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{{DISPLAYTITLE:Airy and Related Functions - 9.12 Scorer Functions}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! 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
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| [https://dlmf.nist.gov/9.12.E1 9.12.E1] || [[Item:Q2927|<math>\deriv[2]{w}{z}-zw = \frac{1}{\pi}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}-zw = \frac{1}{\pi}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])- z*w = (1)/(Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]- z*w == Divide[1,Pi]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.8183098865-.8660254040*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .5477155179-.5000000004*I
Test Values: {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 [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.8183098861837907, -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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.5477155176006481, -0.49999999999999994]
Test Values: {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>
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| [https://dlmf.nist.gov/9.12.E4 9.12.E4] || [[Item:Q2932|<math>\ScorerGi@{z} = \AiryBi@{z}\int_{z}^{\infty}\AiryAi@{t}\diff{t}+\AiryAi@{z}\int_{0}^{z}\AiryBi@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi@{z} = \AiryBi@{z}\int_{z}^{\infty}\AiryAi@{t}\diff{t}+\AiryAi@{z}\int_{0}^{z}\AiryBi@{t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = AiryBi(z)*int(AiryAi(t), t = z..infinity)+ AiryAi(z)*int(AiryBi(t), t = 0..z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerGi[z] == AiryBi[z]*Integrate[AiryAi[t], {t, z, Infinity}, GenerateConditions->None]+ AiryAi[z]*Integrate[AiryBi[t], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E5 9.12.E5] || [[Item:Q2933|<math>\ScorerHi@{z} = \AiryBi@{z}\int_{-\infty}^{z}\AiryAi@{t}\diff{t}-\AiryAi@{z}\int_{-\infty}^{z}\AiryBi@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerHi@{z} = \AiryBi@{z}\int_{-\infty}^{z}\AiryAi@{t}\diff{t}-\AiryAi@{z}\int_{-\infty}^{z}\AiryBi@{t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = AiryBi(z)*int(AiryAi(t), t = - infinity..z)- AiryAi(z)*int(AiryBi(t), t = - infinity..z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerHi[z] == AiryBi[z]*Integrate[AiryAi[t], {t, - Infinity, z}, GenerateConditions->None]- AiryAi[z]*Integrate[AiryBi[t], {t, - Infinity, z}, GenerateConditions->None]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E6 9.12.E6] || [[Item:Q2934|<math>\ScorerGi@{0} = \tfrac{1}{2}\ScorerHi@{0}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi@{0} = \tfrac{1}{2}\ScorerHi@{0}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(0)*(int(AiryAi(t), t = (0) .. infinity))+AiryAi(0)*(int(AiryBi(t), t = 0 .. (0))) = (1)/(2)*AiryBi(0)*(int(AiryAi(t), t = -infinity .. (0)))-AiryAi(0)*(int(AiryBi(t), t = -infinity .. (0)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerGi[0] == Divide[1,2]*ScorerHi[0]</syntaxhighlight> || Failure || Successful || Skip - No test values generated || Successful [Tested: 1]
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| [https://dlmf.nist.gov/9.12.E6 9.12.E6] || [[Item:Q2934|<math>\tfrac{1}{2}\ScorerHi@{0} = \tfrac{1}{3}\AiryBi@{0}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{2}\ScorerHi@{0} = \tfrac{1}{3}\AiryBi@{0}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(2)*AiryBi(0)*(int(AiryAi(t), t = -infinity .. (0)))-AiryAi(0)*(int(AiryBi(t), t = -infinity .. (0))) = (1)/(3)*AiryBi(0)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*ScorerHi[0] == Divide[1,3]*AiryBi[0]</syntaxhighlight> || Failure || Successful || Skip - No test values generated || Successful [Tested: 1]
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| [https://dlmf.nist.gov/9.12.E6 9.12.E6] || [[Item:Q2934|<math>\tfrac{1}{3}\AiryBi@{0} = {1\Big{/}\!\left(3^{7/6}\EulerGamma@{\tfrac{2}{3}}\right)=0.20497\;55424\ldots,}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{3}\AiryBi@{0} = {1\Big{/}\!\left(3^{7/6}\EulerGamma@{\tfrac{2}{3}}\right)=0.20497\;55424\ldots,}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(3)*AiryBi(0) = 1/((3)^(7/6)* GAMMA((2)/(3))) = 0.2049755424</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,3]*AiryBi[0] == 1/((3)^(7/6)* Gamma[Divide[2,3]]) == 0.2049755424</syntaxhighlight> || Error || Failure || Skip - symbolical successful subtest || Error
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| [https://dlmf.nist.gov/9.12.E7 9.12.E7] || [[Item:Q2935|<math>\ScorerGi'@{0} = \tfrac{1}{2}\ScorerHi'@{0}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi'@{0} = \tfrac{1}{2}\ScorerHi'@{0}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=0, diff( AiryBi(temp)*(int(AiryAi(t), t = (temp) .. infinity))+AiryAi(temp)*(int(AiryBi(t), t = 0 .. (temp))), temp$(1) ) ) = (1)/(2)*subs( temp=0, diff( AiryBi(temp)*(int(AiryAi(t), t = -infinity .. (temp)))-AiryAi(temp)*(int(AiryBi(t), t = -infinity .. (temp))), temp$(1) ) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[ScorerGi[temp], {temp, 1}]/.temp-> 0) == Divide[1,2]*(D[ScorerHi[temp], {temp, 1}]/.temp-> 0)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/9.12.E7 9.12.E7] || [[Item:Q2935|<math>\tfrac{1}{2}\ScorerHi'@{0} = \tfrac{1}{3}\AiryBi'@{0}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{2}\ScorerHi'@{0} = \tfrac{1}{3}\AiryBi'@{0}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(2)*subs( temp=0, diff( AiryBi(temp)*(int(AiryAi(t), t = -infinity .. (temp)))-AiryAi(temp)*(int(AiryBi(t), t = -infinity .. (temp))), temp$(1) ) ) = (1)/(3)*subs( temp=0, diff( AiryBi(temp), temp$(1) ) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*(D[ScorerHi[temp], {temp, 1}]/.temp-> 0) == Divide[1,3]*(D[AiryBi[temp], {temp, 1}]/.temp-> 0)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/9.12.E7 9.12.E7] || [[Item:Q2935|<math>\tfrac{1}{3}\AiryBi'@{0} = 1\Big{/}\left(3^{5/6}\EulerGamma@{\tfrac{1}{3}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{3}\AiryBi'@{0} = 1\Big{/}\left(3^{5/6}\EulerGamma@{\tfrac{1}{3}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(3)*subs( temp=0, diff( AiryBi(temp), temp$(1) ) ) = 1/((3)^(5/6)* GAMMA((1)/(3)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,3]*(D[AiryBi[temp], {temp, 1}]/.temp-> 0) == 1/((3)^(5/6)* Gamma[Divide[1,3]])</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/9.12.E7 9.12.E7] || [[Item:Q2935|<math>1\Big{/}\left(3^{5/6}\EulerGamma@{\tfrac{1}{3}}\right) = 0.14942\;94524\ldots</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1\Big{/}\left(3^{5/6}\EulerGamma@{\tfrac{1}{3}}\right) = 0.14942\;94524\ldots</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>1/((3)^(5/6)* GAMMA((1)/(3))) = 0.1494294524</syntaxhighlight> || <syntaxhighlight lang=mathematica>1/((3)^(5/6)* Gamma[Divide[1,3]]) == 0.1494294524</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/9.12.E11 9.12.E11] || [[Item:Q2939|<math>\ScorerGi@{z}+\ScorerHi@{z} = \AiryBi@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi@{z}+\ScorerHi@{z} = \AiryBi@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z)))+ AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = AiryBi(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerGi[z]+ ScorerHi[z] == AiryBi[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E12 9.12.E12] || [[Item:Q2940|<math>\ScorerGi@{z} = \tfrac{1}{2}e^{\pi i/3}\ScorerHi@{ze^{-2\pi i/3}}+\tfrac{1}{2}e^{-\pi i/3}\ScorerHi@{ze^{2\pi i/3}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi@{z} = \tfrac{1}{2}e^{\pi i/3}\ScorerHi@{ze^{-2\pi i/3}}+\tfrac{1}{2}e^{-\pi i/3}\ScorerHi@{ze^{2\pi i/3}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = (1)/(2)*exp(Pi*I/3)*AiryBi(z*exp(- 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))-AiryAi(z*exp(- 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))+(1)/(2)*exp(- Pi*I/3)*AiryBi(z*exp(2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(2*Pi*I/3))))-AiryAi(z*exp(2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(2*Pi*I/3))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerGi[z] == Divide[1,2]*Exp[Pi*I/3]*ScorerHi[z*Exp[- 2*Pi*I/3]]+Divide[1,2]*Exp[- Pi*I/3]*ScorerHi[z*Exp[2*Pi*I/3]]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.2356545741-.3070803572*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1024598659-.3465846956*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E13 9.12.E13] || [[Item:Q2941|<math>\ScorerGi@{z} = e^{-\pi i/3}\ScorerHi@{ze^{+ 2\pi i/3}}+ i\AiryAi@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi@{z} = e^{-\pi i/3}\ScorerHi@{ze^{+ 2\pi i/3}}+ i\AiryAi@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = exp(- Pi*I/3)*AiryBi(z*exp(+ 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(+ 2*Pi*I/3))))-AiryAi(z*exp(+ 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(+ 2*Pi*I/3))))+ I*AiryAi(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerGi[z] == Exp[- Pi*I/3]*ScorerHi[z*Exp[+ 2*Pi*I/3]]+ I*AiryAi[z]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1900131227-.1739897867*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1844815903-.2874294645*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E13 9.12.E13] || [[Item:Q2941|<math>\ScorerGi@{z} = e^{+\pi i/3}\ScorerHi@{ze^{- 2\pi i/3}}- i\AiryAi@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi@{z} = e^{+\pi i/3}\ScorerHi@{ze^{- 2\pi i/3}}- i\AiryAi@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = exp(+ Pi*I/3)*AiryBi(z*exp(- 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))-AiryAi(z*exp(- 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))- I*AiryAi(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerGi[z] == Exp[+ Pi*I/3]*ScorerHi[z*Exp[- 2*Pi*I/3]]- I*AiryAi[z]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1672613648-.2485864233*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .590395216e-1-.1022594507*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E14 9.12.E14] || [[Item:Q2942|<math>\ScorerHi@{z} = e^{+ 2\pi i/3}\ScorerHi@{ze^{+ 2\pi i/3}}+2e^{-\pi i/6}\AiryAi@{ze^{- 2\pi i/3}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerHi@{z} = e^{+ 2\pi i/3}\ScorerHi@{ze^{+ 2\pi i/3}}+2e^{-\pi i/6}\AiryAi@{ze^{- 2\pi i/3}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = exp(+ 2*Pi*I/3)*AiryBi(z*exp(+ 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(+ 2*Pi*I/3))))-AiryAi(z*exp(+ 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(+ 2*Pi*I/3))))+ 2*exp(- Pi*I/6)*AiryAi(z*exp(- 2*Pi*I/3))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerHi[z] == Exp[+ 2*Pi*I/3]*ScorerHi[z*Exp[+ 2*Pi*I/3]]+ 2*Exp[- Pi*I/6]*AiryAi[z*Exp[- 2*Pi*I/3]]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.3013591505+.3291123823*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4978424366-.3195314878*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E14 9.12.E14] || [[Item:Q2942|<math>\ScorerHi@{z} = e^{- 2\pi i/3}\ScorerHi@{ze^{- 2\pi i/3}}+2e^{+\pi i/6}\AiryAi@{ze^{+ 2\pi i/3}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerHi@{z} = e^{- 2\pi i/3}\ScorerHi@{ze^{- 2\pi i/3}}+2e^{+\pi i/6}\AiryAi@{ze^{+ 2\pi i/3}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = exp(- 2*Pi*I/3)*AiryBi(z*exp(- 2*Pi*I/3))*(int(AiryAi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))-AiryAi(z*exp(- 2*Pi*I/3))*(int(AiryBi(t), t = -infinity .. (z*exp(- 2*Pi*I/3))))+ 2*exp(+ Pi*I/6)*AiryAi(z*exp(+ 2*Pi*I/3))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerHi[z] == Exp[- 2*Pi*I/3]*ScorerHi[z*Exp[- 2*Pi*I/3]]+ 2*Exp[+ Pi*I/6]*AiryAi[z*Exp[+ 2*Pi*I/3]]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .430564314-.2897051813*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1771185635+.1022594505*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E15 9.12.E15] || [[Item:Q2943|<math>\ScorerGi@{z} = \frac{3^{-2/3}}{\pi}\*\sum_{k=0}^{\infty}\cos@{\frac{2k-1}{3}\pi}\EulerGamma@{\frac{k+1}{3}}\frac{(3^{1/3}z)^{k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi@{z} = \frac{3^{-2/3}}{\pi}\*\sum_{k=0}^{\infty}\cos@{\frac{2k-1}{3}\pi}\EulerGamma@{\frac{k+1}{3}}\frac{(3^{1/3}z)^{k}}{k!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = ((3)^(- 2/3))/(Pi)* sum(cos((2*k - 1)/(3)*Pi)*GAMMA((k + 1)/(3))*(((3)^(1/3)* z)^(k))/(factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerGi[z] == Divide[(3)^(- 2/3),Pi]* Sum[Cos[Divide[2*k - 1,3]*Pi]*Gamma[Divide[k + 1,3]]*Divide[((3)^(1/3)* z)^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E16 9.12.E16] || [[Item:Q2944|<math>\ScorerGi'@{z} = \frac{3^{-1/3}}{\pi}\*\sum_{k=0}^{\infty}\cos@{\frac{2k+1}{3}\pi}\EulerGamma@{\frac{k+2}{3}}\frac{(3^{1/3}z)^{k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi'@{z} = \frac{3^{-1/3}}{\pi}\*\sum_{k=0}^{\infty}\cos@{\frac{2k+1}{3}\pi}\EulerGamma@{\frac{k+2}{3}}\frac{(3^{1/3}z)^{k}}{k!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))), z$(1) ) = ((3)^(- 1/3))/(Pi)* sum(cos((2*k + 1)/(3)*Pi)*GAMMA((k + 2)/(3))*(((3)^(1/3)* z)^(k))/(factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[ScorerGi[z], {z, 1}] == Divide[(3)^(- 1/3),Pi]* Sum[Cos[Divide[2*k + 1,3]*Pi]*Gamma[Divide[k + 2,3]]*Divide[((3)^(1/3)* z)^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E17 9.12.E17] || [[Item:Q2945|<math>\ScorerHi@{z} = \frac{3^{-2/3}}{\pi}\sum_{k=0}^{\infty}\EulerGamma@{\frac{k+1}{3}}\frac{(3^{1/3}z)^{k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerHi@{z} = \frac{3^{-2/3}}{\pi}\sum_{k=0}^{\infty}\EulerGamma@{\frac{k+1}{3}}\frac{(3^{1/3}z)^{k}}{k!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = ((3)^(- 2/3))/(Pi)*sum(GAMMA((k + 1)/(3))*(((3)^(1/3)* z)^(k))/(factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerHi[z] == Divide[(3)^(- 2/3),Pi]*Sum[Gamma[Divide[k + 1,3]]*Divide[((3)^(1/3)* z)^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E18 9.12.E18] || [[Item:Q2946|<math>\ScorerHi'@{z} = \frac{3^{-1/3}}{\pi}\sum_{k=0}^{\infty}\EulerGamma@{\frac{k+2}{3}}\frac{(3^{1/3}z)^{k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerHi'@{z} = \frac{3^{-1/3}}{\pi}\sum_{k=0}^{\infty}\EulerGamma@{\frac{k+2}{3}}\frac{(3^{1/3}z)^{k}}{k!}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))), z$(1) ) = ((3)^(- 1/3))/(Pi)*sum(GAMMA((k + 2)/(3))*(((3)^(1/3)* z)^(k))/(factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[ScorerHi[z], {z, 1}] == Divide[(3)^(- 1/3),Pi]*Sum[Gamma[Divide[k + 2,3]]*Divide[((3)^(1/3)* z)^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E19 9.12.E19] || [[Item:Q2947|<math>\ScorerGi@{x} = \frac{1}{\pi}\int_{0}^{\infty}\sin@{\tfrac{1}{3}t^{3}+xt}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi@{x} = \frac{1}{\pi}\int_{0}^{\infty}\sin@{\tfrac{1}{3}t^{3}+xt}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)/(Pi)*int(sin((1)/(3)*(t)^(3)+ x*t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerGi[x] == Divide[1,Pi]*Integrate[Sin[Divide[1,3]*(t)^(3)+ x*t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/9.12.E20 9.12.E20] || [[Item:Q2948|<math>\ScorerHi@{z} = \frac{1}{\pi}\int_{0}^{\infty}\exp@{-\tfrac{1}{3}t^{3}+zt}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerHi@{z} = \frac{1}{\pi}\int_{0}^{\infty}\exp@{-\tfrac{1}{3}t^{3}+zt}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = (1)/(Pi)*int(exp(-(1)/(3)*(t)^(3)+ z*t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerHi[z] == Divide[1,Pi]*Integrate[Exp[-Divide[1,3]*(t)^(3)+ z*t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .4525872086+.6186053865*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.510759173-.1408206709*I
Test Values: {z = 1.5}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.12.E21 9.12.E21] || [[Item:Q2949|<math>\ScorerGi@{z} = -\frac{1}{\pi}\int_{0}^{\infty}\exp@{-\tfrac{1}{3}t^{3}-\tfrac{1}{2}zt}\cos@{\tfrac{1}{2}\sqrt{3}zt+\tfrac{2}{3}\pi}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerGi@{z} = -\frac{1}{\pi}\int_{0}^{\infty}\exp@{-\tfrac{1}{3}t^{3}-\tfrac{1}{2}zt}\cos@{\tfrac{1}{2}\sqrt{3}zt+\tfrac{2}{3}\pi}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))) = -(1)/(Pi)*int(exp(-(1)/(3)*(t)^(3)-(1)/(2)*z*t)*cos((1)/(2)*sqrt(3)*z*t +(2)/(3)*Pi), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerGi[z] == -Divide[1,Pi]*Integrate[Exp[-Divide[1,3]*(t)^(3)-Divide[1,2]*z*t]*Cos[Divide[1,2]*Sqrt[3]*z*t +Divide[2,3]*Pi], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Skipped - Because timed out
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| [https://dlmf.nist.gov/9.12.E22 9.12.E22] || [[Item:Q2950|<math>\ScorerHi@{-z} = \frac{4z^{2}}{3^{3/2}\pi^{2}}\int_{0}^{\infty}\frac{\modBesselK{1/3}@{t}}{\zeta^{2}+t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerHi@{-z} = \frac{4z^{2}}{3^{3/2}\pi^{2}}\int_{0}^{\infty}\frac{\modBesselK{1/3}@{t}}{\zeta^{2}+t^{2}}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{3}\pi</math> || <syntaxhighlight lang=mathematica>AiryBi(- z)*(int(AiryAi(t), t = -infinity .. (- z)))-AiryAi(- z)*(int(AiryBi(t), t = -infinity .. (- z))) = (4*(z)^(2))/((3)^(3/2)* (Pi)^(2))*int((BesselK(1/3, t))/((2)/(3)*((z)^((3)/(2)))^(2)+ (t)^(2)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerHi[- z] == Divide[4*(z)^(2),(3)^(3/2)* (Pi)^(2)]*Integrate[Divide[BesselK[1/3, t],Divide[2,3]*((z)^(Divide[3,2]))^(2)+ (t)^(2)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 4]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .660208669e-1-.1388055037e-1*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .528823090e-1
Test Values: {z = 1.5}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 4]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.06602086668543175, -0.01388055052265768]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0.05288230872547964
Test Values: {Rule[z, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/9.12.E24 9.12.E24] || [[Item:Q2952|<math>\ScorerHi@{z} = \frac{3^{-2/3}}{2\pi^{2}i}\int_{-i\infty}^{i\infty}\EulerGamma@{\tfrac{1}{3}+\tfrac{1}{3}t}\EulerGamma@{-t}(3^{1/3}e^{\pi i}z)^{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ScorerHi@{z} = \frac{3^{-2/3}}{2\pi^{2}i}\int_{-i\infty}^{i\infty}\EulerGamma@{\tfrac{1}{3}+\tfrac{1}{3}t}\EulerGamma@{-t}(3^{1/3}e^{\pi i}z)^{t}\diff{t}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{3}+\tfrac{1}{3}t)} > 0, \realpart@@{(-t)} > 0</math> || <syntaxhighlight lang=mathematica>AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))) = ((3)^(- 2/3))/(2*(Pi)^(2)* I)*int(GAMMA((1)/(3)+(1)/(3)*t)*GAMMA(- t)*((3)^(1/3)* exp(Pi*I)*z)^(t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ScorerHi[z] == Divide[(3)^(- 2/3),2*(Pi)^(2)* I]*Integrate[Gamma[Divide[1,3]+Divide[1,3]*t]*Gamma[- t]*((3)^(1/3)* Exp[Pi*I]*z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Manual Skip! || Skipped - Because timed out
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