Parabolic Cylinder Functions - 12.7 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
12.7.E1 U ⁑ ( - 1 2 , z ) = D 0 ⁑ ( z ) parabolic-U 1 2 𝑧 Whittaker-D 0 𝑧 {\displaystyle{\displaystyle U\left(-\tfrac{1}{2},z\right)=D_{0}\left(z\right)}}
\paraU@{-\tfrac{1}{2}}{z} = \WhittakerparaD{0}@{z}		 

CylinderU(-(1)/(2), z) = CylinderD(0, z)

ParabolicCylinderD[- 1/2 -(-Divide[1,2]), z] == ParabolicCylinderD[0, z]
Successful Successful - Successful [Tested: 7]
12.7.E1 D 0 ⁑ ( z ) = e - 1 4 ⁒ z 2 Whittaker-D 0 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle D_{0}\left(z\right)=e^{-\frac{1}{4}z^{2}}}}
\WhittakerparaD{0}@{z} = e^{-\frac{1}{4}z^{2}}		 

CylinderD(0, z) = exp(-(1)/(4)*(z)^(2))

ParabolicCylinderD[0, z] == Exp[-Divide[1,4]*(z)^(2)]
Successful Successful - Successful [Tested: 7]
12.7.E2 U ⁑ ( - n - 1 2 , z ) = D n ⁑ ( z ) parabolic-U 𝑛 1 2 𝑧 Whittaker-D 𝑛 𝑧 {\displaystyle{\displaystyle U\left(-n-\tfrac{1}{2},z\right)=D_{n}\left(z% \right)}}
\paraU@{-n-\tfrac{1}{2}}{z} = \WhittakerparaD{n}@{z}		 

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

ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), z] == ParabolicCylinderD[n, z]
Successful Successful - Successful [Tested: 7]
12.7.E4 V ⁑ ( - 1 2 , z ) = ( 2 / Ο€ ) ⁒ e 1 4 ⁒ z 2 ⁒ F ⁑ ( z / 2 ) parabolic-V 1 2 𝑧 2 πœ‹ superscript 𝑒 1 4 superscript 𝑧 2 Dawsons-integral 𝑧 2 {\displaystyle{\displaystyle V\left(-\tfrac{1}{2},z\right)=(\ifrac{2}{\sqrt{% \pi}}\,)e^{\frac{1}{4}z^{2}}F\left(z/\sqrt{2}\right)}}
\paraV@{-\tfrac{1}{2}}{z} = (\ifrac{2}{\sqrt{\pi}}\,)e^{\frac{1}{4}z^{2}}\DawsonsintF@{z/\sqrt{2}}		 

CylinderV(-(1)/(2), z) = ((2)/(sqrt(Pi)))*exp((1)/(4)*(z)^(2))*dawson(z/(sqrt(2)))

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])]
Successful Failure -
Failed [7 / 7]
Result: Complex[-0.6813729414422256, -0.33849358809725466]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.4709386394349885, -0.6804499612300876]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

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

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

ParabolicCylinderD[- 1/2 -(Divide[1,2]), z] == ParabolicCylinderD[- 1, z]
Successful Successful - Successful [Tested: 7]
12.7.E5 D - 1 ⁑ ( z ) = 1 2 ⁒ Ο€ ⁒ e 1 4 ⁒ z 2 ⁒ erfc ⁑ ( z / 2 ) Whittaker-D 1 𝑧 1 2 πœ‹ superscript 𝑒 1 4 superscript 𝑧 2 complementary-error-function 𝑧 2 {\displaystyle{\displaystyle D_{-1}\left(z\right)=\sqrt{\tfrac{1}{2}\pi}\,e^{% \frac{1}{4}z^{2}}\operatorname{erfc}\left(z/\sqrt{2}\right)}}
\WhittakerparaD{-1}@{z} = \sqrt{\tfrac{1}{2}\pi}\,e^{\frac{1}{4}z^{2}}\erfc@{z/\sqrt{2}}		 

CylinderD(- 1, z) = sqrt((1)/(2)*Pi)*exp((1)/(4)*(z)^(2))*erfc(z/(sqrt(2)))

ParabolicCylinderD[- 1, z] == Sqrt[Divide[1,2]*Pi]*Exp[Divide[1,4]*(z)^(2)]*Erfc[z/(Sqrt[2])]
Successful Successful - Successful [Tested: 7]
12.7.E6 U ⁑ ( n + 1 2 , z ) = D - n - 1 ⁑ ( z ) parabolic-U 𝑛 1 2 𝑧 Whittaker-D 𝑛 1 𝑧 {\displaystyle{\displaystyle U\left(n+\tfrac{1}{2},z\right)=D_{-n-1}\left(z% \right)}}
\paraU@{n+\tfrac{1}{2}}{z} = \WhittakerparaD{-n-1}@{z}		 

CylinderU(n +(1)/(2), z) = CylinderD(- n - 1, z)

ParabolicCylinderD[- 1/2 -(n +Divide[1,2]), z] == ParabolicCylinderD[- n - 1, z]
Successful Successful Manual Skip! Successful [Tested: 7]
12.7.E6 D - n - 1 ⁑ ( z ) = Ο€ 2 ⁒ ( - 1 ) n n ! ⁒ e - 1 4 ⁒ z 2 ⁒ d n ( e 1 2 ⁒ z 2 ⁒ erfc ⁑ ( z / 2 ) ) d z n Whittaker-D 𝑛 1 𝑧 πœ‹ 2 superscript 1 𝑛 𝑛 superscript 𝑒 1 4 superscript 𝑧 2 derivative superscript 𝑒 1 2 superscript 𝑧 2 complementary-error-function 𝑧 2 𝑧 𝑛 {\displaystyle{\displaystyle D_{-n-1}\left(z\right)=\sqrt{\frac{\pi}{2}}\frac{% (-1)^{n}}{n!}e^{-\frac{1}{4}z^{2}}\frac{{\mathrm{d}}^{n}\left(e^{\frac{1}{2}z^% {2}}\operatorname{erfc}\left(z/\sqrt{2}\right)\right)}{{\mathrm{d}z}^{n}}}}
\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}		 

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)])

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}]
Failure Failure Manual Skip!
Failed [2 / 7]
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]}


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]}

12.7.E8 U ⁑ ( - 2 , z ) = z 5 / 2 4 ⁒ 2 ⁒ Ο€ ⁒ ( 2 ⁒ K 1 4 ⁑ ( 1 4 ⁒ z 2 ) + 3 ⁒ K 3 4 ⁑ ( 1 4 ⁒ z 2 ) - K 5 4 ⁑ ( 1 4 ⁒ z 2 ) ) parabolic-U 2 𝑧 superscript 𝑧 5 2 4 2 πœ‹ 2 modified-Bessel-second-kind 1 4 1 4 superscript 𝑧 2 3 modified-Bessel-second-kind 3 4 1 4 superscript 𝑧 2 modified-Bessel-second-kind 5 4 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle U\left(-2,z\right)=\frac{z^{5/2}}{4\sqrt{2\pi}}% \left(2K_{\frac{1}{4}}\left(\tfrac{1}{4}z^{2}\right)+3K_{\frac{3}{4}}\left(% \tfrac{1}{4}z^{2}\right)-K_{\frac{5}{4}}\left(\tfrac{1}{4}z^{2}\right)\right)}}
\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)		 

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)))

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)])
Failure Failure
Failed [2 / 7]
Result: -2.928712959+.1903824416*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: -1.578570932+.7263102924*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Failed [2 / 7]
Result: Complex[-2.928712959362369, 0.19038244130086163]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-1.5785709321816723, 0.7263102922437361]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

12.7.E9 U ⁑ ( - 1 , z ) = z 3 / 2 2 ⁒ 2 ⁒ Ο€ ⁒ ( K 1 4 ⁑ ( 1 4 ⁒ z 2 ) + K 3 4 ⁑ ( 1 4 ⁒ z 2 ) ) parabolic-U 1 𝑧 superscript 𝑧 3 2 2 2 πœ‹ modified-Bessel-second-kind 1 4 1 4 superscript 𝑧 2 modified-Bessel-second-kind 3 4 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle U\left(-1,z\right)=\frac{z^{3/2}}{2\sqrt{2\pi}}% \left(K_{\frac{1}{4}}\left(\tfrac{1}{4}z^{2}\right)+K_{\frac{3}{4}}\left(% \tfrac{1}{4}z^{2}\right)\right)}}
\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)		 

CylinderU(- 1, z) = ((z)^(3/2))/(2*sqrt(2*Pi))*(BesselK((1)/(4), (1)/(4)*(z)^(2))+ BesselK((3)/(4), (1)/(4)*(z)^(2)))

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)])
Failure Failure
Failed [2 / 7]
Result: .5254625443+1.913964596*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: -.1061142274-1.367750447*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Failed [2 / 7]
Result: Complex[0.5254625445137794, 1.9139645960722755]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-0.10611422720224939, -1.3677504477251]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

12.7.E10 U ⁑ ( 0 , z ) = z 2 ⁒ Ο€ ⁒ K 1 4 ⁑ ( 1 4 ⁒ z 2 ) parabolic-U 0 𝑧 𝑧 2 πœ‹ modified-Bessel-second-kind 1 4 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle U\left(0,z\right)=\sqrt{\frac{z}{2\pi}}K_{\frac{1% }{4}}\left(\tfrac{1}{4}z^{2}\right)}}
\paraU@{0}{z} = \sqrt{\frac{z}{2\pi}}\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}		 

CylinderU(0, z) = sqrt((z)/(2*Pi))*BesselK((1)/(4), (1)/(4)*(z)^(2))

ParabolicCylinderD[- 1/2 -(0), z] == Sqrt[Divide[z,2*Pi]]*BesselK[Divide[1,4], Divide[1,4]*(z)^(2)]
Failure Failure
Failed [2 / 7]
Result: 2.016879450-1.384601654*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: 1.973186649+1.022506910*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Failed [2 / 7]
Result: Complex[2.0168794499257325, -1.3846016541017099]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[1.9731866495584476, 1.0225069102497304]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

12.7.E11 U ⁑ ( 1 , z ) = z 3 / 2 2 ⁒ Ο€ ⁒ ( K 3 4 ⁑ ( 1 4 ⁒ z 2 ) - K 1 4 ⁑ ( 1 4 ⁒ z 2 ) ) parabolic-U 1 𝑧 superscript 𝑧 3 2 2 πœ‹ modified-Bessel-second-kind 3 4 1 4 superscript 𝑧 2 modified-Bessel-second-kind 1 4 1 4 superscript 𝑧 2 {\displaystyle{\displaystyle U\left(1,z\right)=\frac{z^{3/2}}{\sqrt{2\pi}}% \left(K_{\frac{3}{4}}\left(\tfrac{1}{4}z^{2}\right)-K_{\frac{1}{4}}\left(% \tfrac{1}{4}z^{2}\right)\right)}}
\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)		 

CylinderU(1, z) = ((z)^(3/2))/(sqrt(2*Pi))*(BesselK((3)/(4), (1)/(4)*(z)^(2))- BesselK((1)/(4), (1)/(4)*(z)^(2)))

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)])
Failure Failure
Failed [2 / 7]
Result: .6696041257-1.050010143*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: 2.182924166+1.008719675*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Failed [2 / 7]
Result: Complex[0.6696041258052213, -1.050010141970097]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[2.1829241651976083, 1.0087196737510498]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

12.7.E14 U ⁑ ( a , z ) = 2 - 1 4 - 1 2 ⁒ a ⁒ e - 1 4 ⁒ z 2 ⁒ U ⁑ ( 1 2 ⁒ a + 1 4 , 1 2 , 1 2 ⁒ z 2 ) parabolic-U π‘Ž 𝑧 superscript 2 1 4 1 2 π‘Ž superscript 𝑒 1 4 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 2 π‘Ž 1 4 1 2 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle U\left(a,z\right)=2^{-\frac{1}{4}-\frac{1}{2}a}e^% {-\frac{1}{4}z^{2}}U\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}% z^{2}\right)}}
\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}}		 

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))

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)]
Failure Failure
Failed [10 / 42]
Result: -1.528312538+1.673428352*I
Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}


Result: -1.682421259-.5335370987*I
Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [10 / 42]
Result: Complex[-1.5283125381510665, 1.6734283529572487]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-1.6824212600186188, -0.5335370991065028]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
12.7.E14 2 - 1 4 - 1 2 ⁒ a ⁒ e - 1 4 ⁒ z 2 ⁒ U ⁑ ( 1 2 ⁒ a + 1 4 , 1 2 , 1 2 ⁒ z 2 ) = 2 - 3 4 - 1 2 ⁒ a ⁒ z ⁒ e - 1 4 ⁒ z 2 ⁒ U ⁑ ( 1 2 ⁒ a + 3 4 , 3 2 , 1 2 ⁒ z 2 ) superscript 2 1 4 1 2 π‘Ž superscript 𝑒 1 4 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 2 π‘Ž 1 4 1 2 1 2 superscript 𝑧 2 superscript 2 3 4 1 2 π‘Ž 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 2 π‘Ž 3 4 3 2 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle 2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}% }U\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2},\tfrac{1}{2}z^{2}\right)=2^{-% \frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}U\left(\tfrac{1}{2}a+\tfrac{3}{% 4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)}}
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}}		 

(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))

(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)]
Failure Failure
Failed [12 / 42]
Result: 1.528312538-1.673428353*I
Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}


Result: 1.682421260+.5335370988*I
Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [12 / 42]
Result: Complex[1.5283125381510665, -1.673428352957249]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[1.6824212600186188, 0.5335370991065027]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
12.7.E14 2 - 3 4 - 1 2 ⁒ a ⁒ z ⁒ e - 1 4 ⁒ z 2 ⁒ U ⁑ ( 1 2 ⁒ a + 3 4 , 3 2 , 1 2 ⁒ z 2 ) = 2 - 1 2 ⁒ a ⁒ z - 1 2 ⁒ W - 1 2 ⁒ a , + 1 4 ⁑ ( 1 2 ⁒ z 2 ) superscript 2 3 4 1 2 π‘Ž 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 2 π‘Ž 3 4 3 2 1 2 superscript 𝑧 2 superscript 2 1 2 π‘Ž superscript 𝑧 1 2 Whittaker-confluent-hypergeometric-W 1 2 π‘Ž 1 4 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle 2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2% }}U\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=2^{-% \frac{1}{2}a}z^{-\frac{1}{2}}W_{-\frac{1}{2}a,+\frac{1}{4}}\left(\tfrac{1}{2}z% ^{2}\right)}}
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}}		 

(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))

(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)]
Failure Failure
Failed [12 / 42]
Result: .725579081e-1+1.600870446*I
Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}


Result: -.5744420805-1.107979180*I
Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [12 / 42]
Result: Complex[0.0725579074030912, 1.600870445554158]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-0.574442080456058, -1.1079791795625606]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
12.7.E14 2 - 3 4 - 1 2 ⁒ a ⁒ z ⁒ e - 1 4 ⁒ z 2 ⁒ U ⁑ ( 1 2 ⁒ a + 3 4 , 3 2 , 1 2 ⁒ z 2 ) = 2 - 1 2 ⁒ a ⁒ z - 1 2 ⁒ W - 1 2 ⁒ a , - 1 4 ⁑ ( 1 2 ⁒ z 2 ) superscript 2 3 4 1 2 π‘Ž 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 2 π‘Ž 3 4 3 2 1 2 superscript 𝑧 2 superscript 2 1 2 π‘Ž superscript 𝑧 1 2 Whittaker-confluent-hypergeometric-W 1 2 π‘Ž 1 4 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle 2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2% }}U\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{2},\tfrac{1}{2}z^{2}\right)=2^{-% \frac{1}{2}a}z^{-\frac{1}{2}}W_{-\frac{1}{2}a,-\frac{1}{4}}\left(\tfrac{1}{2}z% ^{2}\right)}}
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}}		 

(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))

(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)]
Failure Failure
Failed [12 / 42]
Result: .725579081e-1+1.600870446*I
Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}


Result: -.5744420804-1.107979180*I
Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [12 / 42]
Result: Complex[0.0725579074030912, 1.600870445554158]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-0.5744420804560579, -1.1079791795625609]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
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