Results of Parabolic Cylinder Functions

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DLMF Formula 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}} diff(w, [z$(2)])-((1)/(4)*(z)^(2)+ a)* w = 0 D[w, {z, 2}]-(Divide[1,4]*(z)^(2)+ a)* w = 0 Failure Failure
Fail
1.414213562-5.414213560*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-1.414213561-2.585786437*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.414213562-5.414213560*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-1.414213561-2.585786437*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.4142135623730945, -5.414213562373096] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730954, -2.5857864376269055] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730945, -5.414213562373096] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730954, -2.5857864376269055] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
12.2.E3 d 2 w d z 2 + ( 1 4 ⁒ z 2 - a ) ⁒ w = 0 derivative 𝑀 𝑧 2 1 4 superscript 𝑧 2 π‘Ž 𝑀 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% \tfrac{1}{4}z^{2}-a\right)w=0}} diff(w, [z$(2)])+((1)/(4)*(z)^(2)- a)* w = 0 D[w, {z, 2}]+(Divide[1,4]*(z)^(2)- a)* w = 0 Failure Failure
Fail
-1.414213561-2.585786437*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.414213562-5.414213560*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.414213561-2.585786437*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
1.414213562-5.414213560*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-1.4142135623730954, -2.5857864376269055] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730945, -5.414213562373096] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730954, -2.5857864376269055] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730945, -5.414213562373096] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... 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}} 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
Fail
2.121320342+3.292893218*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-.7071067810+6.121320341*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
2.121320342+3.292893218*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-.7071067810+6.121320341*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[2.121320343559643, 3.292893218813453] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[6.121320343559643, -0.707106781186547] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.121320343559642, -4.707106781186548] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.8786796564403578, -0.7071067811865477] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... 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)}} CylinderD(nu, z)= CylinderU(-(1)/(2)- nu, z) ParabolicCylinderD[\[Nu], z]= ParabolicCylinderD[--Divide[1,2]- \[Nu] - 1/2, z] Successful Failure -
Fail
Complex[0.6373277951223405, 0.24855575768220314] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[5.494742013097394, -8.693674717807955] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.2264312008959464, -0.021546439055701437] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.3291655729814447, 3.7734764780430785] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ½, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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)}}} CylinderU(a, 0)=(sqrt(Pi))/((2)^((1)/(2)*a +(1)/(4))* GAMMA((3)/(4)+(1)/(2)*a)) ParabolicCylinderD[-a - 1/2, 0]=Divide[Sqrt[Pi],(2)^(Divide[1,2]*a +Divide[1,4])* Gamma[Divide[3,4]+Divide[1,2]*a]] Successful Successful - -
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)}}} subs( temp=0, diff( CylinderU(a, temp), temp$(1) ) )= -(sqrt(Pi))/((2)^((1)/(2)*a -(1)/(4))* GAMMA((1)/(4)+(1)/(2)*a)) (D[ParabolicCylinderD[-a - 1/2, temp], {temp, 1}]/.temp-> 0)= -Divide[Sqrt[Pi],(2)^(Divide[1,2]*a -Divide[1,4])* Gamma[Divide[1,4]+Divide[1,2]*a]] Successful Successful - -
12.2.E8 V ⁑ ( a , 0 ) = Ο€ ⁒ 2 1 2 ⁒ a + 1 4 ( Ξ“ ⁑ ( 3 4 - 1 2 ⁒ a ) ) 2 ⁒ Ξ“ ⁑ ( 1 4 + 1 2 ⁒ a ) parabolic-V π‘Ž 0 πœ‹ superscript 2 1 2 π‘Ž 1 4 superscript Euler-Gamma 3 4 1 2 π‘Ž 2 Euler-Gamma 1 4 1 2 π‘Ž {\displaystyle{\displaystyle V\left(a,0\right)=\frac{\pi 2^{\frac{1}{2}a+\frac% {1}{4}}}{\left(\Gamma\left(\frac{3}{4}-\frac{1}{2}a\right)\right)^{2}\Gamma% \left(\frac{1}{4}+\frac{1}{2}a\right)}}} CylinderV(a, 0)=(Pi*(2)^((1)/(2)*a +(1)/(4)))/((GAMMA((3)/(4)-(1)/(2)*a))^(2)* GAMMA((1)/(4)+(1)/(2)*a)) Error Successful Error - -
12.2.E9 V β€² ⁑ ( a , 0 ) = Ο€ ⁒ 2 1 2 ⁒ a + 3 4 ( Ξ“ ⁑ ( 1 4 - 1 2 ⁒ a ) ) 2 ⁒ Ξ“ ⁑ ( 3 4 + 1 2 ⁒ a ) diffop parabolic-V 1 π‘Ž 0 πœ‹ superscript 2 1 2 π‘Ž 3 4 superscript Euler-Gamma 1 4 1 2 π‘Ž 2 Euler-Gamma 3 4 1 2 π‘Ž {\displaystyle{\displaystyle V'\left(a,0\right)=\frac{\pi 2^{\frac{1}{2}a+% \frac{3}{4}}}{\left(\Gamma\left(\frac{1}{4}-\frac{1}{2}a\right)\right)^{2}% \Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)}}} subs( temp=0, diff( CylinderV(a, temp), temp$(1) ) )=(Pi*(2)^((1)/(2)*a +(3)/(4)))/((GAMMA((1)/(4)-(1)/(2)*a))^(2)* GAMMA((3)/(4)+(1)/(2)*a)) Error Successful Error - -
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}}} (CylinderU(a, z))*diff(CylinderV(a, z), z)-diff(CylinderU(a, z), z)*(CylinderV(a, z))=sqrt(2/ Pi) Error Failure Error Successful -
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)}}} (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[-a - 1/2, z], ParabolicCylinderD[-a - 1/2, - z]}, z]=Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]] Failure Failure Successful Successful
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})}}} (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[-a - 1/2, z], ParabolicCylinderD[-- a - 1/2, + I*z]}, z]= - I*Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])] Failure Failure Successful
Fail
Complex[0.14811550020669734, -0.23555829917293641] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.05275467566832733, -0.08637346425615816] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.393334833694336, -1.5801979189958324] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-124.21447949802922, -41.9141177725009] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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})}}} (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[-a - 1/2, z], ParabolicCylinderD[-- a - 1/2, - I*z]}, z]= + I*Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])] Failure Failure Successful
Fail
Complex[1.7828893409022697, 7.309124598403374] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.6470896010983296, 11.133201917893151] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[12.667352367459173, 6.462811262967247] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.2682027250441195, 0.1306629371451986] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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)}} CylinderU(- n -(1)/(2), - z)=(- 1)^(n)* CylinderU(- n -(1)/(2), z) ParabolicCylinderD[-- n -Divide[1,2] - 1/2, - z]=(- 1)^(n)* ParabolicCylinderD[-- n -Divide[1,2] - 1/2, z] Failure Failure Successful Successful
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)}} CylinderV(n +(1)/(2), - z)=(- 1)^(n)* CylinderV(n +(1)/(2), z) Error Failure Error Successful -
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)}} CylinderU(a, - z)= - sin(Pi*a)*CylinderU(a, z)+(Pi)/(GAMMA((1)/(2)+ a))*CylinderV(a, z) Error Successful Error - -
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)}} CylinderV(a, - z)=(cos(Pi*a))/(GAMMA((1)/(2)- a))*CylinderU(a, z)+ sin(Pi*a)*CylinderV(a, z) Error Failure Error Successful -
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)*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[-- a - 1/2, + I*z]= Gamma[Divide[1,2]+ a]*(Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[-a - 1/2, z]+ Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[-a - 1/2, - z]) Failure Failure Successful
Fail
Complex[-2.155433665722218, -2.0290531891335233] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.3367861286344187, -0.6580032872978093] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.914157355854969, -11.153359556689384] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[92.56613149692683, 18.07525013091991] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... 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)*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[-- a - 1/2, - I*z]= Gamma[Divide[1,2]+ a]*(Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[-a - 1/2, z]+ Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[-a - 1/2, - z]) Failure Failure Successful
Fail
Complex[-5.914157355854969, -11.153359556689384] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[92.56613149692683, 18.07525013091991] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.155433665722218, -2.0290531891335233] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.3367861286344187, -0.6580032872978093] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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)*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[-a - 1/2, z]= Gamma[Divide[1,2]- a]*(Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[-- a - 1/2, + I*z]+ Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[-- a - 1/2, - I*z]) Failure Failure Successful
Fail
Complex[0.10912779491905389, 1.6001420083926765] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.5516631154927725, 0.012853611485611899] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.6505649452472415, 6.316478461243763] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-28.79709943083936, -39.22297340462678] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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)*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[-a - 1/2, z]= Gamma[Divide[1,2]- a]*(Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[-- a - 1/2, - I*z]+ Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[-- a - 1/2, + I*z]) Failure Failure Successful
Fail
Complex[0.10912779491905389, 1.6001420083926765] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.5516631154927725, 0.012853611485611899] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.6505649452472415, 6.316478461243763] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-28.79709943083936, -39.22297340462678] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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)}} 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[-a - 1/2, z]= + I*Exp[+ I*Pi*a]*ParabolicCylinderD[-a - 1/2, - z]+Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]*Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[-- a - 1/2, + I*z] Failure Failure Successful
Fail
Complex[0.04310956766516494, 0.6077329076178599] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.040829587948855356, 0.14655410699953625] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.6609728840735412, 2.5128092684920165] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-11.485901419364549, -15.648449818478493] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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)}} 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[-a - 1/2, z]= - I*Exp[- I*Pi*a]*ParabolicCylinderD[-a - 1/2, - z]+Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]*Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[-- a - 1/2, - I*z] Failure Failure Successful
Fail
Complex[111.04952307011636, -190.96406860603162] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[587.0445111625338, 1542.4071633053381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[10.224359018803703, -50.78000597255555] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[6.66355099948441, -11.085954332579782] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... 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)}} 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) Error Failure Error Successful -
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)}} 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) Error Failure Error Successful -
12.4.E1 U ⁑ ( a , z ) = U ⁑ ( a , 0 ) ⁒ u 1 ⁒ ( a , z ) + U β€² ⁑ ( a , 0 ) ⁒ u 2 ⁒ ( a , z ) parabolic-U π‘Ž 𝑧 parabolic-U π‘Ž 0 subscript 𝑒 1 π‘Ž 𝑧 diffop parabolic-U 1 π‘Ž 0 subscript 𝑒 2 π‘Ž 𝑧 {\displaystyle{\displaystyle U\left(a,z\right)=U\left(a,0\right)u_{1}(a,z)+U'% \left(a,0\right)u_{2}(a,z)}} CylinderU(a, z)= CylinderU(a, 0)*u[1]*(a , z)+ subs( temp=0, diff( CylinderU(a, temp), temp$(1) ) )*u[2]*(a , z) ParabolicCylinderD[-a - 1/2, z]= ParabolicCylinderD[-a - 1/2, 0]*Subscript[u, 1]*(a , z)+ (D[ParabolicCylinderD[-a - 1/2, temp], {temp, 1}]/.temp-> 0)*Subscript[u, 2]*(a , z) Failure Failure Error Error
12.4.E2 V ⁑ ( a , z ) = V ⁑ ( a , 0 ) ⁒ u 1 ⁒ ( a , z ) + V β€² ⁑ ( a , 0 ) ⁒ u 2 ⁒ ( a , z ) parabolic-V π‘Ž 𝑧 parabolic-V π‘Ž 0 subscript 𝑒 1 π‘Ž 𝑧 diffop parabolic-V 1 π‘Ž 0 subscript 𝑒 2 π‘Ž 𝑧 {\displaystyle{\displaystyle V\left(a,z\right)=V\left(a,0\right)u_{1}(a,z)+V'% \left(a,0\right)u_{2}(a,z)}} CylinderV(a, z)= CylinderV(a, 0)*u[1]*(a , z)+ subs( temp=0, diff( CylinderV(a, temp), temp$(1) ) )*u[2]*(a , z) Error Failure Error Error -
12.5.E1 U ⁑ ( a , z ) = e - 1 4 ⁒ z 2 Ξ“ ⁑ ( 1 2 + a ) ⁒ ∫ 0 ∞ t a - 1 2 ⁒ e - 1 2 ⁒ t 2 - z ⁒ t ⁒ d t parabolic-U π‘Ž 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Euler-Gamma 1 2 π‘Ž superscript subscript 0 superscript 𝑑 π‘Ž 1 2 superscript 𝑒 1 2 superscript 𝑑 2 𝑧 𝑑 𝑑 {\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}} 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[-a - 1/2, 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}] Successful Successful - -
12.5.E2 U ⁑ ( a , z ) = z ⁒ e - 1 4 ⁒ z 2 Ξ“ ⁑ ( 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 π‘Ž 𝑧 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Euler-Gamma 1 4 1 2 π‘Ž superscript subscript 0 superscript 𝑑 1 2 π‘Ž 3 4 superscript 𝑒 𝑑 superscript superscript 𝑧 2 2 𝑑 1 2 π‘Ž 3 4 𝑑 {\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}} 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[-a - 1/2, 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}] Failure Failure Skip Skip
12.5.E3 U ⁑ ( a , z ) = e - 1 4 ⁒ z 2 Ξ“ ⁑ ( 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 π‘Ž 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Euler-Gamma 3 4 1 2 π‘Ž superscript subscript 0 superscript 𝑑 1 2 π‘Ž 1 4 superscript 𝑒 𝑑 superscript superscript 𝑧 2 2 𝑑 1 2 π‘Ž 1 4 𝑑 {\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}} 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[-a - 1/2, 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}] Failure Failure Skip Skip
12.5.E4 U ⁑ ( a , z ) = 2 Ο€ ⁒ e 1 4 ⁒ z 2 ⁒ ∫ 0 ∞ t - a - 1 2 ⁒ e - 1 2 ⁒ t 2 ⁒ cos ⁑ ( z ⁒ t + ( 1 2 ⁒ a + 1 4 ) ⁒ Ο€ ) ⁒ d t parabolic-U π‘Ž 𝑧 2 πœ‹ superscript 𝑒 1 4 superscript 𝑧 2 superscript subscript 0 superscript 𝑑 π‘Ž 1 2 superscript 𝑒 1 2 superscript 𝑑 2 𝑧 𝑑 1 2 π‘Ž 1 4 πœ‹ 𝑑 {\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}} 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[-a - 1/2, 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}] Successful Failure - Successful
12.5.E5 U ⁑ ( a , z ) = Ξ“ ⁑ ( 1 2 - a ) 2 ⁒ Ο€ ⁒ i ⁒ e - 1 4 ⁒ z 2 ⁒ ∫ - ∞ ( 0 + ) e z ⁒ t - 1 2 ⁒ t 2 ⁒ t a - 1 2 ⁒ d t parabolic-U π‘Ž 𝑧 Euler-Gamma 1 2 π‘Ž 2 πœ‹ 𝑖 superscript 𝑒 1 4 superscript 𝑧 2 superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑑 1 2 superscript 𝑑 2 superscript 𝑑 π‘Ž 1 2 𝑑 {\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}} 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[-a - 1/2, 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 +)}] Error Failure - Error
12.5.E6 U ⁑ ( a , z ) = e 1 4 ⁒ z 2 i ⁒ 2 ⁒ Ο€ ⁒ ∫ c - i ⁒ ∞ c + i ⁒ ∞ e - z ⁒ t + 1 2 ⁒ t 2 ⁒ t - a - 1 2 ⁒ d t parabolic-U π‘Ž 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 𝑖 2 πœ‹ superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript 𝑒 𝑧 𝑑 1 2 superscript 𝑑 2 superscript 𝑑 π‘Ž 1 2 𝑑 {\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}} 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[-a - 1/2, 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}] Failure Failure Skip Error
12.5.E8 U ⁑ ( a , z ) = e - 1 4 ⁒ z 2 ⁒ z - a - 1 2 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 2 + a ) ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ξ“ ⁑ ( t ) ⁒ Ξ“ ⁑ ( 1 2 + a - 2 ⁒ t ) ⁒ 2 t ⁒ z 2 ⁒ t ⁒ d t parabolic-U π‘Ž 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 π‘Ž 1 2 2 πœ‹ 𝑖 Euler-Gamma 1 2 π‘Ž superscript subscript 𝑖 𝑖 Euler-Gamma 𝑑 Euler-Gamma 1 2 π‘Ž 2 𝑑 superscript 2 𝑑 superscript 𝑧 2 𝑑 𝑑 {\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}} 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[-a - 1/2, 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}] Failure Failure Skip Error
12.5.E9 V ⁑ ( a , z ) = 2 Ο€ ⁒ e 1 4 ⁒ z 2 ⁒ z a - 1 2 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 2 - a ) ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ξ“ ⁑ ( t ) ⁒ Ξ“ ⁑ ( 1 2 - a - 2 ⁒ t ) ⁒ 2 t ⁒ z 2 ⁒ t ⁒ cos ⁑ ( Ο€ ⁒ t ) ⁒ d t parabolic-V π‘Ž 𝑧 2 πœ‹ superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 π‘Ž 1 2 2 πœ‹ 𝑖 Euler-Gamma 1 2 π‘Ž superscript subscript 𝑖 𝑖 Euler-Gamma 𝑑 Euler-Gamma 1 2 π‘Ž 2 𝑑 superscript 2 𝑑 superscript 𝑧 2 𝑑 πœ‹ 𝑑 𝑑 {\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}} 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) Error Failure Error Skip -
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)}} CylinderU(-(1)/(2), z)= CylinderD(0, z) ParabolicCylinderD[--Divide[1,2] - 1/2, z]= ParabolicCylinderD[0, z] Successful Failure -
Fail
Complex[-0.5535325382896138, 0.3790800612672758] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.5535325382896137, -0.3790800612672758] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.8272649632907405, 3.413116871149585] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.8272649632907406, -3.413116871149585] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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}}}} CylinderD(0, z)= exp(-(1)/(4)*(z)^(2)) ParabolicCylinderD[0, z]= Exp[-Divide[1,4]*(z)^(2)] Successful Successful - -
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)}} CylinderU(- n -(1)/(2), z)= CylinderD(n, z) ParabolicCylinderD[-- n -Divide[1,2] - 1/2, z]= ParabolicCylinderD[n, z] Successful Failure - Successful
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)}} CylinderV(-(1)/(2), z)=((2)/(sqrt(Pi)))* exp((1)/(4)*(z)^(2))*dawson(z/sqrt(2)) Error Successful Error - -
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)}} CylinderU((1)/(2), z)= CylinderD(- 1, z) ParabolicCylinderD[-Divide[1,2] - 1/2, z]= ParabolicCylinderD[- 1, z] Successful Successful - -
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)}} 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 - -
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)}} CylinderU(n +(1)/(2), z)= CylinderD(- n - 1, z) ParabolicCylinderD[-n +Divide[1,2] - 1/2, z]= ParabolicCylinderD[- n - 1, z] Successful Failure -
Fail
Complex[-0.033951461448720965, -0.03725090437629219] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.033951461448720965, 0.03725090437629219] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.34180963565085953, 2.078059729615049] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.34180963565085953, -2.078059729615049] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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}}}} 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 Successful Successful
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(2\!K_{\frac{1}{4}}\left(\tfrac{1}{4}z^{2}\right)+3\!K_{\frac{3}{4}}\left% (\tfrac{1}{4}z^{2}\right)-K_{\frac{5}{4}}\left(\tfrac{1}{4}z^{2}\right)\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[-- 2 - 1/2, 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
Fail
-1.522148495+5.325348960*I <- {z = -2^(1/2)-I*2^(1/2)}
-1.522148495-5.325348960*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-2.176847672351804, -0.5824405608621696] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.9645562602658246, -7.19064614909563] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[2], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.138711633327063, -0.6435109071691986] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.642441416717864, -0.8480286890415233] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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)}} 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 - 1/2, 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
Fail
-1.876943385-2.185260402*I <- {z = -2^(1/2)-I*2^(1/2)}
-1.876943385+2.185260402*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.5944827156757435, 1.0193657184065432] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.38219130358976394, -5.588839869826917] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.5563466766510021, 0.9582953720995141] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.0600764600418031, 0.7537775902271894] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
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)}} CylinderU(0, z)=sqrt((z)/(2*Pi))*BesselK((1)/(4), (1)/(4)*(z)^(2)) ParabolicCylinderD[-0 - 1/2, z]=Sqrt[Divide[z,2*Pi]]*BesselK[Divide[1,4], Divide[1,4]*(z)^(2)] Failure Failure
Fail
2.172244779+.8389494551*I <- {z = -2^(1/2)-I*2^(1/2)}
2.172244779-.8389494551*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[2.172244778329958, 0.8389494547782839] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.172244778329958, -0.8389494547782839] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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)}} 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 - 1/2, 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
Fail
.172418861e-1+4.146422642*I <- {z = -2^(1/2)-I*2^(1/2)}
.172418861e-1-4.146422642*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.017241884225636828, 4.146422642593135] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.017241884225636828, -4.146422642593135] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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)}} 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[-a - 1/2, 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
Fail
-1.552891817+2.595743470*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-1.468308225-15.26302300*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
-1.468308225+15.26302300*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-1.552891817-2.595743470*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-1.5528918212023513, 2.595743468355251] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4683082379662649, -15.263023009398044] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.4683082379662649, 15.263023009398044] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.5528918212023513, -2.595743468355251] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... 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)^(-(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
Fail
-.8240048131+.2272943481*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
.1685356209+.1817412146*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
.1685356209-.1817412146*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-.8240048131-.2272943481*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.8240048125018726, 0.22729434917761066] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.1685356207165261, 0.1817412145670384] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.1685356207165261, -0.1817412145670384] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.8240048125018726, -0.22729434917761066] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... 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)^(-(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 Skip
Fail
Complex[0.5256495808397417, 0.2983552316621309] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.17513841764178226, -0.006602796925256171] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.17513841764178226, 0.006602796925256171] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.5256495808397417, -0.2983552316621309] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... 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)^(-(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 Skip
Fail
Complex[0.5256495808397417, 0.2983552316621309] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.17513841764178226, -0.0066027969252561575] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.17513841764178226, 0.0066027969252561575] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.5256495808397417, -0.2983552316621309] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
12.8.E1 z ⁒ U ⁑ ( a , z ) - U ⁑ ( a - 1 , z ) + ( a + 1 2 ) ⁒ U ⁑ ( a + 1 , z ) = 0 𝑧 parabolic-U π‘Ž 𝑧 parabolic-U π‘Ž 1 𝑧 π‘Ž 1 2 parabolic-U π‘Ž 1 𝑧 0 {\displaystyle{\displaystyle zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{% 1}{2})U\left(a+1,z\right)=0}} z*CylinderU(a, z)- CylinderU(a - 1, z)+(a +(1)/(2))* CylinderU(a + 1, z)= 0 z*ParabolicCylinderD[-a - 1/2, z]- ParabolicCylinderD[-a - 1 - 1/2, z]+(a +Divide[1,2])* ParabolicCylinderD[-a + 1 - 1/2, z]= 0 Successful Failure -
Fail
Complex[-2.212392415520596, -2.7835289048360865] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.47884522381711175, 0.5350814275558994] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.550805211617881, -3.493113886500579] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[73.63864596732188, 31.232726056290662] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
12.8.E2 U β€² ⁑ ( a , z ) + 1 2 ⁒ z ⁒ U ⁑ ( a , z ) + ( a + 1 2 ) ⁒ U ⁑ ( a + 1 , z ) = 0 diffop parabolic-U 1 π‘Ž 𝑧 1 2 𝑧 parabolic-U π‘Ž 𝑧 π‘Ž 1 2 parabolic-U π‘Ž 1 𝑧 0 {\displaystyle{\displaystyle U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)% +(a+\tfrac{1}{2})U\left(a+1,z\right)=0}} subs( temp=z, diff( CylinderU(a, temp), temp$(1) ) )+(1)/(2)*z*CylinderU(a, z)+(a +(1)/(2))* CylinderU(a + 1, z)= 0 (D[ParabolicCylinderD[-a - 1/2, temp], {temp, 1}]/.temp-> z)+Divide[1,2]*z*ParabolicCylinderD[-a - 1/2, z]+(a +Divide[1,2])* ParabolicCylinderD[-a + 1 - 1/2, z]= 0 Successful Failure -
Fail
Complex[-1.2227570491036914, -2.308624400336341] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2737322573556834, 0.45100801175309346] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.0354158227028893, -2.0443578293881037] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[48.97674308818175, 32.48331095383318] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
12.8.E3 U β€² ⁑ ( a , z ) - 1 2 ⁒ z ⁒ U ⁑ ( a , z ) + U ⁑ ( a - 1 , z ) = 0 diffop parabolic-U 1 π‘Ž 𝑧 1 2 𝑧 parabolic-U π‘Ž 𝑧 parabolic-U π‘Ž 1 𝑧 0 {\displaystyle{\displaystyle U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)% +U\left(a-1,z\right)=0}} subs( temp=z, diff( CylinderU(a, temp), temp$(1) ) )-(1)/(2)*z*CylinderU(a, z)+ CylinderU(a - 1, z)= 0 (D[ParabolicCylinderD[-a - 1/2, temp], {temp, 1}]/.temp-> z)-Divide[1,2]*z*ParabolicCylinderD[-a - 1/2, z]+ ParabolicCylinderD[-a - 1 - 1/2, z]= 0 Successful Failure -
Fail
Complex[0.9896353664169044, 0.4749045044997455] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.20511296646142838, -0.0840734158028059] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.5153893889149921, 1.4487560571124753] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-24.66190287914013, 1.250584897542514] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
12.8.E4 2 ⁒ U β€² ⁑ ( a , z ) + U ⁑ ( a - 1 , z ) + ( a + 1 2 ) ⁒ U ⁑ ( a + 1 , z ) = 0 2 diffop parabolic-U 1 π‘Ž 𝑧 parabolic-U π‘Ž 1 𝑧 π‘Ž 1 2 parabolic-U π‘Ž 1 𝑧 0 {\displaystyle{\displaystyle 2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac% {1}{2})U\left(a+1,z\right)=0}} 2*subs( temp=z, diff( CylinderU(a, temp), temp$(1) ) )+ CylinderU(a - 1, z)+(a +(1)/(2))* CylinderU(a + 1, z)= 0 2*(D[ParabolicCylinderD[-a - 1/2, temp], {temp, 1}]/.temp-> z)+ ParabolicCylinderD[-a - 1 - 1/2, z]+(a +Divide[1,2])* ParabolicCylinderD[-a + 1 - 1/2, z]= 0 Successful Failure -
Fail
Complex[-0.233121682686787, -1.8337198958365952] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.06861929089425498, 0.36693459595028755] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.520026433787896, -0.5956017722756277] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[24.314840209041613, 33.733895851375685] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
12.8.E5 z ⁒ V ⁑ ( a , z ) - V ⁑ ( a + 1 , z ) + ( a - 1 2 ) ⁒ V ⁑ ( a - 1 , z ) = 0 𝑧 parabolic-V π‘Ž 𝑧 parabolic-V π‘Ž 1 𝑧 π‘Ž 1 2 parabolic-V π‘Ž 1 𝑧 0 {\displaystyle{\displaystyle zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{% 1}{2})V\left(a-1,z\right)=0}} z*CylinderV(a, z)- CylinderV(a + 1, z)+(a -(1)/(2))* CylinderV(a - 1, z)= 0 Error Successful Error - -
12.8.E6 V β€² ⁑ ( a , z ) - 1 2 ⁒ z ⁒ V ⁑ ( a , z ) - ( a - 1 2 ) ⁒ V ⁑ ( a - 1 , z ) = 0 diffop parabolic-V 1 π‘Ž 𝑧 1 2 𝑧 parabolic-V π‘Ž 𝑧 π‘Ž 1 2 parabolic-V π‘Ž 1 𝑧 0 {\displaystyle{\displaystyle V'\left(a,z\right)-\tfrac{1}{2}zV\left(a,z\right)% -(a-\tfrac{1}{2})V\left(a-1,z\right)=0}} subs( temp=z, diff( CylinderV(a, temp), temp$(1) ) )-(1)/(2)*z*CylinderV(a, z)-(a -(1)/(2))* CylinderV(a - 1, z)= 0 Error Successful Error - -
12.8.E7 V β€² ⁑ ( a , z ) + 1 2 ⁒ z ⁒ V ⁑ ( a , z ) - V ⁑ ( a + 1 , z ) = 0 diffop parabolic-V 1 π‘Ž 𝑧 1 2 𝑧 parabolic-V π‘Ž 𝑧 parabolic-V π‘Ž 1 𝑧 0 {\displaystyle{\displaystyle V'\left(a,z\right)+\tfrac{1}{2}zV\left(a,z\right)% -V\left(a+1,z\right)=0}} subs( temp=z, diff( CylinderV(a, temp), temp$(1) ) )+(1)/(2)*z*CylinderV(a, z)- CylinderV(a + 1, z)= 0 Error Successful Error - -
12.8.E8 2 ⁒ V β€² ⁑ ( a , z ) - V ⁑ ( a + 1 , z ) - ( a - 1 2 ) ⁒ V ⁑ ( a - 1 , z ) = 0 2 diffop parabolic-V 1 π‘Ž 𝑧 parabolic-V π‘Ž 1 𝑧 π‘Ž 1 2 parabolic-V π‘Ž 1 𝑧 0 {\displaystyle{\displaystyle 2V'\left(a,z\right)-V\left(a+1,z\right)-(a-\tfrac% {1}{2})V\left(a-1,z\right)=0}} 2*subs( temp=z, diff( CylinderV(a, temp), temp$(1) ) )- CylinderV(a + 1, z)-(a -(1)/(2))* CylinderV(a - 1, z)= 0 Error Successful Error - -
12.8.E9 d m d z m ⁑ ( e 1 4 ⁒ z 2 ⁒ U ⁑ ( a , z ) ) = ( - 1 ) m ⁒ ( 1 2 + a ) m ⁒ e 1 4 ⁒ z 2 ⁒ U ⁑ ( a + m , z ) derivative 𝑧 π‘š superscript 𝑒 1 4 superscript 𝑧 2 parabolic-U π‘Ž 𝑧 superscript 1 π‘š Pochhammer 1 2 π‘Ž π‘š superscript 𝑒 1 4 superscript 𝑧 2 parabolic-U π‘Ž π‘š 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^% {\frac{1}{4}z^{2}}U\left(a,z\right)\right)=(-1)^{m}{\left(\tfrac{1}{2}+a\right% )_{m}}e^{\frac{1}{4}z^{2}}U\left(a+m,z\right)}} diff(exp((1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)])=(- 1)^(m)* pochhammer((1)/(2)+ a, m)*exp((1)/(4)*(z)^(2))*CylinderU(a + m, z) D[Exp[Divide[1,4]*(z)^(2)]*ParabolicCylinderD[-a - 1/2, z], {z, m}]=(- 1)^(m)* Pochhammer[Divide[1,2]+ a, m]*Exp[Divide[1,4]*(z)^(2)]*ParabolicCylinderD[-a + m - 1/2, z] Failure Failure Skip Skip
12.8.E10 d m d z m ⁑ ( e - 1 4 ⁒ z 2 ⁒ U ⁑ ( a , z ) ) = ( - 1 ) m ⁒ e - 1 4 ⁒ z 2 ⁒ U ⁑ ( a - m , z ) derivative 𝑧 π‘š superscript 𝑒 1 4 superscript 𝑧 2 parabolic-U π‘Ž 𝑧 superscript 1 π‘š superscript 𝑒 1 4 superscript 𝑧 2 parabolic-U π‘Ž π‘š 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^% {-\frac{1}{4}z^{2}}U\left(a,z\right)\right)=(-1)^{m}e^{-\frac{1}{4}z^{2}}U% \left(a-m,z\right)}} diff(exp(-(1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)])=(- 1)^(m)* exp(-(1)/(4)*(z)^(2))*CylinderU(a - m, z) D[Exp[-Divide[1,4]*(z)^(2)]*ParabolicCylinderD[-a - 1/2, z], {z, m}]=(- 1)^(m)* Exp[-Divide[1,4]*(z)^(2)]*ParabolicCylinderD[-a - m - 1/2, z] Failure Failure Skip Skip
12.8.E11 d m d z m ⁑ ( e 1 4 ⁒ z 2 ⁒ V ⁑ ( a , z ) ) = e 1 4 ⁒ z 2 ⁒ V ⁑ ( a + m , z ) derivative 𝑧 π‘š superscript 𝑒 1 4 superscript 𝑧 2 parabolic-V π‘Ž 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 parabolic-V π‘Ž π‘š 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^% {\frac{1}{4}z^{2}}V\left(a,z\right)\right)=e^{\frac{1}{4}z^{2}}V\left(a+m,z% \right)}} diff(exp((1)/(4)*(z)^(2))*CylinderV(a, z), [z$(m)])= exp((1)/(4)*(z)^(2))*CylinderV(a + m, z) Error Failure Error Skip -
12.8.E12 d m d z m ⁑ ( e - 1 4 ⁒ z 2 ⁒ V ⁑ ( a , z ) ) = ( - 1 ) m ⁒ ( 1 2 - a ) m ⁒ e - 1 4 ⁒ z 2 ⁒ V ⁑ ( a - m , z ) derivative 𝑧 π‘š superscript 𝑒 1 4 superscript 𝑧 2 parabolic-V π‘Ž 𝑧 superscript 1 π‘š Pochhammer 1 2 π‘Ž π‘š superscript 𝑒 1 4 superscript 𝑧 2 parabolic-V π‘Ž π‘š 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^% {-\frac{1}{4}z^{2}}V\left(a,z\right)\right)=(-1)^{m}{\left(\tfrac{1}{2}-a% \right)_{m}}e^{-\frac{1}{4}z^{2}}V\left(a-m,z\right)}} diff(exp(-(1)/(4)*(z)^(2))*CylinderV(a, z), [z$(m)])=(- 1)^(m)* pochhammer((1)/(2)- a, m)*exp(-(1)/(4)*(z)^(2))*CylinderV(a - m, z) Error Failure Error Skip -
12.10.E2 d 2 w d t 2 = ΞΌ 4 ⁒ ( t 2 + 1 ) ⁒ w derivative 𝑀 𝑑 2 superscript πœ‡ 4 superscript 𝑑 2 1 𝑀 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\mu^{4% }(t^{2}+1)w}} diff(w, [t$(2)])= (mu)^(4)*((t)^(2)+ 1)* w D[w, {t, 2}]= (\[Mu])^(4)*((t)^(2)+ 1)* w Failure Failure
Fail
-67.88225086+113.1370848*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
113.1370848+67.88225084*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
67.88225084-113.1370848*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
-113.1370848-67.88225084*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-67.88225099390857, 113.13708498984761] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-67.88225099390857, 113.13708498984761] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-67.88225099390857, 113.13708498984761] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-67.88225099390857, 113.13708498984761] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
12.10.E2 d 2 w d t 2 = ΞΌ 4 ⁒ ( t 2 - 1 ) ⁒ w derivative 𝑀 𝑑 2 superscript πœ‡ 4 superscript 𝑑 2 1 𝑀 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\mu^{4% }(t^{2}-1)w}} diff(w, [t$(2)])= (mu)^(4)*((t)^(2)- 1)* w D[w, {t, 2}]= (\[Mu])^(4)*((t)^(2)- 1)* w Failure Failure
Fail
-113.1370848+67.88225086*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
67.88225084+113.1370848*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
113.1370848-67.88225084*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
-67.88225084-113.1370848*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-113.13708498984761, 67.88225099390857] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-113.13708498984761, 67.88225099390857] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-113.13708498984761, 67.88225099390857] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-113.13708498984761, 67.88225099390857] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
12.12.E1 ∫ 0 ∞ e - 1 4 ⁒ t 2 ⁒ t ΞΌ - 1 ⁒ U ⁑ ( a , t ) ⁒ d t = Ο€ ⁒ 2 - 1 2 ⁒ ( ΞΌ + a + 1 2 ) ⁒ Ξ“ ⁑ ( ΞΌ ) Ξ“ ⁑ ( 1 2 ⁒ ( ΞΌ + a + 3 2 ) ) superscript subscript 0 superscript 𝑒 1 4 superscript 𝑑 2 superscript 𝑑 πœ‡ 1 parabolic-U π‘Ž 𝑑 𝑑 πœ‹ superscript 2 1 2 πœ‡ π‘Ž 1 2 Euler-Gamma πœ‡ Euler-Gamma 1 2 πœ‡ π‘Ž 3 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{\mu-1}U% \left(a,t\right)\mathrm{d}t=\frac{\sqrt{\pi}2^{-\frac{1}{2}(\mu+a+\frac{1}{2})% }\Gamma\left(\mu\right)}{\Gamma\left(\frac{1}{2}(\mu+a+\frac{3}{2})\right)}}} 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)))) Integrate[Exp[-Divide[1,4]*(t)^(2)]*(t)^(\[Mu]- 1)* ParabolicCylinderD[-a - 1/2, t], {t, 0, Infinity}]=Divide[Sqrt[Pi]*(2)^(-Divide[1,2]*(\[Mu]+ a +Divide[1,2]))* Gamma[\[Mu]],Gamma[Divide[1,2]*(\[Mu]+ a +Divide[3,2])]] Failure Failure Skip Error
12.12.E2 ∫ 0 ∞ e - 3 4 ⁒ t 2 ⁒ t - a - 3 2 ⁒ U ⁑ ( a , t ) ⁒ d t = 2 1 4 + 1 2 ⁒ a ⁒ Ξ“ ⁑ ( - a - 1 2 ) ⁒ cos ⁑ ( ( 1 4 ⁒ a + 1 8 ) ⁒ Ο€ ) superscript subscript 0 superscript 𝑒 3 4 superscript 𝑑 2 superscript 𝑑 π‘Ž 3 2 parabolic-U π‘Ž 𝑑 𝑑 superscript 2 1 4 1 2 π‘Ž Euler-Gamma π‘Ž 1 2 1 4 π‘Ž 1 8 πœ‹ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{3}{4}t^{2}}t^{-a-\frac{% 3}{2}}U\left(a,t\right)\mathrm{d}t=2^{\frac{1}{4}+\frac{1}{2}a}\Gamma\left(-a-% \tfrac{1}{2}\right)\cos\left((\tfrac{1}{4}a+\tfrac{1}{8})\pi\right)}} 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) Integrate[Exp[-Divide[3,4]*(t)^(2)]*(t)^(- a -Divide[3,2])* ParabolicCylinderD[-a - 1/2, t], {t, 0, Infinity}]= (2)^(Divide[1,4]+Divide[1,2]*a)* Gamma[- a -Divide[1,2]]*Cos[(Divide[1,4]*a +Divide[1,8])* Pi] Failure Failure Skip Successful
12.12.E3 ∫ 0 ∞ e - 1 4 ⁒ t 2 ⁒ t - a - 1 2 ⁒ ( x 2 + t 2 ) - 1 ⁒ U ⁑ ( a , t ) ⁒ d t = Ο€ / 2 ⁒ Ξ“ ⁑ ( 1 2 - a ) ⁒ x - a - 3 2 ⁒ e 1 4 ⁒ x 2 ⁒ U ⁑ ( - a , x ) superscript subscript 0 superscript 𝑒 1 4 superscript 𝑑 2 superscript 𝑑 π‘Ž 1 2 superscript superscript π‘₯ 2 superscript 𝑑 2 1 parabolic-U π‘Ž 𝑑 𝑑 πœ‹ 2 Euler-Gamma 1 2 π‘Ž superscript π‘₯ π‘Ž 3 2 superscript 𝑒 1 4 superscript π‘₯ 2 parabolic-U π‘Ž π‘₯ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{% 1}{2}}(x^{2}+t^{2})^{-1}U\left(a,t\right)\mathrm{d}t=\sqrt{\pi/2}\Gamma\left(% \tfrac{1}{2}-a\right)x^{-a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}U\left(-a,x\right)}} 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) Integrate[Exp[-Divide[1,4]*(t)^(2)]*(t)^(- a -Divide[1,2])*((x)^(2)+ (t)^(2))^(- 1)* ParabolicCylinderD[-a - 1/2, t], {t, 0, Infinity}]=Sqrt[Pi/ 2]*Gamma[Divide[1,2]- a]*(x)^(- a -Divide[3,2])* Exp[Divide[1,4]*(x)^(2)]*ParabolicCylinderD[-- a - 1/2, x] Failure Failure Skip Error
12.13.E1 U ⁑ ( a , x + y ) = e 1 2 ⁒ x ⁒ y + 1 4 ⁒ y 2 ⁒ βˆ‘ m = 0 ∞ ( - y ) m m ! ⁒ U ⁑ ( a - m , x ) parabolic-U π‘Ž π‘₯ 𝑦 superscript 𝑒 1 2 π‘₯ 𝑦 1 4 superscript 𝑦 2 superscript subscript π‘š 0 superscript 𝑦 π‘š π‘š parabolic-U π‘Ž π‘š π‘₯ {\displaystyle{\displaystyle U\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y% ^{2}}\sum_{m=0}^{\infty}\frac{(-y)^{m}}{m!}U\left(a-m,x\right)}} 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) ParabolicCylinderD[-a - 1/2, x + y]= Exp[Divide[1,2]*x*y +Divide[1,4]*(y)^(2)]*Sum[Divide[(- y)^(m),(m)!]*ParabolicCylinderD[-a - m - 1/2, x], {m, 0, Infinity}] Failure Failure Skip Skip
12.13.E5 U ⁑ ( a , x ⁒ cos ⁑ t + y ⁒ sin ⁑ t ) = e 1 4 ⁒ ( x ⁒ sin ⁑ t - y ⁒ cos ⁑ t ) 2 ⁒ βˆ‘ m = 0 ∞ ( - a - 1 2 m ) ⁒ ( tan ⁑ t ) m ⁒ U ⁑ ( m + a , x ) ⁒ U ⁑ ( - m - 1 2 , y ) parabolic-U π‘Ž π‘₯ 𝑑 𝑦 𝑑 superscript 𝑒 1 4 superscript π‘₯ 𝑑 𝑦 𝑑 2 superscript subscript π‘š 0 binomial π‘Ž 1 2 π‘š superscript 𝑑 π‘š parabolic-U π‘š π‘Ž π‘₯ parabolic-U π‘š 1 2 𝑦 {\displaystyle{\displaystyle U\left(a,x\cos t+y\sin t\right)\\ =e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt% }{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m+a,x\right)U\left(-m-\tfrac{1}{2},y% \right)}} 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) ParabolicCylinderD[-a - 1/2, 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[-m + a - 1/2, x]*ParabolicCylinderD[-- m -Divide[1,2] - 1/2, y], {m, 0, Infinity}] Error Error - -
12.14.E15 w 1 ⁒ ( a , x ) = e - 1 4 ⁒ i ⁒ x 2 ⁒ M ⁑ ( 1 4 - 1 2 ⁒ i ⁒ a , 1 2 , 1 2 ⁒ i ⁒ x 2 ) subscript 𝑀 1 π‘Ž π‘₯ superscript 𝑒 1 4 𝑖 superscript π‘₯ 2 Kummer-confluent-hypergeometric-M 1 4 1 2 𝑖 π‘Ž 1 2 1 2 𝑖 superscript π‘₯ 2 {\displaystyle{\displaystyle w_{1}(a,x)=e^{-\frac{1}{4}ix^{2}}M\left(\tfrac{1}% {4}-\tfrac{1}{2}ia,\tfrac{1}{2},\tfrac{1}{2}ix^{2}\right)}} w[1]*(a , x)= exp(-(1)/(4)*I*(x)^(2))*KummerM((1)/(4)-(1)/(2)*I*a, (1)/(2), (1)/(2)*I*(x)^(2)) Subscript[w, 1]*(a , x)= 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)] Failure Failure Error Error
12.14.E15 e - 1 4 ⁒ i ⁒ x 2 ⁒ M ⁑ ( 1 4 - 1 2 ⁒ i ⁒ a , 1 2 , 1 2 ⁒ i ⁒ x 2 ) = e 1 4 ⁒ i ⁒ x 2 ⁒ M ⁑ ( 1 4 + 1 2 ⁒ i ⁒ a , 1 2 , - 1 2 ⁒ i ⁒ x 2 ) superscript 𝑒 1 4 𝑖 superscript π‘₯ 2 Kummer-confluent-hypergeometric-M 1 4 1 2 𝑖 π‘Ž 1 2 1 2 𝑖 superscript π‘₯ 2 superscript 𝑒 1 4 𝑖 superscript π‘₯ 2 Kummer-confluent-hypergeometric-M 1 4 1 2 𝑖 π‘Ž 1 2 1 2 𝑖 superscript π‘₯ 2 {\displaystyle{\displaystyle e^{-\frac{1}{4}ix^{2}}M\left(\tfrac{1}{4}-\tfrac{% 1}{2}ia,\tfrac{1}{2},\tfrac{1}{2}ix^{2}\right)=e^{\frac{1}{4}ix^{2}}M\left(% \tfrac{1}{4}+\tfrac{1}{2}ia,\tfrac{1}{2},-\tfrac{1}{2}ix^{2}\right)}} 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)) 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)] Failure Successful Successful -
12.14.E16 w 2 ⁒ ( a , x ) = x ⁒ e - 1 4 ⁒ i ⁒ x 2 ⁒ M ⁑ ( 3 4 - 1 2 ⁒ i ⁒ a , 3 2 , 1 2 ⁒ i ⁒ x 2 ) subscript 𝑀 2 π‘Ž π‘₯ π‘₯ superscript 𝑒 1 4 𝑖 superscript π‘₯ 2 Kummer-confluent-hypergeometric-M 3 4 1 2 𝑖 π‘Ž 3 2 1 2 𝑖 superscript π‘₯ 2 {\displaystyle{\displaystyle w_{2}(a,x)=xe^{-\frac{1}{4}ix^{2}}M\left(\tfrac{3% }{4}-\tfrac{1}{2}ia,\tfrac{3}{2},\tfrac{1}{2}ix^{2}\right)}} w[2]*(a , x)= x*exp(-(1)/(4)*I*(x)^(2))*KummerM((3)/(4)-(1)/(2)*I*a, (3)/(2), (1)/(2)*I*(x)^(2)) Subscript[w, 2]*(a , x)= 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)] Failure Failure Error Error
12.14.E16 x ⁒ e - 1 4 ⁒ i ⁒ x 2 ⁒ M ⁑ ( 3 4 - 1 2 ⁒ i ⁒ a , 3 2 , 1 2 ⁒ i ⁒ x 2 ) = x ⁒ e 1 4 ⁒ i ⁒ x 2 ⁒ M ⁑ ( 3 4 + 1 2 ⁒ i ⁒ a , 3 2 , - 1 2 ⁒ i ⁒ x 2 ) π‘₯ superscript 𝑒 1 4 𝑖 superscript π‘₯ 2 Kummer-confluent-hypergeometric-M 3 4 1 2 𝑖 π‘Ž 3 2 1 2 𝑖 superscript π‘₯ 2 π‘₯ superscript 𝑒 1 4 𝑖 superscript π‘₯ 2 Kummer-confluent-hypergeometric-M 3 4 1 2 𝑖 π‘Ž 3 2 1 2 𝑖 superscript π‘₯ 2 {\displaystyle{\displaystyle xe^{-\frac{1}{4}ix^{2}}M\left(\tfrac{3}{4}-\tfrac% {1}{2}ia,\tfrac{3}{2},\tfrac{1}{2}ix^{2}\right)=xe^{\frac{1}{4}ix^{2}}M\left(% \tfrac{3}{4}+\tfrac{1}{2}ia,\tfrac{3}{2},-\tfrac{1}{2}ix^{2}\right)}} 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)) 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)] Failure Successful Successful -
12.14.E24 d 2 w d t 2 = ΞΌ 4 ⁒ ( 1 - t 2 ) ⁒ w derivative 𝑀 𝑑 2 superscript πœ‡ 4 1 superscript 𝑑 2 𝑀 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\mu^{4% }(1-t^{2})w}} diff(w, [t$(2)])= (mu)^(4)*(1 - (t)^(2))* w D[w, {t, 2}]= (\[Mu])^(4)*(1 - (t)^(2))* w Failure Failure
Fail
113.1370848-67.88225086*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
-67.88225084-113.1370848*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
-113.1370848+67.88225084*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
67.88225084+113.1370848*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[113.13708498984761, -67.88225099390857] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[113.13708498984761, -67.88225099390857] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[113.13708498984761, -67.88225099390857] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[113.13708498984761, -67.88225099390857] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
12.15.E1 d 2 w d z 2 + ( Ξ½ + Ξ» - 1 - Ξ» - 2 ⁒ z Ξ» ) ⁒ w = 0 derivative 𝑀 𝑧 2 𝜈 superscript πœ† 1 superscript πœ† 2 superscript 𝑧 πœ† 𝑀 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% \nu+\lambda^{-1}-\lambda^{-2}z^{\lambda}\right)w=0}} diff(w, [z$(2)])+(nu + (lambda)^(- 1)- (lambda)^(- 2)* (z)^(lambda))* w = 0 D[w, {z, 2}]+(\[Nu]+ (\[Lambda])^(- 1)- (\[Lambda])^(- 2)* (z)^(\[Lambda]))* w = 0 Failure Failure
Fail
.8849712009+3.576499096*I <- {lambda = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-1.464705312+7.209161093*I <- {lambda = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
38.31009603+4.160077343*I <- {lambda = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
1.044102247+4.017893257*I <- {lambda = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Skip
12.17.E4 1 ΞΎ 2 + Ξ· 2 ⁒ ( βˆ‚ 2 ⁑ w βˆ‚ ⁑ ΞΎ 2 + βˆ‚ 2 ⁑ w βˆ‚ ⁑ Ξ· 2 ) + βˆ‚ 2 ⁑ w βˆ‚ ⁑ ΞΆ 2 + k 2 ⁒ w = 0 1 superscript πœ‰ 2 superscript πœ‚ 2 partial-derivative 𝑀 πœ‰ 2 partial-derivative 𝑀 πœ‚ 2 partial-derivative 𝑀 𝜁 2 superscript π‘˜ 2 𝑀 0 {\displaystyle{\displaystyle\frac{1}{\xi^{2}+\eta^{2}}\left(\frac{{\partial}^{% 2}w}{{\partial\xi}^{2}}+\frac{{\partial}^{2}w}{{\partial\eta}^{2}}\right)+% \frac{{\partial}^{2}w}{{\partial\zeta}^{2}}+k^{2}w=0}} (1)/((xi)^(2)+ (eta)^(2))*(diff(w, [xi$(2)])+ diff(w, [eta$(2)]))+ diff(w, [zeta$(2)])+ (k)^(2)* w = 0 Divide[1,(\[Xi])^(2)+ (\[Eta])^(2)]*(D[w, {\[Xi], 2}]+ D[w, {\[Eta], 2}])+ D[w, {\[zeta], 2}]+ (k)^(2)* w = 0 Failure Failure
Fail
1.414213562+1.414213562*I <- {w = 2^(1/2)+I*2^(1/2), k = 1}
5.656854248+5.656854248*I <- {w = 2^(1/2)+I*2^(1/2), k = 2}
12.72792206+12.72792206*I <- {w = 2^(1/2)+I*2^(1/2), k = 3}
1.414213562-1.414213562*I <- {w = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Error
12.17#Ex4 d 2 U d ΞΎ 2 + ( Οƒ ⁒ ΞΎ 2 + Ξ» ) ⁒ U = 0 derivative parabolic-U πœ‰ 2 𝜎 superscript πœ‰ 2 πœ† parabolic-U 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}\xi}^{2}}+% \left(\sigma\xi^{2}+\lambda\right)U=0}} diff(CylinderU(xi, +), [(sigma*(xi)^(2)+ lambda)*$(2)])*CylinderU(=, 0) D[ParabolicCylinderD[-\[Xi] - 1/2, +], {(\[Sigma]*(\[Xi])^(2)+ \[Lambda])*, 2}]*ParabolicCylinderD[-= - 1/2, 0] Error Error - -
12.17#Ex5 d 2 V d Ξ· 2 + ( Οƒ ⁒ Ξ· 2 - Ξ» ) ⁒ V = 0 derivative parabolic-V πœ‚ 2 𝜎 superscript πœ‚ 2 πœ† parabolic-V 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}V}{{\mathrm{d}\eta}^{2}}+% \left(\sigma\eta^{2}-\lambda\right)V=0}} diff(CylinderV(eta, +), [(sigma*(eta)^(2)- lambda)*$(2)])*CylinderV(=, 0) Error Error Error - -
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