0, Times[Complex[3.533949646070574*^-17, 0.0], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.0, Times[Complex[1.0601848938211722*^-16, 3.533949646070574*^-17], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.5.E1 12.5.E1] || [[Item:Q4115|<math>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{1}{2}+a}}\int_{0}^{\infty}t^{a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}-zt}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{1}{2}+a}}\int_{0}^{\infty}t^{a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}-zt}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > -\tfrac{1}{2}, \realpart@@{(\frac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (exp(-(1)/(4)*(z)^(2)))/(GAMMA((1)/(2)+ a))*int((t)^(a -(1)/(2))* exp(-(1)/(2)*(t)^(2)- z*t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Exp[-Divide[1,4]*(z)^(2)],Gamma[Divide[1,2]+ a]]*Integrate[(t)^(a -Divide[1,2])* Exp[-Divide[1,2]*(t)^(2)- z*t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
-|-
-| [https://dlmf.nist.gov/12.5.E2 12.5.E2] || [[Item:Q4116|<math>\paraU@{a}{z} = \frac{ze^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{3}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{3}{4}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{ze^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{3}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{3}{4}}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi, \realpart@@{a} > -\tfrac{1}{2}, \realpart@@{(\frac{1}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (z*exp(-(1)/(4)*(z)^(2)))/(GAMMA((1)/(4)+(1)/(2)*a))* int((t)^((1)/(2)*a -(3)/(4))* exp(- t)*((z)^(2)+ 2*t)^(-(1)/(2)*a -(3)/(4)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[z*Exp[-Divide[1,4]*(z)^(2)],Gamma[Divide[1,4]+Divide[1,2]*a]]* Integrate[(t)^(Divide[1,2]*a -Divide[3,4])* Exp[- t]*((z)^(2)+ 2*t)^(-Divide[1,2]*a -Divide[3,4]), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 15]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
-Test Values: {a = 2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
-Test Values: {a = 2, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br></div></div> || Successful [Tested: 15]
-|-
-| [https://dlmf.nist.gov/12.5.E3 12.5.E3] || [[Item:Q4117|<math>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{1}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{1}{4}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{1}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{1}{4}}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi, \realpart@@{a} > -\tfrac{3}{2}, \realpart@@{(\frac{3}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (exp(-(1)/(4)*(z)^(2)))/(GAMMA((3)/(4)+(1)/(2)*a))* int((t)^((1)/(2)*a -(1)/(4))* exp(- t)*((z)^(2)+ 2*t)^(-(1)/(2)*a -(1)/(4)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Exp[-Divide[1,4]*(z)^(2)],Gamma[Divide[3,4]+Divide[1,2]*a]]* Integrate[(t)^(Divide[1,2]*a -Divide[1,4])* Exp[- t]*((z)^(2)+ 2*t)^(-Divide[1,2]*a -Divide[1,4]), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 20]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
-Test Values: {a = 3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
-Test Values: {a = 3/2, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 20]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
-Test Values: {Rule[a, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
-Test Values: {Rule[a, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.5.E4 12.5.E4] || [[Item:Q4118|<math>\paraU@{a}{z} = \sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}\*\int_{0}^{\infty}t^{-a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}}\cos@{zt+\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)\pi}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}\*\int_{0}^{\infty}t^{-a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}}\cos@{zt+\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)\pi}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = sqrt((2)/(Pi))*exp((1)/(4)*(z)^(2))* int((t)^(- a -(1)/(2))* exp(-(1)/(2)*(t)^(2))*cos(z*t +((1)/(2)*a +(1)/(4))*Pi), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Sqrt[Divide[2,Pi]]*Exp[Divide[1,4]*(z)^(2)]* Integrate[(t)^(- a -Divide[1,2])* Exp[-Divide[1,2]*(t)^(2)]*Cos[z*t +(Divide[1,2]*a +Divide[1,4])*Pi], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.5.E5 12.5.E5] || [[Item:Q4119|<math>\paraU@{a}{z} = \frac{\EulerGamma@{\frac{1}{2}-a}}{2\pi i}e^{-\frac{1}{4}z^{2}}\int_{-\infty}^{(0+)}e^{zt-\frac{1}{2}t^{2}}t^{a-\frac{1}{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{\EulerGamma@{\frac{1}{2}-a}}{2\pi i}e^{-\frac{1}{4}z^{2}}\int_{-\infty}^{(0+)}e^{zt-\frac{1}{2}t^{2}}t^{a-\frac{1}{2}}\diff{t}</syntaxhighlight> || <math>a \neq \frac{1}{2}, -\pi < \phase@@{t}, \phase@@{t} < \pi, \realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (GAMMA((1)/(2)- a))/(2*Pi*I)*exp(-(1)/(4)*(z)^(2))*int(exp(z*t -(1)/(2)*(t)^(2))*(t)^(a -(1)/(2)), t = - infinity..(0 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Gamma[Divide[1,2]- a],2*Pi*I]*Exp[-Divide[1,4]*(z)^(2)]*Integrate[Exp[z*t -Divide[1,2]*(t)^(2)]*(t)^(a -Divide[1,2]), {t, - Infinity, (0 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
-|-
-| [https://dlmf.nist.gov/12.5.E6 12.5.E6] || [[Item:Q4120|<math>\paraU@{a}{z} = \frac{e^{\frac{1}{4}z^{2}}}{i\sqrt{2\pi}}\int_{c-i\infty}^{c+i\infty}e^{-zt+\frac{1}{2}t^{2}}t^{-a-\frac{1}{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{e^{\frac{1}{4}z^{2}}}{i\sqrt{2\pi}}\int_{c-i\infty}^{c+i\infty}e^{-zt+\frac{1}{2}t^{2}}t^{-a-\frac{1}{2}}\diff{t}</syntaxhighlight> || <math>-\tfrac{1}{2}\pi < \phase@@{t}, \phase@@{t} < \tfrac{1}{2}\pi, c > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (exp((1)/(4)*(z)^(2)))/(I*sqrt(2*Pi))*int(exp(- z*t +(1)/(2)*(t)^(2))*(t)^(- a -(1)/(2)), t = c - I*infinity..c + I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Exp[Divide[1,4]*(z)^(2)],I*Sqrt[2*Pi]]*Integrate[Exp[- z*t +Divide[1,2]*(t)^(2)]*(t)^(- a -Divide[1,2]), {t, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [126 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8412106295+.2667685493*I
-Test Values: {a = -3/2, c = 3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.7641562685+.8367141760*I
-Test Values: {a = -3/2, c = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.5.E8 12.5.E8] || [[Item:Q4122|<math>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}}{2\pi i\EulerGamma@{\frac{1}{2}+a}}\*\int_{-i\infty}^{i\infty}\EulerGamma@{t}\EulerGamma@{\tfrac{1}{2}+a-2t}2^{t}z^{2t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}}{2\pi i\EulerGamma@{\frac{1}{2}+a}}\*\int_{-i\infty}^{i\infty}\EulerGamma@{t}\EulerGamma@{\tfrac{1}{2}+a-2t}2^{t}z^{2t}\diff{t}</syntaxhighlight> || <math>a \neq -\frac{1}{2}, |\phase@@{z}| < \tfrac{3}{4}\pi, \realpart@@{t} > 0, \realpart@@{(\tfrac{1}{2}+a-2t)} > 0, \realpart@@{(\frac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (exp(-(1)/(4)*(z)^(2))*(z)^(- a -(1)/(2)))/(2*Pi*I*GAMMA((1)/(2)+ a))* int(GAMMA(t)*GAMMA((1)/(2)+ a - 2*t)*(2)^(t)* (z)^(2*t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Exp[-Divide[1,4]*(z)^(2)]*(z)^(- a -Divide[1,2]),2*Pi*I*Gamma[Divide[1,2]+ a]]* Integrate[Gamma[t]*Gamma[Divide[1,2]+ a - 2*t]*(2)^(t)* (z)^(2*t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.5.E9 12.5.E9] || [[Item:Q4123|<math>\paraV@{a}{z} = \sqrt{\frac{2}{\pi}}\frac{e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}}{2\pi i\EulerGamma@{\frac{1}{2}-a}}\*\int_{-i\infty}^{i\infty}\EulerGamma@{t}\EulerGamma@{\tfrac{1}{2}-a-2t}2^{t}z^{2t}\cos@{\pi t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{z} = \sqrt{\frac{2}{\pi}}\frac{e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}}{2\pi i\EulerGamma@{\frac{1}{2}-a}}\*\int_{-i\infty}^{i\infty}\EulerGamma@{t}\EulerGamma@{\tfrac{1}{2}-a-2t}2^{t}z^{2t}\cos@{\pi t}\diff{t}</syntaxhighlight> || <math>a \neq \frac{1}{2}, |\phase@@{z}| < \tfrac{1}{4}\pi, \realpart@@{t} > 0, \realpart@@{(\tfrac{1}{2}-a-2t)} > 0, \realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, z) = sqrt((2)/(Pi))*(exp((1)/(4)*(z)^(2))*(z)^(a -(1)/(2)))/(2*Pi*I*GAMMA((1)/(2)- a))* int(GAMMA(t)*GAMMA((1)/(2)- a - 2*t)*(2)^(t)* (z)^(2*t)* cos(Pi*t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Sqrt[Divide[2,Pi]]*Divide[Exp[Divide[1,4]*(z)^(2)]*(z)^(a -Divide[1,2]),2*Pi*I*Gamma[Divide[1,2]- a]]* Integrate[Gamma[t]*Gamma[Divide[1,2]- a - 2*t]*(2)^(t)* (z)^(2*t)* Cos[Pi*t], {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.7.E1 12.7.E1] || [[Item:Q4124|<math>\paraU@{-\tfrac{1}{2}}{z} = \WhittakerparaD{0}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-\tfrac{1}{2}}{z} = \WhittakerparaD{0}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(-(1)/(2), z) = CylinderD(0, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(-Divide[1,2]), z] == ParabolicCylinderD[0, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.7.E1 12.7.E1] || [[Item:Q4124|<math>\WhittakerparaD{0}@{z} = e^{-\frac{1}{4}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerparaD{0}@{z} = e^{-\frac{1}{4}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderD(0, z) = exp(-(1)/(4)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[0, z] == Exp[-Divide[1,4]*(z)^(2)]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.7.E2 12.7.E2] || [[Item:Q4125|<math>\paraU@{-n-\tfrac{1}{2}}{z} = \WhittakerparaD{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-n-\tfrac{1}{2}}{z} = \WhittakerparaD{n}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(- n -(1)/(2), z) = CylinderD(n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), z] == ParabolicCylinderD[n, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.7.E4 12.7.E4] || [[Item:Q4127|<math>\paraV@{-\tfrac{1}{2}}{z} = (\ifrac{2}{\sqrt{\pi}}\,)e^{\frac{1}{4}z^{2}}\DawsonsintF@{z/\sqrt{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{-\tfrac{1}{2}}{z} = (\ifrac{2}{\sqrt{\pi}}\,)e^{\frac{1}{4}z^{2}}\DawsonsintF@{z/\sqrt{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(-(1)/(2), z) = ((2)/(sqrt(Pi)))*exp((1)/(4)*(z)^(2))*dawson(z/(sqrt(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + -Divide[1,2]], Pi]*(Sin[Pi*(-Divide[1,2])] * ParabolicCylinderD[-(-Divide[1,2]) - 1/2, z] + ParabolicCylinderD[-(-Divide[1,2]) - 1/2, -(z)]) == (Divide[2,Sqrt[Pi]])*Exp[Divide[1,4]*(z)^(2)]*DawsonF[z/(Sqrt[2])]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.6813729414422256, -0.33849358809725466]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.4709386394349885, -0.6804499612300876]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E5 12.7.E5] || [[Item:Q4128|<math>\paraU@{\tfrac{1}{2}}{z} = \WhittakerparaD{-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{\tfrac{1}{2}}{z} = \WhittakerparaD{-1}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU((1)/(2), z) = CylinderD(- 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(Divide[1,2]), z] == ParabolicCylinderD[- 1, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.7.E5 12.7.E5] || [[Item:Q4128|<math>\WhittakerparaD{-1}@{z} = \sqrt{\tfrac{1}{2}\pi}\,e^{\frac{1}{4}z^{2}}\erfc@{z/\sqrt{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerparaD{-1}@{z} = \sqrt{\tfrac{1}{2}\pi}\,e^{\frac{1}{4}z^{2}}\erfc@{z/\sqrt{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderD(- 1, z) = sqrt((1)/(2)*Pi)*exp((1)/(4)*(z)^(2))*erfc(z/(sqrt(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1, z] == Sqrt[Divide[1,2]*Pi]*Exp[Divide[1,4]*(z)^(2)]*Erfc[z/(Sqrt[2])]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.7.E6 12.7.E6] || [[Item:Q4129|<math>\paraU@{n+\tfrac{1}{2}}{z} = \WhittakerparaD{-n-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{n+\tfrac{1}{2}}{z} = \WhittakerparaD{-n-1}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(n +(1)/(2), z) = CylinderD(- n - 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(n +Divide[1,2]), z] == ParabolicCylinderD[- n - 1, z]</syntaxhighlight> || Successful || Successful || Manual Skip! || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.7.E6 12.7.E6] || [[Item:Q4129|<math>\WhittakerparaD{-n-1}@{z} = \sqrt{\frac{\pi}{2}}\frac{(-1)^{n}}{n!}e^{-\frac{1}{4}z^{2}}\deriv[n]{\left(e^{\frac{1}{2}z^{2}}\erfc@{z/\sqrt{2}}\right)}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerparaD{-n-1}@{z} = \sqrt{\frac{\pi}{2}}\frac{(-1)^{n}}{n!}e^{-\frac{1}{4}z^{2}}\deriv[n]{\left(e^{\frac{1}{2}z^{2}}\erfc@{z/\sqrt{2}}\right)}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderD(- n - 1, z) = sqrt((Pi)/(2))*((- 1)^(n))/(factorial(n))*exp(-(1)/(4)*(z)^(2))*diff(exp((1)/(2)*(z)^(2))*erfc(z/(sqrt(2))), [z$(n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- n - 1, z] == Sqrt[Divide[Pi,2]]*Divide[(- 1)^(n),(n)!]*Exp[-Divide[1,4]*(z)^(2)]*D[Exp[Divide[1,2]*(z)^(2)]*Erfc[z/(Sqrt[2])], {z, n}]</syntaxhighlight> || Failure || Failure || Manual Skip! || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.017848622575954935, Times[0.7141168694348256, DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[-1, []], Times[-1, 1.5, [Plus[1, ]]], Times[Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 2], Power[1.5, 2]]], Erfc[Times[Power[2, Rational[-1, 2]], 1.5]]]], Equal[[1], Plus[Times[-1, Power[Times[2, Power[Pi, -1]], Rational[1, 2]]], Times[Power[E, Times[Rational[1, 2], Power[1.5, 2]]], 1.5, Erfc[Times[Power[2, Rational[-1, 2]], 1.5]]]]]}]][3.0]]], {Rule[n, 3], Rule[z, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.1293114227985036, Times[1.1773796724029832, DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[-1, []], Times[-1, 0.5, [Plus[1, ]]], Times[Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 2], Power[0.5, 2]]], Erfc[Times[Power[2, Rational[-1, 2]], 0.5]]]], Equal[[1], Plus[Times[-1, Power[Times[2, Power[Pi, -1]], Rational[1, 2]]], Times[Power[E, Times[Rational[1, 2], Power[0.5, 2]]], 0.5, Erfc[Times[Power[2, Rational[-1, 2]], 0.5]]]]]}]][3.0]]], {Rule[n, 3], Rule[z, 0.5]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E8 12.7.E8] || [[Item:Q4131|<math>\paraU@{-2}{z} = \frac{z^{5/2}}{4\sqrt{2\pi}}\left(2\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}+3\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}-\modBesselK{\frac{5}{4}}@{\tfrac{1}{4}z^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-2}{z} = \frac{z^{5/2}}{4\sqrt{2\pi}}\left(2\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}+3\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}-\modBesselK{\frac{5}{4}}@{\tfrac{1}{4}z^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(- 2, z) = ((z)^(5/2))/(4*sqrt(2*Pi))*(2*BesselK((1)/(4), (1)/(4)*(z)^(2))+ 3*BesselK((3)/(4), (1)/(4)*(z)^(2))- BesselK((5)/(4), (1)/(4)*(z)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(- 2), z] == Divide[(z)^(5/2),4*Sqrt[2*Pi]]*(2*BesselK[Divide[1,4], Divide[1,4]*(z)^(2)]+ 3*BesselK[Divide[3,4], Divide[1,4]*(z)^(2)]- BesselK[Divide[5,4], Divide[1,4]*(z)^(2)])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -2.928712959+.1903824416*I
-Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.578570932+.7263102924*I
-Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-2.928712959362369, 0.19038244130086163]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.5785709321816723, 0.7263102922437361]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E9 12.7.E9] || [[Item:Q4132|<math>\paraU@{-1}{z} = \frac{z^{3/2}}{2\sqrt{2\pi}}\left(\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}+\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-1}{z} = \frac{z^{3/2}}{2\sqrt{2\pi}}\left(\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}+\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(- 1, z) = ((z)^(3/2))/(2*sqrt(2*Pi))*(BesselK((1)/(4), (1)/(4)*(z)^(2))+ BesselK((3)/(4), (1)/(4)*(z)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(- 1), z] == Divide[(z)^(3/2),2*Sqrt[2*Pi]]*(BesselK[Divide[1,4], Divide[1,4]*(z)^(2)]+ BesselK[Divide[3,4], Divide[1,4]*(z)^(2)])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .5254625443+1.913964596*I
-Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1061142274-1.367750447*I
-Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.5254625445137794, 1.9139645960722755]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.10611422720224939, -1.3677504477251]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E10 12.7.E10] || [[Item:Q4133|<math>\paraU@{0}{z} = \sqrt{\frac{z}{2\pi}}\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{0}{z} = \sqrt{\frac{z}{2\pi}}\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(0, z) = sqrt((z)/(2*Pi))*BesselK((1)/(4), (1)/(4)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(0), z] == Sqrt[Divide[z,2*Pi]]*BesselK[Divide[1,4], Divide[1,4]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.016879450-1.384601654*I
-Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.973186649+1.022506910*I
-Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.0168794499257325, -1.3846016541017099]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.9731866495584476, 1.0225069102497304]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E11 12.7.E11] || [[Item:Q4134|<math>\paraU@{1}{z} = \frac{z^{3/2}}{\sqrt{2\pi}}\left(\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}-\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{1}{z} = \frac{z^{3/2}}{\sqrt{2\pi}}\left(\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}-\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(1, z) = ((z)^(3/2))/(sqrt(2*Pi))*(BesselK((3)/(4), (1)/(4)*(z)^(2))- BesselK((1)/(4), (1)/(4)*(z)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(1), z] == Divide[(z)^(3/2),Sqrt[2*Pi]]*(BesselK[Divide[3,4], Divide[1,4]*(z)^(2)]- BesselK[Divide[1,4], Divide[1,4]*(z)^(2)])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .6696041257-1.050010143*I
-Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.182924166+1.008719675*I
-Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.6696041258052213, -1.050010141970097]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.1829241651976083, 1.0087196737510498]
-Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E14 12.7.E14] || [[Item:Q4137|<math>\paraU@{a}{z} = 2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = 2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (2)^(-(1)/(4)-(1)/(2)*a)* exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(1)/(4), (1)/(2), (1)/(2)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == (2)^(-Divide[1,4]-Divide[1,2]*a)* Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[1,4], Divide[1,2], Divide[1,2]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.528312538+1.673428352*I
-Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.682421259-.5335370987*I
-Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.5283125381510665, 1.6734283529572487]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.6824212600186188, -0.5335370991065028]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E14 12.7.E14] || [[Item:Q4137|<math>2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)^(-(1)/(4)-(1)/(2)*a)* exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(1)/(4), (1)/(2), (1)/(2)*(z)^(2)) = (2)^(-(3)/(4)-(1)/(2)*a)* z*exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(3)/(4), (3)/(2), (1)/(2)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(2)^(-Divide[1,4]-Divide[1,2]*a)* Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[1,4], Divide[1,2], Divide[1,2]*(z)^(2)] == (2)^(-Divide[3,4]-Divide[1,2]*a)* z*Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[3,4], Divide[3,2], Divide[1,2]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.528312538-1.673428353*I
-Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.682421260+.5335370988*I
-Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.5283125381510665, -1.673428352957249]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.6824212600186188, 0.5335370991065027]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E14 12.7.E14] || [[Item:Q4137|<math>2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{1}{2}a}z^{-\frac{1}{2}}\WhittakerconfhyperW{-\frac{1}{2}a}{+\frac{1}{4}}@{\tfrac{1}{2}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{1}{2}a}z^{-\frac{1}{2}}\WhittakerconfhyperW{-\frac{1}{2}a}{+\frac{1}{4}}@{\tfrac{1}{2}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)^(-(3)/(4)-(1)/(2)*a)* z*exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(3)/(4), (3)/(2), (1)/(2)*(z)^(2)) = (2)^(-(1)/(2)*a)* (z)^(-(1)/(2))* WhittakerW(-(1)/(2)*a, +(1)/(4), (1)/(2)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(2)^(-Divide[3,4]-Divide[1,2]*a)* z*Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[3,4], Divide[3,2], Divide[1,2]*(z)^(2)] == (2)^(-Divide[1,2]*a)* (z)^(-Divide[1,2])* WhittakerW[-Divide[1,2]*a, +Divide[1,4], Divide[1,2]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .725579081e-1+1.600870446*I
-Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5744420805-1.107979180*I
-Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0725579074030912, 1.600870445554158]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.574442080456058, -1.1079791795625606]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E14 12.7.E14] || [[Item:Q4137|<math>2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{1}{2}a}z^{-\frac{1}{2}}\WhittakerconfhyperW{-\frac{1}{2}a}{-\frac{1}{4}}@{\tfrac{1}{2}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{1}{2}a}z^{-\frac{1}{2}}\WhittakerconfhyperW{-\frac{1}{2}a}{-\frac{1}{4}}@{\tfrac{1}{2}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)^(-(3)/(4)-(1)/(2)*a)* z*exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(3)/(4), (3)/(2), (1)/(2)*(z)^(2)) = (2)^(-(1)/(2)*a)* (z)^(-(1)/(2))* WhittakerW(-(1)/(2)*a, -(1)/(4), (1)/(2)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(2)^(-Divide[3,4]-Divide[1,2]*a)* z*Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[3,4], Divide[3,2], Divide[1,2]*(z)^(2)] == (2)^(-Divide[1,2]*a)* (z)^(-Divide[1,2])* WhittakerW[-Divide[1,2]*a, -Divide[1,4], Divide[1,2]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .725579081e-1+1.600870446*I
-Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5744420804-1.107979180*I
-Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0725579074030912, 1.600870445554158]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.5744420804560579, -1.1079791795625609]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E1 12.8.E1] || [[Item:Q4138|<math>z\paraU@{a}{z}-\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z\paraU@{a}{z}-\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z*CylinderU(a, z)- CylinderU(a - 1, z)+(a +(1)/(2))*CylinderU(a + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>z*ParabolicCylinderD[- 1/2 -(a), z]- ParabolicCylinderD[- 1/2 -(a - 1), z]+(a +Divide[1,2])*ParabolicCylinderD[- 1/2 -(a + 1), z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
-|-
-| [https://dlmf.nist.gov/12.8.E2 12.8.E2] || [[Item:Q4139|<math>\paraU'@{a}{z}+\tfrac{1}{2}z\paraU@{a}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU'@{a}{z}+\tfrac{1}{2}z\paraU@{a}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( CylinderU(a, z), z$(1) )+(1)/(2)*z*CylinderU(a, z)+(a +(1)/(2))*CylinderU(a + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]+Divide[1,2]*z*ParabolicCylinderD[- 1/2 -(a), z]+(a +Divide[1,2])*ParabolicCylinderD[- 1/2 -(a + 1), z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
-|-
-| [https://dlmf.nist.gov/12.8.E3 12.8.E3] || [[Item:Q4140|<math>\paraU'@{a}{z}-\tfrac{1}{2}z\paraU@{a}{z}+\paraU@{a-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU'@{a}{z}-\tfrac{1}{2}z\paraU@{a}{z}+\paraU@{a-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( CylinderU(a, z), z$(1) )-(1)/(2)*z*CylinderU(a, z)+ CylinderU(a - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]-Divide[1,2]*z*ParabolicCylinderD[- 1/2 -(a), z]+ ParabolicCylinderD[- 1/2 -(a - 1), z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
-|-
-| [https://dlmf.nist.gov/12.8.E4 12.8.E4] || [[Item:Q4141|<math>2\paraU'@{a}{z}+\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2\paraU'@{a}{z}+\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>2*diff( CylinderU(a, z), z$(1) )+ CylinderU(a - 1, z)+(a +(1)/(2))*CylinderU(a + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]+ ParabolicCylinderD[- 1/2 -(a - 1), z]+(a +Divide[1,2])*ParabolicCylinderD[- 1/2 -(a + 1), z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
-|-
-| [https://dlmf.nist.gov/12.8.E5 12.8.E5] || [[Item:Q4142|<math>z\paraV@{a}{z}-\paraV@{a+1}{z}+(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z\paraV@{a}{z}-\paraV@{a+1}{z}+(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z*CylinderV(a, z)- CylinderV(a + 1, z)+(a -(1)/(2))*CylinderV(a - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)])+(a -Divide[1,2])*Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[3.060490169192143*^-17, 1.7669748230352868*^-17], GAMMA[-1.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-8.361414638298002*^-17, 7.414495684541142*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E6 12.8.E6] || [[Item:Q4143|<math>\paraV'@{a}{z}-\tfrac{1}{2}z\paraV@{a}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV'@{a}{z}-\tfrac{1}{2}z\paraV@{a}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( CylinderV(a, z), z$(1) )-(1)/(2)*z*CylinderV(a, z)-(a -(1)/(2))*CylinderV(a - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]-Divide[1,2]*z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])-(a -Divide[1,2])*Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [39 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[-7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[-1.5302450845960716*^-17, -8.834874115176434*^-18], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-9.955091265133296*^-17, -1.7329819619999673*^-18], GAMMA[-1.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E7 12.8.E7] || [[Item:Q4144|<math>\paraV'@{a}{z}+\tfrac{1}{2}z\paraV@{a}{z}-\paraV@{a+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV'@{a}{z}+\tfrac{1}{2}z\paraV@{a}{z}-\paraV@{a+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( CylinderV(a, z), z$(1) )+(1)/(2)*z*CylinderV(a, z)- CylinderV(a + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]+Divide[1,2]*z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)]) == 0</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[1.5302450845960716*^-17, 8.834874115176434*^-18], GAMMA[-1.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-1.83165059034313*^-16, 7.241197488341145*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E8 12.8.E8] || [[Item:Q4145|<math>2\paraV'@{a}{z}-\paraV@{a+1}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2\paraV'@{a}{z}-\paraV@{a+1}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>2*diff( CylinderV(a, z), z$(1) )- CylinderV(a + 1, z)-(a -(1)/(2))*CylinderV(a - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)])-(a -Divide[1,2])*Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[-7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-2.8271597168564594*^-16, 7.067899292141149*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]]
-Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E9 12.8.E9] || [[Item:Q4146|<math>\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}+a}{m}e^{\frac{1}{4}z^{2}}\paraU@{a+m}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}+a}{m}e^{\frac{1}{4}z^{2}}\paraU@{a+m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp((1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)]) = (- 1)^(m)* pochhammer((1)/(2)+ a, m)*exp((1)/(4)*(z)^(2))*CylinderU(a + m, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a), z], {z, m}] == (- 1)^(m)* Pochhammer[Divide[1,2]+ a, m]*Exp[Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a + m), z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.0, 0.0], DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][1.0]], {Rule[a, -1.5], Rule[m, 1], Rule[z, Power[E, Times<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[2.0, DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E10 12.8.E10] || [[Item:Q4147|<math>\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}e^{-\frac{1}{4}z^{2}}\paraU@{a-m}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}e^{-\frac{1}{4}z^{2}}\paraU@{a-m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp(-(1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)]) = (- 1)^(m)* exp(-(1)/(4)*(z)^(2))*CylinderU(a - m, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[-Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a), z], {z, m}] == (- 1)^(m)* Exp[-Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a - m), z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.07045205979755337, 0.7756076114781977], DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]}]][1.0]], {Rule[a, -1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[2.000032302229117, -0.49556574541480647], Times[2.0, DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E11 12.8.E11] || [[Item:Q4148|<math>\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = e^{\frac{1}{4}z^{2}}\paraV@{a+m}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = e^{\frac{1}{4}z^{2}}\paraV@{a+m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp((1)/(4)*(z)^(2))*CylinderV(a, z), [z$(m)]) = exp((1)/(4)*(z)^(2))*CylinderV(a + m, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, m}] == Exp[Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a + m], Pi]*(Sin[Pi*(a + m)] * ParabolicCylinderD[-(a + m) - 1/2, z] + ParabolicCylinderD[-(a + m) - 1/2, -(z)])</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [126 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[2.150599663294456*^-18, -9.777500999643939*^-18], GAMMA[0.0]], Times[0.3183098861837907, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], <syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-0.9299481905237211, -0.4298894311242862], GAMMA[1.0]], Times[0.6366197723675814, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0], Times[1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E12 12.8.E12] || [[Item:Q4149|<math>\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}-a}{m}e^{-\frac{1}{4}z^{2}}\paraV@{a-m}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}-a}{m}e^{-\frac{1}{4}z^{2}}\paraV@{a-m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp(-(1)/(4)*(z)^(2))*CylinderV(a, z), [z$(m)]) = (- 1)^(m)* pochhammer((1)/(2)- a, m)*exp(-(1)/(4)*(z)^(2))*CylinderV(a - m, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[-Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, m}] == (- 1)^(m)* Pochhammer[Divide[1,2]- a, m]*Exp[-Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a - m], Pi]*(Sin[Pi*(a - m)] * ParabolicCylinderD[-(a - m) - 1/2, z] + ParabolicCylinderD[-(a - m) - 1/2, -(z)])</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [126 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-6.091780348003315*^-17, 1.3399109614774574*^-17], GAMMA[-2.0]], Times[0.3183098861837907, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][1.0], Times[1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Ti<syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[1.6052302675286988*^-15, 3.2948039393826443*^-16], GAMMA[-3.0]], Times[0.6366197723675814, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
-Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][2.0], Times[1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]}]][2.0]]]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex1 12.10#Ex1] || [[Item:Q4154|<math>a = +\tfrac{1}{2}\mu^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a = +\tfrac{1}{2}\mu^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a = +(1)/(2)*(mu)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a == +Divide[1,2]*\[Mu]^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex2 12.10#Ex2] || [[Item:Q4155|<math>x = \mu t\sqrt{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>x = \mu t\sqrt{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">x = mu*t*sqrt(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">x == \[Mu]*t*Sqrt[2]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.10.E2 12.10.E2] || [[Item:Q4156|<math>\deriv[2]{w}{t} = \mu^{4}(t^{2}+ 1)w</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{t} = \mu^{4}(t^{2}+ 1)w</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [t$(2)]) = (mu)^(4)*((t)^(2)+ 1)*w</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {t, 2}] == \[Mu]^(4)*((t)^(2)+ 1)*w</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.814582564-1.625000003*I
-Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.625000003+2.814582564*I
-Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.814582562299425, -1.6250000000000009]
-Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.814582562299425, -1.6250000000000009]
-Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.10.E2 12.10.E2] || [[Item:Q4156|<math>\deriv[2]{w}{t} = \mu^{4}(t^{2}- 1)w</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{t} = \mu^{4}(t^{2}- 1)w</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [t$(2)]) = (mu)^(4)*((t)^(2)- 1)*w</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {t, 2}] == \[Mu]^(4)*((t)^(2)- 1)*w</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.082531755-.6250000011*I
-Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .6250000011+1.082531755*I
-Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.0825317547305482, -0.6250000000000002]
-Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.0825317547305482, -0.6250000000000002]
-Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10.E13 12.10.E13] || [[Item:Q4171|<math>v_{s}(t) = u_{s}(t)+\tfrac{1}{2}tu_{s-1}(t)-r_{s-2}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>v_{s}(t) = u_{s}(t)+\tfrac{1}{2}tu_{s-1}(t)-r_{s-2}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">v[s](t) = u[s](t)+(1)/(2)*tu[s - 1](t)- r[s - 2](t)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[v, s][t] == Subscript[u, s][t]+Divide[1,2]*Subscript[tu, s - 1][t]- Subscript[r, s - 2][t]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex9 12.10#Ex9] || [[Item:Q4175|<math>\gamma_{0} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{0} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[0] = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 0] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex10 12.10#Ex10] || [[Item:Q4176|<math>\gamma_{1} = -\tfrac{1}{24}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{1} = -\tfrac{1}{24}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[1] = -(1)/(24)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 1] == -Divide[1,24]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex11 12.10#Ex11] || [[Item:Q4177|<math>\gamma_{2} = \tfrac{1}{1152}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{2} = \tfrac{1}{1152}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[2] = (1)/(1152)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 2] == Divide[1,1152]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex12 12.10#Ex12] || [[Item:Q4178|<math>\gamma_{3} = \tfrac{1003}{4\;14720}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{3} = \tfrac{1003}{4\;14720}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[3] = (1003)/(414720)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 3] == Divide[1003,414720]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex13 12.10#Ex13] || [[Item:Q4179|<math>\gamma_{4} = -\tfrac{4027}{398\;13120}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{4} = -\tfrac{4027}{398\;13120}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[4] = -(4027)/(39813120)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 4] == -Divide[4027,39813120]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex18 12.10#Ex18] || [[Item:Q4198|<math>\mathsf{A}_{1}(\tau) = -\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathsf{A}_{1}(\tau) = -\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">A[1]*(((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))) = -(1)/(12)*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))*(20*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(2)+ 30*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))+ 9)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[A, 1]*((Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))) == -Divide[1,12]*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))*(20*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(2)+ 30*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))+ 9)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex19 12.10#Ex19] || [[Item:Q4199|<math>\mathsf{A}_{2}(\tau) = \tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404\tau^{2}+8028\tau+945)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathsf{A}_{2}(\tau) = \tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404\tau^{2}+8028\tau+945)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">A[2]*(((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))) = (1)/(288)*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(2)*(6160*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(4)+ 18480*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(3)+ 19404*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(2)+ 8028*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))+ 945)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[A, 2]*((Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))) == Divide[1,288]*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(2)*(6160*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(4)+ 18480*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(3)+ 19404*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(2)+ 8028*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))+ 945)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex22 12.10#Ex22] || [[Item:Q4208|<math>A_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\beta_{m}(\phi(\zeta))^{6(2s-m)}u_{2s-m}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>A_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\beta_{m}(\phi(\zeta))^{6(2s-m)}u_{2s-m}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">A[s](zeta) = (zeta)^(- 3*s)* sum((-(6*m + 1)/(6*m - 1)*alpha[m])*((((zeta)/((t)^(2)- 1))^((1)/(4))))^(6*(2*s - m))* u[2*s - m](t), m = 0..2*s)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[A, s][\[Zeta]] == \[Zeta]^(- 3*s)* Sum[(-Divide[6*m + 1,6*m - 1]*Subscript[\[Alpha], m])*(((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4])))^(6*(2*s - m))* Subscript[u, 2*s - m][t], {m, 0, 2*s}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex23 12.10#Ex23] || [[Item:Q4209|<math>\zeta^{2}B_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\alpha_{m}(\phi(\zeta))^{6(2s-m+1)}u_{2s-m+1}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\zeta^{2}B_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\alpha_{m}(\phi(\zeta))^{6(2s-m+1)}u_{2s-m+1}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(zeta)^(2)* B[s](zeta) = - (zeta)^(- 3*s)* sum((alpha[m]((((zeta)/((t)^(2)- 1))^((1)/(4)))))^(6*(2*s - m + 1))* u[2*s - m + 1](t), m = 0..2*s + 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Zeta]^(2)* Subscript[B, s][\[Zeta]] == - \[Zeta]^(- 3*s)* Sum[(Subscript[\[Alpha], m][((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4]))])^(6*(2*s - m + 1))* Subscript[u, 2*s - m + 1][t], {m, 0, 2*s + 1}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex28 12.10#Ex28] || [[Item:Q4215|<math>\zeta C_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\beta_{m}(\phi(\zeta))^{6(2s-m+1)}v_{2s-m+1}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\zeta C_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\beta_{m}(\phi(\zeta))^{6(2s-m+1)}v_{2s-m+1}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">zeta*C[s](zeta) = - (zeta)^(- 3*s)* sum((-(6*m + 1)/(6*m - 1)*alpha[m])*((((zeta)/((t)^(2)- 1))^((1)/(4))))^(6*(2*s - m + 1))* v[2*s - m + 1](t), m = 0..2*s + 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Zeta]*Subscript[C, s][\[Zeta]] == - \[Zeta]^(- 3*s)* Sum[(-Divide[6*m + 1,6*m - 1]*Subscript[\[Alpha], m])*(((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4])))^(6*(2*s - m + 1))* Subscript[v, 2*s - m + 1][t], {m, 0, 2*s + 1}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.10#Ex29 12.10#Ex29] || [[Item:Q4216|<math>D_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\alpha_{m}(\phi(\zeta))^{6(2s-m)}v_{2s-m}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>D_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\alpha_{m}(\phi(\zeta))^{6(2s-m)}v_{2s-m}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">D[s](zeta) = (zeta)^(- 3*s)* sum((alpha[m]((((zeta)/((t)^(2)- 1))^((1)/(4)))))^(6*(2*s - m))* v[2*s - m](t), m = 0..2*s)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[D, s][\[Zeta]] == \[Zeta]^(- 3*s)* Sum[(Subscript[\[Alpha], m][((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4]))])^(6*(2*s - m))* Subscript[v, 2*s - m][t], {m, 0, 2*s}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.11.E5 12.11.E5] || [[Item:Q4221|<math>p_{0}(\zeta) = t(\zeta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{0}(\zeta) = t(\zeta)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[0](zeta) = t(zeta)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, 0][\[Zeta]] == t[\[Zeta]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.11.E6 12.11.E6] || [[Item:Q4222|<math>p_{1}(\zeta) = \frac{t^{3}-6t}{24(t^{2}-1)^{2}}+\frac{5}{48((t^{2}-1)\zeta^{3})^{\frac{1}{2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{1}(\zeta) = \frac{t^{3}-6t}{24(t^{2}-1)^{2}}+\frac{5}{48((t^{2}-1)\zeta^{3})^{\frac{1}{2}}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[1](zeta) = ((t)^(3)- 6*t)/(24*((t)^(2)- 1)^(2))+(5)/(48*(((t)^(2)- 1)*(zeta)^(3))^((1)/(2)))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, 1][\[Zeta]] == Divide[(t)^(3)- 6*t,24*((t)^(2)- 1)^(2)]+Divide[5,48*(((t)^(2)- 1)*\[Zeta]^(3))^(Divide[1,2])]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.11.E8 12.11.E8] || [[Item:Q4224|<math>q_{0}(\zeta) = t(\zeta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>q_{0}(\zeta) = t(\zeta)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">q[0](zeta) = t(zeta)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[q, 0][\[Zeta]] == t[\[Zeta]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.12.E1 12.12.E1] || [[Item:Q4226|<math>\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{\mu-1}\paraU@{a}{t}\diff{t} = \frac{\sqrt{\pi}2^{-\frac{1}{2}(\mu+a+\frac{1}{2})}\EulerGamma@{\mu}}{\EulerGamma@{\frac{1}{2}(\mu+a+\frac{3}{2})}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{\mu-1}\paraU@{a}{t}\diff{t} = \frac{\sqrt{\pi}2^{-\frac{1}{2}(\mu+a+\frac{1}{2})}\EulerGamma@{\mu}}{\EulerGamma@{\frac{1}{2}(\mu+a+\frac{3}{2})}}</syntaxhighlight> || <math>\realpart@@{\mu} > 0, \realpart@@{(\mu)} > 0, \realpart@@{(\frac{1}{2}(\mu+a+\frac{3}{2}))} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(-(1)/(4)*(t)^(2))*(t)^(mu - 1)* CylinderU(a, t), t = 0..infinity) = (sqrt(Pi)*(2)^(-(1)/(2)*(mu + a +(1)/(2)))* GAMMA(mu))/(GAMMA((1)/(2)*(mu + a +(3)/(2))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[1,4]*(t)^(2)]*(t)^(\[Mu]- 1)* ParabolicCylinderD[- 1/2 -(a), t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi]*(2)^(-Divide[1,2]*(\[Mu]+ a +Divide[1,2]))* Gamma[\[Mu]],Gamma[Divide[1,2]*(\[Mu]+ a +Divide[3,2])]]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.12.E2 12.12.E2] || [[Item:Q4227|<math>\int_{0}^{\infty}e^{-\frac{3}{4}t^{2}}t^{-a-\frac{3}{2}}\paraU@{a}{t}\diff{t} = 2^{\frac{1}{4}+\frac{1}{2}a}\EulerGamma@{-a-\tfrac{1}{2}}\cos@{(\tfrac{1}{4}a+\tfrac{1}{8})\pi}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-\frac{3}{4}t^{2}}t^{-a-\frac{3}{2}}\paraU@{a}{t}\diff{t} = 2^{\frac{1}{4}+\frac{1}{2}a}\EulerGamma@{-a-\tfrac{1}{2}}\cos@{(\tfrac{1}{4}a+\tfrac{1}{8})\pi}</syntaxhighlight> || <math>\realpart@@{a} < -\tfrac{1}{2}, \realpart@@{(-a-\tfrac{1}{2})} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(-(3)/(4)*(t)^(2))*(t)^(- a -(3)/(2))* CylinderU(a, t), t = 0..infinity) = (2)^((1)/(4)+(1)/(2)*a)* GAMMA(- a -(1)/(2))*cos(((1)/(4)*a +(1)/(8))*Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[3,4]*(t)^(2)]*(t)^(- a -Divide[3,2])* ParabolicCylinderD[- 1/2 -(a), t], {t, 0, Infinity}, GenerateConditions->None] == (2)^(Divide[1,4]+Divide[1,2]*a)* Gamma[- a -Divide[1,2]]*Cos[(Divide[1,4]*a +Divide[1,8])*Pi]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Successful [Tested: 2]
-|-
-| [https://dlmf.nist.gov/12.12.E3 12.12.E3] || [[Item:Q4228|<math>\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{1}{2}}(x^{2}+t^{2})^{-1}\paraU@{a}{t}\diff{t} = \sqrt{\pi/2}\EulerGamma@{\tfrac{1}{2}-a}x^{-a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}\paraU@{-a}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{1}{2}}(x^{2}+t^{2})^{-1}\paraU@{a}{t}\diff{t} = \sqrt{\pi/2}\EulerGamma@{\tfrac{1}{2}-a}x^{-a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}\paraU@{-a}{x}</syntaxhighlight> || <math>x > 0, \realpart@@{a} < \tfrac{1}{2}, \realpart@@{(\tfrac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(-(1)/(4)*(t)^(2))*(t)^(- a -(1)/(2))*((x)^(2)+ (t)^(2))^(- 1)* CylinderU(a, t), t = 0..infinity) = sqrt(Pi/2)*GAMMA((1)/(2)- a)*(x)^(- a -(3)/(2))* exp((1)/(4)*(x)^(2))*CylinderU(- a, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[1,4]*(t)^(2)]*(t)^(- a -Divide[1,2])*((x)^(2)+ (t)^(2))^(- 1)* ParabolicCylinderD[- 1/2 -(a), t], {t, 0, Infinity}, GenerateConditions->None] == Sqrt[Pi/2]*Gamma[Divide[1,2]- a]*(x)^(- a -Divide[3,2])* Exp[Divide[1,4]*(x)^(2)]*ParabolicCylinderD[- 1/2 -(- a), x]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.13.E1 12.13.E1] || [[Item:Q4230|<math>\paraU@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{(-y)^{m}}{m!}\paraU@{a-m}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{(-y)^{m}}{m!}\paraU@{a-m}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, x + y) = exp((1)/(2)*x*y +(1)/(4)*(y)^(2))*sum(((- y)^(m))/(factorial(m))*CylinderU(a - m, x), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), x + y] == Exp[Divide[1,2]*x*y +Divide[1,4]*(y)^(2)]*Sum[Divide[(- y)^(m),(m)!]*ParabolicCylinderD[- 1/2 -(a - m), x], {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.13.E2 12.13.E2] || [[Item:Q4231|<math>\paraU@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}y^{m}\paraU@{a+m}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}y^{m}\paraU@{a+m}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, x + y) = exp(-(1)/(2)*x*y -(1)/(4)*(y)^(2))*sum(binomial(- a -(1)/(2),m)*(y)^(m)* CylinderU(a + m, x), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), x + y] == Exp[-Divide[1,2]*x*y -Divide[1,4]*(y)^(2)]*Sum[Binomial[- a -Divide[1,2],m]*(y)^(m)* ParabolicCylinderD[- 1/2 -(a + m), x], {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.13.E3 12.13.E3] || [[Item:Q4232|<math>\paraV@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{a-\tfrac{1}{2}}{m}y^{m}\paraV@{a-m}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{a-\tfrac{1}{2}}{m}y^{m}\paraV@{a-m}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(a, x + y) = exp((1)/(2)*x*y +(1)/(4)*(y)^(2))*sum(binomial(a -(1)/(2),m)*(y)^(m)* CylinderV(a - m, x), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, x + y] + ParabolicCylinderD[-(a) - 1/2, -(x + y)]) == Exp[Divide[1,2]*x*y +Divide[1,4]*(y)^(2)]*Sum[Binomial[a -Divide[1,2],m]*(y)^(m)* Divide[GAMMA[1/2 + a - m], Pi]*(Sin[Pi*(a - m)] * ParabolicCylinderD[-(a - m) - 1/2, x] + ParabolicCylinderD[-(a - m) - 1/2, -(x)]), {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.13.E4 12.13.E4] || [[Item:Q4233|<math>\paraV@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{y^{m}}{m!}\paraV@{a+m}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{y^{m}}{m!}\paraV@{a+m}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(a, x + y) = exp(-(1)/(2)*x*y -(1)/(4)*(y)^(2))*sum(((y)^(m))/(factorial(m))*CylinderV(a + m, x), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, x + y] + ParabolicCylinderD[-(a) - 1/2, -(x + y)]) == Exp[-Divide[1,2]*x*y -Divide[1,4]*(y)^(2)]*Sum[Divide[(y)^(m),(m)!]*Divide[GAMMA[1/2 + a + m], Pi]*(Sin[Pi*(a + m)] * ParabolicCylinderD[-(a + m) - 1/2, x] + ParabolicCylinderD[-(a + m) - 1/2, -(x)]), {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.13.E5 12.13.E5] || [[Item:Q4234|<math>\paraU@{a}{x\cos@@{t}+y\sin@@{t}}\\ = e^{\frac{1}{4}(x\sin@@{t}-y\cos@@{t})^{2}}\*\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}(\tan@@{t})^{m}\paraU@{m+a}{x}\paraU@{-m-\tfrac{1}{2}}{y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{x\cos@@{t}+y\sin@@{t}}\\ = e^{\frac{1}{4}(x\sin@@{t}-y\cos@@{t})^{2}}\*\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}(\tan@@{t})^{m}\paraU@{m+a}{x}\paraU@{-m-\tfrac{1}{2}}{y}</syntaxhighlight> || <math>\realpart@@{a} \leq -\tfrac{1}{2}, 0 \leq t, t \leq \tfrac{1}{4}\pi</math> || <syntaxhighlight lang=mathematica>CylinderU(a, x*cos(t)+ y*sin(t)) = exp((1)/(4)*(x*sin(t)- y*cos(t))^(2))* sum(binomial(- a -(1)/(2),m)*(tan(t))^(m)* CylinderU(m + a, x)*CylinderU(- m -(1)/(2), y), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), x*Cos[t]+ y*Sin[t]] == Exp[Divide[1,4]*(x*Sin[t]- y*Cos[t])^(2)]* Sum[Binomial[- a -Divide[1,2],m]*(Tan[t])^(m)* ParabolicCylinderD[- 1/2 -(m + a), x]*ParabolicCylinderD[- 1/2 -(- m -Divide[1,2]), y], {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skip - No test values generated
-|-
-| [https://dlmf.nist.gov/12.14.E1 12.14.E1] || [[Item:Q4236|<math>\paraW@{a}{0} = 2^{-\frac{3}{4}}\left|\frac{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}ia}}{\EulerGamma@{\tfrac{3}{4}+\tfrac{1}{2}ia}}\right|^{\frac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{0} = 2^{-\frac{3}{4}}\left|\frac{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}ia}}{\EulerGamma@{\tfrac{3}{4}+\tfrac{1}{2}ia}}\right|^{\frac{1}{2}}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{4}+\tfrac{1}{2}\iunit a)} > 0, \realpart@@{(\tfrac{3}{4}+\tfrac{1}{2}\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), 0 * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), 0 * Exp[Divide[Pi*I,4]]] ) == (2)^(-Divide[3,4])*(Abs[Divide[Gamma[Divide[1,4]+Divide[1,2]*I*a],Gamma[Divide[3,4]+Divide[1,2]*I*a]]])^(Divide[1,2])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.6502446611528931, Times[0.2167171091323973, Plus[Times[Complex[2.1101734540747557, 0.09157129889910319], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[2.1101734540747557, -0.09157129889910319], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]
-Test Values: {Rule[a, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.6502446611528931, Times[0.15393043293932354, Plus[Times[Complex[2.1101734540747557, -0.09157129889910319], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, 1.5]]]]]]]], Times[Complex[2.1101734540747557, 0.09157129889910319], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, 1.5]]]]]]]]]]]
-Test Values: {Rule[a, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.14.E2 12.14.E2] || [[Item:Q4237|<math>\paraW'@{a}{0} = -2^{-\frac{1}{4}}\left|\frac{\EulerGamma@{\tfrac{3}{4}+\tfrac{1}{2}ia}}{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}ia}}\right|^{\frac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW'@{a}{0} = -2^{-\frac{1}{4}}\left|\frac{\EulerGamma@{\tfrac{3}{4}+\tfrac{1}{2}ia}}{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}ia}}\right|^{\frac{1}{2}}</syntaxhighlight> || <math>\realpart@@{(\tfrac{3}{4}+\tfrac{1}{2}\iunit a)} > 0, \realpart@@{(\tfrac{1}{4}+\tfrac{1}{2}\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), temp * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), temp * Exp[Divide[Pi*I,4]]] ), {temp, 1}]/.temp-> 0) == - (2)^(-Divide[1,4])*(Abs[Divide[Gamma[Divide[3,4]+Divide[1,2]*I*a],Gamma[Divide[1,4]+Divide[1,2]*I*a]]])^(Divide[1,2])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.7689413383471582, Times[0.2167171091323973, Plus[Times[Complex[-1.704391150531108, -1.8258251253417301], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]], Times[Complex[-1.704391150531108, 1.8258251253417301], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]]
-Test Values: {Rule[a, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.7689413383471582, Times[0.15393043293932354, Plus[Times[Complex[-1.704391150531108, 1.8258251253417301], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, 1.5]]]]]]]]], Times[Complex[-1.704391150531108, -1.8258251253417301], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, 1.5]]]]]]]]]]]]
-Test Values: {Rule[a, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.14.E3 12.14.E3] || [[Item:Q4238|<math>\Wronskian@{\paraW@{a}{x},\paraW@{a}{-x}} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraW@{a}{x},\paraW@{a}{-x}} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), x * Exp[Divide[Pi*I,4]]] ), Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), - x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), - x * Exp[Divide[Pi*I,4]]] )}, x] == 1</syntaxhighlight> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-1.0, Times[0.49552852896181854, Power[2.718281828459045, Plus[-2.356194490192345, Times[Complex[0.0, -1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]], Plus[Complex[6.858735565841029, 8.017762045530217], Times[Complex[-3.325234230733274, 7.771974729433958], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]], Times[Complex[-0.5683445061301408, 1.832896863544324], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]], Plus[Complex[-3.829019967249232, -1.729594934825754], Times[Complex[-1.3099191255337557, -0.33304402326481836], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[-0.6925599504260578, -1.7781797223294367], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]], Plus[Complex[1.1964059764236668, -0.5640566230504777], Times[Complex[1.7883988364886165, 3.791736125757196], Power[2.718281828459045, Times<syntaxhighlight lang=mathematica>Result: Plus[-1.0, Times[0.49552852896181854, Power[2.718281828459045, Plus[-2.356194490192345, Times[Complex[0.0, -1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]], Plus[Complex[6.858735565841018, 8.01776204553021], Times[Complex[2.1162015796075306, -3.4943658291294586], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]], Times[Complex[1.7644828790311722, 1.0958018333501354], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]], Plus[Complex[-1.886026428459961, 2.3891008291554687], Times[Complex[0.20069396529457476, -1.987274101858182], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[1.683947543704261, -1.245446997525291], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]], Plus[Complex[1.1608218100872771, 1.6575881988304835], Times[Complex[3.0041801875943053, -0.5876813330718609], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]
-Test Values: {Rule[a, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.14.E4 12.14.E4] || [[Item:Q4239|<math>\paraW@{a}{x} = \sqrt{k/2}\,e^{\frac{1}{4}\pi a}\left(e^{i\rho}\paraU@{ia}{xe^{-\pi i/4}}+e^{-i\rho}\paraU@{-ia}{xe^{\pi i/4}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{x} = \sqrt{k/2}\,e^{\frac{1}{4}\pi a}\left(e^{i\rho}\paraU@{ia}{xe^{-\pi i/4}}+e^{-i\rho}\paraU@{-ia}{xe^{\pi i/4}}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), x * Exp[Divide[Pi*I,4]]] ) == Sqrt[(Sqrt[1 + Exp[2*Pi*a]]- Exp[Pi*a])/2]*Exp[Divide[1,4]*Pi*a]*(Exp[I*(Divide[1,8]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]]))]*ParabolicCylinderD[- 1/2 -(I*a), x*Exp[- Pi*I/4]]+ Exp[- I*(Divide[1,8]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]]))]*ParabolicCylinderD[- 1/2 -(- I*a), x*Exp[Pi*I/4]])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.7504500073451766, 0.0], Times[0.2167171091323973, Plus[Times[Complex[-0.5683445061301404, -1.832896863544323], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[-0.5683445061301404, 1.832896863544323], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]
-Test Values: {Rule[a, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.17071363418721158, 0.0], Times[0.2167171091323973, Plus[Times[Complex[1.764482879031172, -1.0958018333501351], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[1.764482879031172, 1.0958018333501351], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]
-Test Values: {Rule[a, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.14.E8 12.14.E8] || [[Item:Q4244|<math>\paraW@{a}{x} = \paraW@{a}{0}w_{1}(a,x)+\paraW'@{a}{0}w_{2}(a,x)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{x} = \paraW@{a}{0}w_{1}(a,x)+\paraW'@{a}{0}w_{2}(a,x)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), x * Exp[Divide[Pi*I,4]]] ) == Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), 0 * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), 0 * Exp[Divide[Pi*I,4]]] )*(Sum[Subscript[\[Alpha], n][a]*Divide[(x)^(2*n),(2*n)!], {n, 0, Infinity}, GenerateConditions->None])+ (D[Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), temp * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), temp * Exp[Divide[Pi*I,4]]] ), {temp, 1}]/.temp-> 0)*(Sum[Subscript[\[Beta], n][a]*Divide[(x)^(2*n + 1),(2*n + 1)!], {n, 0, Infinity}, GenerateConditions->None])</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.14#Ex5 12.14#Ex5] || [[Item:Q4249|<math>\alpha_{0}(a) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{0}(a) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[0](a) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], 0][a] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.14#Ex6 12.14#Ex6] || [[Item:Q4250|<math>\alpha_{1}(a) = a</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{1}(a) = a</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[1](a) = a</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], 1][a] == a</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.14#Ex7 12.14#Ex7] || [[Item:Q4251|<math>\beta_{0}(a) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta_{0}(a) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta[0](a) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Beta], 0][a] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|- style="background: #dfe6e9;"
-| [https://dlmf.nist.gov/12.14#Ex8 12.14#Ex8] || [[Item:Q4252|<math>\beta_{1}(a) = a</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta_{1}(a) = a</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta[1](a) = a</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Beta], 1][a] == a</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.14.E13 12.14.E13] || [[Item:Q4253|<math>\paraW@{0}{+ x} = 2^{-\frac{5}{4}}\sqrt{\pi x}\left(\BesselJ{-\frac{1}{4}}@{\tfrac{1}{4}x^{2}}-\BesselJ{\frac{1}{4}}@{\tfrac{1}{4}x^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{0}{+ x} = 2^{-\frac{5}{4}}\sqrt{\pi x}\left(\BesselJ{-\frac{1}{4}}@{\tfrac{1}{4}x^{2}}-\BesselJ{\frac{1}{4}}@{\tfrac{1}{4}x^{2}}\right)</syntaxhighlight> || <math>\realpart@@{((-\frac{1}{4})+k+1)} > 0, \realpart@@{((\frac{1}{4})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), + x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), + x * Exp[Divide[Pi*I,4]]] ) == (2)^(-Divide[5,4])*Sqrt[Pi*x]*(BesselJ[-Divide[1,4], Divide[1,4]*(x)^(2)]- BesselJ[Divide[1,4], Divide[1,4]*(x)^(2)])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.22960009916312846, Times[0.4550898605622274, Plus[Times[Complex[0.5125789656744846, -0.578293218532047], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.5125789656744846, 0.578293218532047], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]
-Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.7771899742615831, Times[0.4550898605622274, Plus[Times[Complex[1.0093127652068992, -0.20538419268274744], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.0093127652068992, 0.20538419268274744], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]
-Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.14.E13 12.14.E13] || [[Item:Q4253|<math>\paraW@{0}{- x} = 2^{-\frac{5}{4}}\sqrt{\pi x}\left(\BesselJ{-\frac{1}{4}}@{\tfrac{1}{4}x^{2}}+\BesselJ{\frac{1}{4}}@{\tfrac{1}{4}x^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{0}{- x} = 2^{-\frac{5}{4}}\sqrt{\pi x}\left(\BesselJ{-\frac{1}{4}}@{\tfrac{1}{4}x^{2}}+\BesselJ{\frac{1}{4}}@{\tfrac{1}{4}x^{2}}\right)</syntaxhighlight> || <math>\realpart@@{((-\frac{1}{4})+k+1)} > 0, \realpart@@{((\frac{1}{4})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), - x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), - x * Exp[Divide[Pi*I,4]]] ) == (2)^(-Divide[5,4])*Sqrt[Pi*x]*(BesselJ[-Divide[1,4], Divide[1,4]*(x)^(2)]+ BesselJ[Divide[1,4], Divide[1,4]*(x)^(2)])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-1.6050209192353964, Times[0.4550898605622274, Plus[Times[Complex[1.669165402738578, 0.5782932185320475], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.669165402738578, -0.5782932185320475], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]
-Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-1.2656786607564097, Times[0.4550898605622274, Plus[Times[Complex[1.4200811505723943, 0.2053841926827476], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.4200811505723943, -0.2053841926827476], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]
-Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.14.E14 12.14.E14] || [[Item:Q4254|<math>\deriv{}{x}\paraW@{0}{+ x} = -2^{-\frac{9}{4}}x\sqrt{\pi x}\left(\BesselJ{\frac{3}{4}}@{\tfrac{1}{4}x^{2}}+\BesselJ{-\frac{3}{4}}@{\tfrac{1}{4}x^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{x}\paraW@{0}{+ x} = -2^{-\frac{9}{4}}x\sqrt{\pi x}\left(\BesselJ{\frac{3}{4}}@{\tfrac{1}{4}x^{2}}+\BesselJ{-\frac{3}{4}}@{\tfrac{1}{4}x^{2}}\right)</syntaxhighlight> || <math>\realpart@@{((\frac{3}{4})+k+1)} > 0, \realpart@@{((-\frac{3}{4})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), + x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), + x * Exp[Divide[Pi*I,4]]] ), x] == - (2)^(-Divide[9,4])* x*Sqrt[Pi*x]*(BesselJ[Divide[3,4], Divide[1,4]*(x)^(2)]+ BesselJ[-Divide[3,4], Divide[1,4]*(x)^(2)])</syntaxhighlight> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.6138624292597322, Times[0.4550898605622274, Plus[Times[Complex[-0.6342811205261311, 0.23110891742402956], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.6342811205261311, -0.23110891742402956], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]
-Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.497609493984496, Times[0.4550898605622274, Plus[Times[Complex[-0.5880519854532475, 0.008953751453165265], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.5880519854532475, -0.008953751453165265], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]
-Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.14.E14 12.14.E14] || [[Item:Q4254|<math>\deriv{}{x}\paraW@{0}{- x} = -2^{-\frac{9}{4}}x\sqrt{\pi x}\left(\BesselJ{\frac{3}{4}}@{\tfrac{1}{4}x^{2}}-\BesselJ{-\frac{3}{4}}@{\tfrac{1}{4}x^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{x}\paraW@{0}{- x} = -2^{-\frac{9}{4}}x\sqrt{\pi x}\left(\BesselJ{\frac{3}{4}}@{\tfrac{1}{4}x^{2}}-\BesselJ{-\frac{3}{4}}@{\tfrac{1}{4}x^{2}}\right)</syntaxhighlight> || <math>\realpart@@{((\frac{3}{4})+k+1)} > 0, \realpart@@{((-\frac{3}{4})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), - x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), - x * Exp[Divide[Pi*I,4]]] ), x] == - (2)^(-Divide[9,4])* x*Sqrt[Pi*x]*(BesselJ[Divide[3,4], Divide[1,4]*(x)^(2)]- BesselJ[-Divide[3,4], Divide[1,4]*(x)^(2)])</syntaxhighlight> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.06418969137726768, Times[0.4550898605622274, Plus[Times[Complex[0.17206328567807166, 0.23110891742402973], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[0.17206328567807166, -0.23110891742402973], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]
-Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.4763137641163559, Times[0.4550898605622274, Plus[Times[Complex[0.5701444825469169, 0.008953751453165182], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[0.5701444825469169, -0.008953751453165182], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]
-Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.14.E15 12.14.E15] || [[Item:Q4255|<math>w_{1}(a,x) = e^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}-\tfrac{1}{2}ia}{\tfrac{1}{2}}{\tfrac{1}{2}ix^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w_{1}(a,x) = e^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}-\tfrac{1}{2}ia}{\tfrac{1}{2}}{\tfrac{1}{2}ix^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(sum(alpha[n](a)*((x)^(2*n))/(factorial(2*n)), n = 0..infinity)) = exp(-(1)/(4)*I*(x)^(2))*KummerM((1)/(4)-(1)/(2)*I*a, (1)/(2), (1)/(2)*I*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Sum[Subscript[\[Alpha], n][a]*Divide[(x)^(2*n),(2*n)!], {n, 0, Infinity}, GenerateConditions->None]) == Exp[-Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[1,4]-Divide[1,2]*I*a, Divide[1,2], Divide[1,2]*I*(x)^(2)]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.14.E15 12.14.E15] || [[Item:Q4255|<math>e^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}-\tfrac{1}{2}ia}{\tfrac{1}{2}}{\tfrac{1}{2}ix^{2}} = e^{\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}+\tfrac{1}{2}ia}{\tfrac{1}{2}}{-\tfrac{1}{2}ix^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}-\tfrac{1}{2}ia}{\tfrac{1}{2}}{\tfrac{1}{2}ix^{2}} = e^{\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}+\tfrac{1}{2}ia}{\tfrac{1}{2}}{-\tfrac{1}{2}ix^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(-(1)/(4)*I*(x)^(2))*KummerM((1)/(4)-(1)/(2)*I*a, (1)/(2), (1)/(2)*I*(x)^(2)) = exp((1)/(4)*I*(x)^(2))*KummerM((1)/(4)+(1)/(2)*I*a, (1)/(2), -(1)/(2)*I*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[-Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[1,4]-Divide[1,2]*I*a, Divide[1,2], Divide[1,2]*I*(x)^(2)] == Exp[Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[1,4]+Divide[1,2]*I*a, Divide[1,2], -Divide[1,2]*I*(x)^(2)]</syntaxhighlight> || Failure || Successful || Successful [Tested: 18] || Successful [Tested: 18]
-|-
-| [https://dlmf.nist.gov/12.14.E16 12.14.E16] || [[Item:Q4256|<math>w_{2}(a,x) = xe^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}-\tfrac{1}{2}ia}{\tfrac{3}{2}}{\tfrac{1}{2}ix^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w_{2}(a,x) = xe^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}-\tfrac{1}{2}ia}{\tfrac{3}{2}}{\tfrac{1}{2}ix^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(sum(beta[n](a)*((x)^(2*n + 1))/(factorial(2*n + 1)), n = 0..infinity)) = x*exp(-(1)/(4)*I*(x)^(2))*KummerM((3)/(4)-(1)/(2)*I*a, (3)/(2), (1)/(2)*I*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Sum[Subscript[\[Beta], n][a]*Divide[(x)^(2*n + 1),(2*n + 1)!], {n, 0, Infinity}, GenerateConditions->None]) == x*Exp[-Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[3,4]-Divide[1,2]*I*a, Divide[3,2], Divide[1,2]*I*(x)^(2)]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.14.E16 12.14.E16] || [[Item:Q4256|<math>xe^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}-\tfrac{1}{2}ia}{\tfrac{3}{2}}{\tfrac{1}{2}ix^{2}} = xe^{\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}+\tfrac{1}{2}ia}{\tfrac{3}{2}}{-\tfrac{1}{2}ix^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>xe^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}-\tfrac{1}{2}ia}{\tfrac{3}{2}}{\tfrac{1}{2}ix^{2}} = xe^{\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}+\tfrac{1}{2}ia}{\tfrac{3}{2}}{-\tfrac{1}{2}ix^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x*exp(-(1)/(4)*I*(x)^(2))*KummerM((3)/(4)-(1)/(2)*I*a, (3)/(2), (1)/(2)*I*(x)^(2)) = x*exp((1)/(4)*I*(x)^(2))*KummerM((3)/(4)+(1)/(2)*I*a, (3)/(2), -(1)/(2)*I*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>x*Exp[-Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[3,4]-Divide[1,2]*I*a, Divide[3,2], Divide[1,2]*I*(x)^(2)] == x*Exp[Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[3,4]+Divide[1,2]*I*a, Divide[3,2], -Divide[1,2]*I*(x)^(2)]</syntaxhighlight> || Failure || Successful || Successful [Tested: 18] || Successful [Tested: 18]
-|-
-| [https://dlmf.nist.gov/12.14.E17 12.14.E17] || [[Item:Q4257|<math>\paraW@{a}{x} = \sqrt{\frac{2k}{x}}\left(s_{1}(a,x)\cos@@{\omega}-s_{2}(a,x)\sin@@{\omega}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{x} = \sqrt{\frac{2k}{x}}\left(s_{1}(a,x)\cos@@{\omega}-s_{2}(a,x)\sin@@{\omega}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), x * Exp[Divide[Pi*I,4]]] ) == Sqrt[Divide[2*(Sqrt[1 + Exp[2*Pi*a]]- Exp[Pi*a]),x]]*(Subscript[s, 1][a , x]* Cos[Divide[1,4]*(x)^(2)- a*Log[x]+Divide[1,4]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]])]- Subscript[s, 2][a , x]* Sin[Divide[1,4]*(x)^(2)- a*Log[x]+Divide[1,4]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]])])</syntaxhighlight> || Missing Macro Error || Failure || - || Error
-|-
-| [https://dlmf.nist.gov/12.14.E18 12.14.E18] || [[Item:Q4258|<math>\paraW@{a}{-x} = \sqrt{\frac{2}{kx}}\left(s_{1}(a,x)\sin@@{\omega}+s_{2}(a,x)\cos@@{\omega}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{-x} = \sqrt{\frac{2}{kx}}\left(s_{1}(a,x)\sin@@{\omega}+s_{2}(a,x)\cos@@{\omega}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), - x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), - x * Exp[Divide[Pi*I,4]]] ) == Sqrt[Divide[2,k*x]]*(Subscript[s, 1][a , x]* Sin[Divide[1,4]*(x)^(2)- a*Log[x]+Divide[1,4]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]])]+ Subscript[s, 2][a , x]* Cos[Divide[1,4]*(x)^(2)- a*Log[x]+Divide[1,4]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]])])</syntaxhighlight> || Missing Macro Error || Failure || - || Error
-|-
-| [https://dlmf.nist.gov/12.14.E24 12.14.E24] || [[Item:Q4264|<math>\deriv[2]{w}{t} = \mu^{4}(1-t^{2})w</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{t} = \mu^{4}(1-t^{2})w</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [t$(2)]) = (mu)^(4)*(1 - (t)^(2))*w</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {t, 2}] == \[Mu]^(4)*(1 - (t)^(2))*w</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.082531755+.6250000011*I
-Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.6250000011-1.082531755*I
-Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.0825317547305482, 0.6250000000000002]
-Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.0825317547305482, 0.6250000000000002]
-Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.15.E1 12.15.E1] || [[Item:Q4281|<math>\deriv[2]{w}{z}+\left(\nu+\lambda^{-1}-\lambda^{-2}z^{\lambda}\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}+\left(\nu+\lambda^{-1}-\lambda^{-2}z^{\lambda}\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])+(nu + (lambda)^(- 1)- (lambda)^(- 2)* (z)^(lambda))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]+(\[Nu]+ \[Lambda]^(- 1)- \[Lambda]^(- 2)* (z)^\[Lambda])*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7322275248+.9199723429*I
-Test Values: {lambda = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.402820433+.5288298490*I
-Test Values: {lambda = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.7322275239543282, 0.91997234266967]
-Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.6337978798301105, 0.5539469388852316]
-Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|-
-| [https://dlmf.nist.gov/12.17.E4 12.17.E4] || [[Item:Q4287|<math>\frac{1}{\xi^{2}+\eta^{2}}\left(\pderiv[2]{w}{\xi}+\pderiv[2]{w}{\eta}\right)+\pderiv[2]{w}{\zeta}+k^{2}w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\xi^{2}+\eta^{2}}\left(\pderiv[2]{w}{\xi}+\pderiv[2]{w}{\eta}\right)+\pderiv[2]{w}{\zeta}+k^{2}w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/((xi)^(2)+ (eta)^(2))*(diff(w, [xi$(2)])+ diff(w, [eta$(2)]))+ diff(w, [zeta$(2)])+ (k)^(2)* w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,\[Xi]^(2)+ \[Eta]^(2)]*(D[w, {\[Xi], 2}]+ D[w, {\[Eta], 2}])+ D[w, {\[Zeta], 2}]+ (k)^(2)* w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
-Test Values: {eta = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.464101616+2.*I
-Test Values: {eta = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, k = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.8660254037844387, 0.49999999999999994]
-Test Values: {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[3.464101615137755, 1.9999999999999998]
-Test Values: {Rule[k, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
-|}
 </div>

Results of Parabolic Cylinder Functions: Difference between revisions

Latest revision as of 18:04, 25 May 2021

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