Struve and Related Functions - 12.2 Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
12.2.E2 d 2 w d z 2 - ( 1 4 ⁒ z 2 + a ) ⁒ w = 0 derivative 𝑀 𝑧 2 1 4 superscript 𝑧 2 π‘Ž 𝑀 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(% \tfrac{1}{4}z^{2}+a\right)w=0}}
\deriv[2]{w}{z}-\left(\tfrac{1}{4}z^{2}+a\right)w = 0		 

diff(w, [z$(2)])-((1)/(4)*(z)^(2)+ a)*w = 0

D[w, {z, 2}]-(Divide[1,4]*(z)^(2)+ a)*w == 0
Failure Failure
Failed [300 / 300]
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}


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)}

... skip entries to safe data
Failed [300 / 300]
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]]]}


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]]]}

... skip entries to safe data
12.2.E3 d 2 w d z 2 + ( 1 4 ⁒ z 2 - a ) ⁒ w = 0 derivative 𝑀 𝑧 2 1 4 superscript 𝑧 2 π‘Ž 𝑀 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% \tfrac{1}{4}z^{2}-a\right)w=0}}
\deriv[2]{w}{z}+\left(\tfrac{1}{4}z^{2}-a\right)w = 0		 

diff(w, [z$(2)])+((1)/(4)*(z)^(2)- a)*w = 0

D[w, {z, 2}]+(Divide[1,4]*(z)^(2)- a)*w == 0
Failure Failure
Failed [300 / 300]
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}


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)}

... skip entries to safe data
Failed [300 / 300]
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]]]}


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]]]}

... skip entries to safe data
12.2.E4 d 2 w d z 2 + ( Ξ½ + 1 2 - 1 4 ⁒ z 2 ) ⁒ w = 0 derivative 𝑀 𝑧 2 𝜈 1 2 1 4 superscript 𝑧 2 𝑀 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% \nu+\tfrac{1}{2}-\tfrac{1}{4}z^{2}\right)w=0}}
\deriv[2]{w}{z}+\left(\nu+\tfrac{1}{2}-\tfrac{1}{4}z^{2}\right)w = 0		 

diff(w, [z$(2)])+(nu +(1)/(2)-(1)/(4)*(z)^(2))*w = 0

D[w, {z, 2}]+(\[Nu]+Divide[1,2]-Divide[1,4]*(z)^(2))*w == 0
Failure Failure
Failed [300 / 300]
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}


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)}

... skip entries to safe data
Failed [296 / 300]
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[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


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[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
12.2.E5 D Ξ½ ⁑ ( z ) = U ⁑ ( - 1 2 - Ξ½ , z ) Whittaker-D 𝜈 𝑧 parabolic-U 1 2 𝜈 𝑧 {\displaystyle{\displaystyle D_{\nu}\left(z\right)=U\left(-\tfrac{1}{2}-\nu,z% \right)}}
\WhittakerparaD{\nu}@{z} = \paraU@{-\tfrac{1}{2}-\nu}{z}		 

CylinderD(nu, z) = CylinderU(-(1)/(2)- nu, z)

ParabolicCylinderD[\[Nu], z] == ParabolicCylinderD[- 1/2 -(-Divide[1,2]- \[Nu]), z]
Successful Successful - Successful [Tested: 70]
12.2.E6 U ⁑ ( a , 0 ) = Ο€ 2 1 2 ⁒ a + 1 4 ⁒ Ξ“ ⁑ ( 3 4 + 1 2 ⁒ a ) parabolic-U π‘Ž 0 πœ‹ superscript 2 1 2 π‘Ž 1 4 Euler-Gamma 3 4 1 2 π‘Ž {\displaystyle{\displaystyle U\left(a,0\right)=\frac{\sqrt{\pi}}{2^{\frac{1}{2% }a+\frac{1}{4}}\Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)}}}
\paraU@{a}{0} = \frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}		 β„œ ⁑ ( 3 4 + 1 2 ⁒ a ) > 0 3 4 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\frac{3}{4}+\frac{1}{2}a)>0}} 
CylinderU(a, 0) = (sqrt(Pi))/((2)^((1)/(2)*a +(1)/(4))* GAMMA((3)/(4)+(1)/(2)*a))

ParabolicCylinderD[- 1/2 -(a), 0] == Divide[Sqrt[Pi],(2)^(Divide[1,2]*a +Divide[1,4])* Gamma[Divide[3,4]+Divide[1,2]*a]]
Successful Successful - Successful [Tested: 4]
12.2.E7 U β€² ⁑ ( a , 0 ) = - Ο€ 2 1 2 ⁒ a - 1 4 ⁒ Ξ“ ⁑ ( 1 4 + 1 2 ⁒ a ) diffop parabolic-U 1 π‘Ž 0 πœ‹ superscript 2 1 2 π‘Ž 1 4 Euler-Gamma 1 4 1 2 π‘Ž {\displaystyle{\displaystyle U'\left(a,0\right)=-\frac{\sqrt{\pi}}{2^{\frac{1}% {2}a-\frac{1}{4}}\Gamma\left(\frac{1}{4}+\frac{1}{2}a\right)}}}
\paraU'@{a}{0} = -\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}		 β„œ ⁑ ( 1 4 + 1 2 ⁒ a ) > 0 1 4 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\frac{1}{4}+\frac{1}{2}a)>0}} 
subs( temp=0, diff( CylinderU(a, temp), temp$(1) ) ) = -(sqrt(Pi))/((2)^((1)/(2)*a -(1)/(4))* GAMMA((1)/(4)+(1)/(2)*a))

(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]]
Successful Successful - Successful [Tested: 3]
12.2.E8 V ⁑ ( a , 0 ) = Ο€ ⁒ 2 1 2 ⁒ a + 1 4 ( Ξ“ ⁑ ( 3 4 - 1 2 ⁒ a ) ) 2 ⁒ Ξ“ ⁑ ( 1 4 + 1 2 ⁒ a ) parabolic-V π‘Ž 0 πœ‹ superscript 2 1 2 π‘Ž 1 4 superscript Euler-Gamma 3 4 1 2 π‘Ž 2 Euler-Gamma 1 4 1 2 π‘Ž {\displaystyle{\displaystyle V\left(a,0\right)=\frac{\pi 2^{\frac{1}{2}a+\frac% {1}{4}}}{\left(\Gamma\left(\frac{3}{4}-\frac{1}{2}a\right)\right)^{2}\Gamma% \left(\frac{1}{4}+\frac{1}{2}a\right)}}}
\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}}		 β„œ ⁑ ( 3 4 - 1 2 ⁒ a ) > 0 , β„œ ⁑ ( 1 4 + 1 2 ⁒ a ) > 0 formulae-sequence 3 4 1 2 π‘Ž 0 1 4 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\frac{3}{4}-\frac{1}{2}a)>0,\Re(\frac{1}{4}+% \frac{1}{2}a)>0}} 
CylinderV(a, 0) = (Pi*(2)^((1)/(2)*a +(1)/(4)))/((GAMMA((3)/(4)-(1)/(2)*a))^(2)* GAMMA((1)/(4)+(1)/(2)*a))

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]]
Successful Failure -
Failed [1 / 1]
Result: Plus[-0.7978845608028653, Times[0.7978845608028655, GAMMA[1.0]]]
Test Values: {Rule[a, 0.5]}

12.2.E9 V β€² ⁑ ( a , 0 ) = Ο€ ⁒ 2 1 2 ⁒ a + 3 4 ( Ξ“ ⁑ ( 1 4 - 1 2 ⁒ a ) ) 2 ⁒ Ξ“ ⁑ ( 3 4 + 1 2 ⁒ a ) diffop parabolic-V 1 π‘Ž 0 πœ‹ superscript 2 1 2 π‘Ž 3 4 superscript Euler-Gamma 1 4 1 2 π‘Ž 2 Euler-Gamma 3 4 1 2 π‘Ž {\displaystyle{\displaystyle V'\left(a,0\right)=\frac{\pi 2^{\frac{1}{2}a+% \frac{3}{4}}}{\left(\Gamma\left(\frac{1}{4}-\frac{1}{2}a\right)\right)^{2}% \Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)}}}
\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}}		 β„œ ⁑ ( 1 4 - 1 2 ⁒ a ) > 0 , β„œ ⁑ ( 3 4 + 1 2 ⁒ a ) > 0 formulae-sequence 1 4 1 2 π‘Ž 0 3 4 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\frac{1}{4}-\frac{1}{2}a)>0,\Re(\frac{3}{4}+% \frac{1}{2}a)>0}} 
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))

(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]]
Successful Failure -
Failed [1 / 1]
Result: -0.7978845608028653
Test Values: {Rule[a, -0.5]}

12.2.E10 𝒲 ⁑ { U ⁑ ( a , z ) , V ⁑ ( a , z ) } = 2 / Ο€ Wronskian parabolic-U π‘Ž 𝑧 parabolic-V π‘Ž 𝑧 2 πœ‹ {\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,z\right),V\left(a,z% \right)\right\}=\sqrt{2/\pi}}}
\Wronskian@{\paraU@{a}{z},\paraV@{a}{z}} = \sqrt{2/\pi}		 

(CylinderU(a, z))*diff(CylinderV(a, z), z)-diff(CylinderU(a, z), z)*(CylinderV(a, z)) = sqrt(2/Pi)

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]
Failure Failure
Failed [14 / 42]
Result: .708254234e-1-.722805450e-2*I
Test Values: {a = 3/2, z = 1/2*3^(1/2)+1/2*I}


Result: .4257865765+.241883787*I
Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [42 / 42]
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]]]}


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]]]}

... skip entries to safe data
12.2.E11 𝒲 ⁑ { U ⁑ ( a , z ) , U ⁑ ( a , - z ) } = 2 ⁒ Ο€ Ξ“ ⁑ ( 1 2 + a ) Wronskian parabolic-U π‘Ž 𝑧 parabolic-U π‘Ž 𝑧 2 πœ‹ Euler-Gamma 1 2 π‘Ž {\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,z\right),U\left(a,-z% \right)\right\}=\frac{\sqrt{2\pi}}{\Gamma\left(\frac{1}{2}+a\right)}}}
\Wronskian@{\paraU@{a}{z},\paraU@{a}{-z}} = \frac{\sqrt{2\pi}}{\EulerGamma@{\frac{1}{2}+a}}		 β„œ ⁑ ( 1 2 + a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}+a)>0}} 
(CylinderU(a, z))*diff(CylinderU(a, - z), z)-diff(CylinderU(a, z), z)*(CylinderU(a, - z)) = (sqrt(2*Pi))/(GAMMA((1)/(2)+ a))

Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(a), - z]}, z] == Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
12.2.E12 𝒲 ⁑ { U ⁑ ( a , z ) , U ⁑ ( - a , + i ⁒ z ) } = - i ⁒ e + i ⁒ Ο€ ⁒ ( 1 2 ⁒ a + 1 4 ) Wronskian parabolic-U π‘Ž 𝑧 parabolic-U π‘Ž 𝑖 𝑧 𝑖 superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 {\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,z\right),U\left(-a,+iz% \right)\right\}=-ie^{+i\pi(\frac{1}{2}a+\frac{1}{4})}}}
\Wronskian@{\paraU@{a}{z},\paraU@{-a}{+ iz}} = - ie^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}		 

(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)))

Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(- a), + I*z]}, z] == - I*Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]
Failure Failure Successful [Tested: 42] Successful [Tested: 42]
12.2.E12 𝒲 ⁑ { U ⁑ ( a , z ) , U ⁑ ( - a , - i ⁒ z ) } = + i ⁒ e - i ⁒ Ο€ ⁒ ( 1 2 ⁒ a + 1 4 ) Wronskian parabolic-U π‘Ž 𝑧 parabolic-U π‘Ž 𝑖 𝑧 𝑖 superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 {\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,z\right),U\left(-a,-iz% \right)\right\}=+ie^{-i\pi(\frac{1}{2}a+\frac{1}{4})}}}
\Wronskian@{\paraU@{a}{z},\paraU@{-a}{- iz}} = + ie^{- i\pi(\frac{1}{2}a+\frac{1}{4})}		 

(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)))

Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(- a), - I*z]}, z] == + I*Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]
Failure Failure Successful [Tested: 42] Successful [Tested: 42]
12.2.E13 U ⁑ ( - n - 1 2 , - z ) = ( - 1 ) n ⁒ U ⁑ ( - n - 1 2 , z ) parabolic-U 𝑛 1 2 𝑧 superscript 1 𝑛 parabolic-U 𝑛 1 2 𝑧 {\displaystyle{\displaystyle U\left(-n-\tfrac{1}{2},-z\right)=(-1)^{n}U\left(-% n-\tfrac{1}{2},z\right)}}
\paraU@{-n-\tfrac{1}{2}}{-z} = (-1)^{n}\paraU@{-n-\tfrac{1}{2}}{z}		 

CylinderU(- n -(1)/(2), - z) = (- 1)^(n)* CylinderU(- n -(1)/(2), z)

ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), - z] == (- 1)^(n)* ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), z]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
12.2.E14 V ⁑ ( n + 1 2 , - z ) = ( - 1 ) n ⁒ V ⁑ ( n + 1 2 , z ) parabolic-V 𝑛 1 2 𝑧 superscript 1 𝑛 parabolic-V 𝑛 1 2 𝑧 {\displaystyle{\displaystyle V\left(n+\tfrac{1}{2},-z\right)=(-1)^{n}V\left(n+% \tfrac{1}{2},z\right)}}
\paraV@{n+\tfrac{1}{2}}{-z} = (-1)^{n}\paraV@{n+\tfrac{1}{2}}{z}		 

CylinderV(n +(1)/(2), - z) = (- 1)^(n)* CylinderV(n +(1)/(2), z)

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)])
Successful Failure - Successful [Tested: 21]
12.2.E15 U ⁑ ( a , - z ) = - sin ⁑ ( Ο€ ⁒ a ) ⁒ U ⁑ ( a , z ) + Ο€ Ξ“ ⁑ ( 1 2 + a ) ⁒ V ⁑ ( a , z ) parabolic-U π‘Ž 𝑧 πœ‹ π‘Ž parabolic-U π‘Ž 𝑧 πœ‹ Euler-Gamma 1 2 π‘Ž parabolic-V π‘Ž 𝑧 {\displaystyle{\displaystyle U\left(a,-z\right)=-\sin\left(\pi a\right)U\left(% a,z\right)+\frac{\pi}{\Gamma\left(\frac{1}{2}+a\right)}V\left(a,z\right)}}
\paraU@{a}{-z} = -\sin@{\pi a}\paraU@{a}{z}+\frac{\pi}{\EulerGamma@{\frac{1}{2}+a}}\paraV@{a}{z}		 β„œ ⁑ ( 1 2 + a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}+a)>0}} 
CylinderU(a, - z) = - sin(Pi*a)*CylinderU(a, z)+(Pi)/(GAMMA((1)/(2)+ a))*CylinderV(a, z)

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)])
Successful Failure -
Failed [21 / 21]
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]]]}


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]]]}

... skip entries to safe data
12.2.E16 V ⁑ ( a , - z ) = cos ⁑ ( Ο€ ⁒ a ) Ξ“ ⁑ ( 1 2 - a ) ⁒ U ⁑ ( a , z ) + sin ⁑ ( Ο€ ⁒ a ) ⁒ V ⁑ ( a , z ) parabolic-V π‘Ž 𝑧 πœ‹ π‘Ž Euler-Gamma 1 2 π‘Ž parabolic-U π‘Ž 𝑧 πœ‹ π‘Ž parabolic-V π‘Ž 𝑧 {\displaystyle{\displaystyle V\left(a,-z\right)=\frac{\cos\left(\pi a\right)}{% \Gamma\left(\frac{1}{2}-a\right)}U\left(a,z\right)+\sin\left(\pi a\right)V% \left(a,z\right)}}
\paraV@{a}{-z} = \frac{\cos@{\pi a}}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sin@{\pi a}\paraV@{a}{z}		 β„œ ⁑ ( 1 2 - a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}-a)>0}} 
CylinderV(a, - z) = (cos(Pi*a))/(GAMMA((1)/(2)- a))*CylinderU(a, z)+ sin(Pi*a)*CylinderV(a, z)

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)])
Failure Failure Successful [Tested: 21]
Failed [7 / 21]
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]]]}


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]]]}

... skip entries to safe data
12.2.E17 2 ⁒ Ο€ ⁒ U ⁑ ( - a , + i ⁒ z ) = Ξ“ ⁑ ( 1 2 + a ) ⁒ ( e - i ⁒ Ο€ ⁒ ( 1 2 ⁒ a - 1 4 ) ⁒ U ⁑ ( a , z ) + e + i ⁒ Ο€ ⁒ ( 1 2 ⁒ a - 1 4 ) ⁒ U ⁑ ( a , - z ) ) 2 πœ‹ parabolic-U π‘Ž 𝑖 𝑧 Euler-Gamma 1 2 π‘Ž superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑧 superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑧 {\displaystyle{\displaystyle\sqrt{2\pi}U\left(-a,+iz\right)=\Gamma\left(\tfrac% {1}{2}+a\right)\left(e^{-i\pi(\frac{1}{2}a-\frac{1}{4})}U\left(a,z\right)+e^{+% i\pi(\frac{1}{2}a-\frac{1}{4})}U\left(a,-z\right)\right)}}
\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)		 β„œ ⁑ ( 1 2 + a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}+a)>0}} 
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))

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])
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
12.2.E17 2 ⁒ Ο€ ⁒ U ⁑ ( - a , - i ⁒ z ) = Ξ“ ⁑ ( 1 2 + a ) ⁒ ( e + i ⁒ Ο€ ⁒ ( 1 2 ⁒ a - 1 4 ) ⁒ U ⁑ ( a , z ) + e - i ⁒ Ο€ ⁒ ( 1 2 ⁒ a - 1 4 ) ⁒ U ⁑ ( a , - z ) ) 2 πœ‹ parabolic-U π‘Ž 𝑖 𝑧 Euler-Gamma 1 2 π‘Ž superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑧 superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑧 {\displaystyle{\displaystyle\sqrt{2\pi}U\left(-a,-iz\right)=\Gamma\left(\tfrac% {1}{2}+a\right)\left(e^{+i\pi(\frac{1}{2}a-\frac{1}{4})}U\left(a,z\right)+e^{-% i\pi(\frac{1}{2}a-\frac{1}{4})}U\left(a,-z\right)\right)}}
\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)		 β„œ ⁑ ( 1 2 + a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}+a)>0}} 
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))

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])
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
12.2.E18 2 ⁒ Ο€ ⁒ U ⁑ ( a , z ) = Ξ“ ⁑ ( 1 2 - a ) ⁒ ( e - i ⁒ Ο€ ⁒ ( 1 2 ⁒ a + 1 4 ) ⁒ U ⁑ ( - a , + i ⁒ z ) + e + i ⁒ Ο€ ⁒ ( 1 2 ⁒ a + 1 4 ) ⁒ U ⁑ ( - a , - i ⁒ z ) ) 2 πœ‹ parabolic-U π‘Ž 𝑧 Euler-Gamma 1 2 π‘Ž superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑖 𝑧 superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑖 𝑧 {\displaystyle{\displaystyle\sqrt{2\pi}U\left(a,z\right)=\Gamma\left(\tfrac{1}% {2}-a\right)\left(e^{-i\pi(\frac{1}{2}a+\frac{1}{4})}U\left(-a,+iz\right)+e^{+% i\pi(\frac{1}{2}a+\frac{1}{4})}U\left(-a,-iz\right)\right)}}
\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)		 β„œ ⁑ ( 1 2 - a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}-a)>0}} 
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))

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])
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
12.2.E18 2 ⁒ Ο€ ⁒ U ⁑ ( a , z ) = Ξ“ ⁑ ( 1 2 - a ) ⁒ ( e + i ⁒ Ο€ ⁒ ( 1 2 ⁒ a + 1 4 ) ⁒ U ⁑ ( - a , - i ⁒ z ) + e - i ⁒ Ο€ ⁒ ( 1 2 ⁒ a + 1 4 ) ⁒ U ⁑ ( - a , + i ⁒ z ) ) 2 πœ‹ parabolic-U π‘Ž 𝑧 Euler-Gamma 1 2 π‘Ž superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑖 𝑧 superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑖 𝑧 {\displaystyle{\displaystyle\sqrt{2\pi}U\left(a,z\right)=\Gamma\left(\tfrac{1}% {2}-a\right)\left(e^{+i\pi(\frac{1}{2}a+\frac{1}{4})}U\left(-a,-iz\right)+e^{-% i\pi(\frac{1}{2}a+\frac{1}{4})}U\left(-a,+iz\right)\right)}}
\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)		 β„œ ⁑ ( 1 2 - a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}-a)>0}} 
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))

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])
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
12.2.E19 U ⁑ ( a , z ) = + i ⁒ e + i ⁒ Ο€ ⁒ a ⁒ U ⁑ ( a , - z ) + 2 ⁒ Ο€ Ξ“ ⁑ ( 1 2 + a ) ⁒ e + i ⁒ Ο€ ⁒ ( 1 2 ⁒ a - 1 4 ) ⁒ U ⁑ ( - a , + i ⁒ z ) parabolic-U π‘Ž 𝑧 𝑖 superscript 𝑒 𝑖 πœ‹ π‘Ž parabolic-U π‘Ž 𝑧 2 πœ‹ Euler-Gamma 1 2 π‘Ž superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑖 𝑧 {\displaystyle{\displaystyle U\left(a,z\right)=+ie^{+i\pi a}U\left(a,-z\right)% +\frac{\sqrt{2\pi}}{\Gamma\left(\tfrac{1}{2}+a\right)}e^{+i\pi(\frac{1}{2}a-% \frac{1}{4})}U\left(-a,+iz\right)}}
\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}		 β„œ ⁑ ( 1 2 + a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}+a)>0}} 
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)

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]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
12.2.E19 U ⁑ ( a , z ) = - i ⁒ e - i ⁒ Ο€ ⁒ a ⁒ U ⁑ ( a , - z ) + 2 ⁒ Ο€ Ξ“ ⁑ ( 1 2 + a ) ⁒ e - i ⁒ Ο€ ⁒ ( 1 2 ⁒ a - 1 4 ) ⁒ U ⁑ ( - a , - i ⁒ z ) parabolic-U π‘Ž 𝑧 𝑖 superscript 𝑒 𝑖 πœ‹ π‘Ž parabolic-U π‘Ž 𝑧 2 πœ‹ Euler-Gamma 1 2 π‘Ž superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑖 𝑧 {\displaystyle{\displaystyle U\left(a,z\right)=-ie^{-i\pi a}U\left(a,-z\right)% +\frac{\sqrt{2\pi}}{\Gamma\left(\tfrac{1}{2}+a\right)}e^{-i\pi(\frac{1}{2}a-% \frac{1}{4})}U\left(-a,-iz\right)}}
\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}		 β„œ ⁑ ( 1 2 + a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}+a)>0}} 
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)

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]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
12.2.E20 V ⁑ ( a , z ) = - i Ξ“ ⁑ ( 1 2 - a ) ⁒ U ⁑ ( a , z ) + 2 Ο€ ⁒ e - i ⁒ Ο€ ⁒ ( 1 2 ⁒ a - 1 4 ) ⁒ U ⁑ ( - a , + i ⁒ z ) parabolic-V π‘Ž 𝑧 𝑖 Euler-Gamma 1 2 π‘Ž parabolic-U π‘Ž 𝑧 2 πœ‹ superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑖 𝑧 {\displaystyle{\displaystyle V\left(a,z\right)=\frac{-i}{\Gamma\left(\frac{1}{% 2}-a\right)}U\left(a,z\right)+\sqrt{\frac{2}{\pi}}e^{-i\pi(\frac{1}{2}a-\frac{% 1}{4})}U\left(-a,+iz\right)}}
\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}		 β„œ ⁑ ( 1 2 - a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}-a)>0}} 
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)

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]
Failure Failure Successful [Tested: 21]
Failed [21 / 21]
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]]]}


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]]]}

... skip entries to safe data
12.2.E20 V ⁑ ( a , z ) = + i Ξ“ ⁑ ( 1 2 - a ) ⁒ U ⁑ ( a , z ) + 2 Ο€ ⁒ e + i ⁒ Ο€ ⁒ ( 1 2 ⁒ a - 1 4 ) ⁒ U ⁑ ( - a , - i ⁒ z ) parabolic-V π‘Ž 𝑧 𝑖 Euler-Gamma 1 2 π‘Ž parabolic-U π‘Ž 𝑧 2 πœ‹ superscript 𝑒 𝑖 πœ‹ 1 2 π‘Ž 1 4 parabolic-U π‘Ž 𝑖 𝑧 {\displaystyle{\displaystyle V\left(a,z\right)=\frac{+i}{\Gamma\left(\frac{1}{% 2}-a\right)}U\left(a,z\right)+\sqrt{\frac{2}{\pi}}e^{+i\pi(\frac{1}{2}a-\frac{% 1}{4})}U\left(-a,-iz\right)}}
\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}		 β„œ ⁑ ( 1 2 - a ) > 0 1 2 π‘Ž 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}-a)>0}} 
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)

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]
Failure Failure Successful [Tested: 21]
Failed [21 / 21]
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]]]}


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]]]}

... skip entries to safe data
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