Parabolic Cylinder Functions - 12.5 Integral Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
12.5.E1 U ⁡ ( a , z ) = e - 1 4 ⁢ z 2 Γ ⁡ ( 1 2 + a ) ⁢ ∫ 0 ∞ t a - 1 2 ⁢ e - 1 2 ⁢ t 2 - z ⁢ t ⁢ d t parabolic-U 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Euler-Gamma 1 2 𝑎 superscript subscript 0 superscript 𝑡 𝑎 1 2 superscript 𝑒 1 2 superscript 𝑡 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{% \Gamma\left(\frac{1}{2}+a\right)}\int_{0}^{\infty}t^{a-\frac{1}{2}}e^{-\frac{1% }{2}t^{2}-zt}\mathrm{d}t}}
\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}		 ℜ ⁡ a > - 1 2 , ℜ ⁡ ( 1 2 + a ) > 0 formulae-sequence 𝑎 1 2 1 2 𝑎 0 {\displaystyle{\displaystyle\Re a>-\tfrac{1}{2},\Re(\frac{1}{2}+a)>0}} 
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)

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]
Successful Successful - Successful [Tested: 21]
12.5.E2 U ⁡ ( a , z ) = z ⁢ e - 1 4 ⁢ z 2 Γ ⁡ ( 1 4 + 1 2 ⁢ a ) ⁢ ∫ 0 ∞ t 1 2 ⁢ a - 3 4 ⁢ e - t ⁢ ( z 2 + 2 ⁢ t ) - 1 2 ⁢ a - 3 4 ⁢ d t parabolic-U 𝑎 𝑧 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Euler-Gamma 1 4 1 2 𝑎 superscript subscript 0 superscript 𝑡 1 2 𝑎 3 4 superscript 𝑒 𝑡 superscript superscript 𝑧 2 2 𝑡 1 2 𝑎 3 4 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{ze^{-\frac{1}{4}z^{2}}}{% \Gamma\left(\frac{1}{4}+\frac{1}{2}a\right)}\*\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}}\mathrm{d}% t}}
\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}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ a > - 1 2 , ℜ ⁡ ( 1 4 + 1 2 ⁢ a ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 𝑎 1 2 1 4 1 2 𝑎 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re a>-\tfrac% {1}{2},\Re(\frac{1}{4}+\frac{1}{2}a)>0}} 
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)

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]
Failure Successful
Failed [2 / 15]
Result: Float(infinity)+Float(infinity)*I
Test Values: {a = 2, z = 1/2*3^(1/2)+1/2*I}


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

Successful [Tested: 15]
12.5.E3 U ⁡ ( a , z ) = e - 1 4 ⁢ z 2 Γ ⁡ ( 3 4 + 1 2 ⁢ a ) ⁢ ∫ 0 ∞ t 1 2 ⁢ a - 1 4 ⁢ e - t ⁢ ( z 2 + 2 ⁢ t ) - 1 2 ⁢ a - 1 4 ⁢ d t parabolic-U 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Euler-Gamma 3 4 1 2 𝑎 superscript subscript 0 superscript 𝑡 1 2 𝑎 1 4 superscript 𝑒 𝑡 superscript superscript 𝑧 2 2 𝑡 1 2 𝑎 1 4 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{% \Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)}\*\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}}\mathrm{d}% t}}
\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}		 | ph ⁡ z | < 1 2 ⁢ π , ℜ ⁡ a > - 3 2 , ℜ ⁡ ( 3 4 + 1 2 ⁢ a ) > 0 formulae-sequence phase 𝑧 1 2 𝜋 formulae-sequence 𝑎 3 2 3 4 1 2 𝑎 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re a>-\tfrac% {3}{2},\Re(\frac{3}{4}+\frac{1}{2}a)>0}} 
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)

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]
Failure Successful
Failed [2 / 20]
Result: Float(infinity)+Float(infinity)*I
Test Values: {a = 3/2, z = 1/2*3^(1/2)+1/2*I}


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

Failed [5 / 20]
Result: Indeterminate
Test Values: {Rule[a, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
12.5.E4 U ⁡ ( a , z ) = 2 π ⁢ e 1 4 ⁢ z 2 ⁢ ∫ 0 ∞ t - a - 1 2 ⁢ e - 1 2 ⁢ t 2 ⁢ cos ⁡ ( z ⁢ t + ( 1 2 ⁢ a + 1 4 ) ⁢ π ) ⁢ d t parabolic-U 𝑎 𝑧 2 𝜋 superscript 𝑒 1 4 superscript 𝑧 2 superscript subscript 0 superscript 𝑡 𝑎 1 2 superscript 𝑒 1 2 superscript 𝑡 2 𝑧 𝑡 1 2 𝑎 1 4 𝜋 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\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\left(% zt+\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)\pi\right)\mathrm{d}t}}
\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}		 

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)

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]
Successful Failure - Successful [Tested: 7]
12.5.E5 U ⁡ ( a , z ) = Γ ⁡ ( 1 2 - a ) 2 ⁢ π ⁢ i ⁢ e - 1 4 ⁢ z 2 ⁢ ∫ - ∞ ( 0 + ) e z ⁢ t - 1 2 ⁢ t 2 ⁢ t a - 1 2 ⁢ d t parabolic-U 𝑎 𝑧 Euler-Gamma 1 2 𝑎 2 𝜋 𝑖 superscript 𝑒 1 4 superscript 𝑧 2 superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑡 1 2 superscript 𝑡 2 superscript 𝑡 𝑎 1 2 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{\Gamma\left(\frac{1}{2}-a% \right)}{2\pi i}e^{-\frac{1}{4}z^{2}}\int_{-\infty}^{(0+)}e^{zt-\frac{1}{2}t^{% 2}}t^{a-\frac{1}{2}}\mathrm{d}t}}
\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}		 a ≠ 1 2 , - π < ph ⁡ t , ph ⁡ t < π , ℜ ⁡ ( 1 2 - a ) > 0 formulae-sequence 𝑎 1 2 formulae-sequence 𝜋 phase 𝑡 formulae-sequence phase 𝑡 𝜋 1 2 𝑎 0 {\displaystyle{\displaystyle a\neq\frac{1}{2},-\pi<\operatorname{ph}t,% \operatorname{ph}t<\pi,\Re(\frac{1}{2}-a)>0}} 
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 +))

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]
Error Failure - Error
12.5.E6 U ⁡ ( a , z ) = e 1 4 ⁢ z 2 i ⁢ 2 ⁢ π ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ e - z ⁢ t + 1 2 ⁢ t 2 ⁢ t - a - 1 2 ⁢ d t parabolic-U 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 𝑖 2 𝜋 superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript 𝑒 𝑧 𝑡 1 2 superscript 𝑡 2 superscript 𝑡 𝑎 1 2 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\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}}\mathrm{d}t}}
\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}		 - 1 2 ⁢ π < ph ⁡ t , ph ⁡ t < 1 2 ⁢ π , c > 0 formulae-sequence 1 2 𝜋 phase 𝑡 formulae-sequence phase 𝑡 1 2 𝜋 𝑐 0 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi<\operatorname{ph}t,\operatorname{% ph}t<\tfrac{1}{2}\pi,c>0}} 
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)

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]
Failure Aborted
Failed [126 / 126]
Result: .8412106295+.2667685493*I
Test Values: {a = -3/2, c = 3/2, z = 1/2*3^(1/2)+1/2*I}


Result: -.7641562685+.8367141760*I
Test Values: {a = -3/2, c = 3/2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
12.5.E8 U ⁡ ( a , z ) = e - 1 4 ⁢ z 2 ⁢ z - a - 1 2 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 2 + a ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( 1 2 + a - 2 ⁢ t ) ⁢ 2 t ⁢ z 2 ⁢ t ⁢ d t parabolic-U 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 𝑎 1 2 2 𝜋 𝑖 Euler-Gamma 1 2 𝑎 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 1 2 𝑎 2 𝑡 superscript 2 𝑡 superscript 𝑧 2 𝑡 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}z^{-% a-\frac{1}{2}}}{2\pi i\Gamma\left(\frac{1}{2}+a\right)}\*\int_{-i\infty}^{i% \infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}+a-2t\right)2^{t}z^{2t}% \mathrm{d}t}}
\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}		 a ≠ - 1 2 , | ph ⁡ z | < 3 4 ⁢ π , ℜ ⁡ t > 0 , ℜ ⁡ ( 1 2 + a - 2 ⁢ t ) > 0 , ℜ ⁡ ( 1 2 + a ) > 0 formulae-sequence 𝑎 1 2 formulae-sequence phase 𝑧 3 4 𝜋 formulae-sequence 𝑡 0 formulae-sequence 1 2 𝑎 2 𝑡 0 1 2 𝑎 0 {\displaystyle{\displaystyle a\neq-\frac{1}{2},|\operatorname{ph}z|<\tfrac{3}{% 4}\pi,\Re t>0,\Re(\tfrac{1}{2}+a-2t)>0,\Re(\frac{1}{2}+a)>0}} 
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)

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]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
12.5.E9 V ⁡ ( a , z ) = 2 π ⁢ e 1 4 ⁢ z 2 ⁢ z a - 1 2 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 2 - a ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( 1 2 - a - 2 ⁢ t ) ⁢ 2 t ⁢ z 2 ⁢ t ⁢ cos ⁡ ( π ⁢ t ) ⁢ d t parabolic-V 𝑎 𝑧 2 𝜋 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 𝑎 1 2 2 𝜋 𝑖 Euler-Gamma 1 2 𝑎 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 1 2 𝑎 2 𝑡 superscript 2 𝑡 superscript 𝑧 2 𝑡 𝜋 𝑡 𝑡 {\displaystyle{\displaystyle V\left(a,z\right)=\sqrt{\frac{2}{\pi}}\frac{e^{% \frac{1}{4}z^{2}}z^{a-\frac{1}{2}}}{2\pi i\Gamma\left(\frac{1}{2}-a\right)}\*% \int_{-i\infty}^{i\infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}-a-2t% \right)2^{t}z^{2t}\cos\left(\pi t\right)\mathrm{d}t}}
\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}		 a ≠ 1 2 , | ph ⁡ z | < 1 4 ⁢ π , ℜ ⁡ t > 0 , ℜ ⁡ ( 1 2 - a - 2 ⁢ t ) > 0 , ℜ ⁡ ( 1 2 - a ) > 0 formulae-sequence 𝑎 1 2 formulae-sequence phase 𝑧 1 4 𝜋 formulae-sequence 𝑡 0 formulae-sequence 1 2 𝑎 2 𝑡 0 1 2 𝑎 0 {\displaystyle{\displaystyle a\neq\frac{1}{2},|\operatorname{ph}z|<\tfrac{1}{4% }\pi,\Re t>0,\Re(\tfrac{1}{2}-a-2t)>0,\Re(\frac{1}{2}-a)>0}} 
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)

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]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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