§12.4 Power-Series Expansions

Notes:: See Miller (1955, pp. 61–63).
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.4

12.4.1 $\mathop{U\/}\nolimits\!\left(a,z\right)=\mathop{U\/}\nolimits\!\left(a,0\right)u_{1}(a,z)+{\mathop{U\/}\nolimits^{{\prime}}}\!\left(a,0\right)u_{2}(a,z),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable, $a$ : real or complex parameter, $u_{1}(a,z)$ : solution and $u_{2}(a,z)$ : solution
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.4.E1
Encodings:: TeX, pMML, png

12.4.2 $\mathop{V\/}\nolimits\!\left(a,z\right)=\mathop{V\/}\nolimits\!\left(a,0\right)u_{1}(a,z)+{\mathop{V\/}\nolimits^{{\prime}}}\!\left(a,0\right)u_{2}(a,z),$

Symbols:: $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $z$ : complex variable, $a$ : real or complex parameter, $u_{1}(a,z)$ : solution and $u_{2}(a,z)$ : solution
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.4.E2
Encodings:: TeX, pMML, png

where the initial values are given by (12.2.6)–(12.2.9), and $u_{1}(a,z)$ and $u_{2}(a,z)$ are the even and odd solutions of (12.2.2) given by

12.4.3 $u_{1}(a,z)=e^{{-\tfrac{1}{4}z^{2}}}\left(1+(a+\tfrac{1}{2})\frac{z^{2}}{2!}+(a+\tfrac{1}{2})(a+\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),$

Symbols:: $e$ : base of exponential function, $!$ : $n!$ : factorial, $z$ : complex variable, $a$ : real or complex parameter and $u_{1}(a,z)$ : solution
A&S Ref:: 19.2.1 (modification of)
Referenced by:: §12.14(v), §12.7(iv)
Permalink:: http://dlmf.nist.gov/12.4.E3
Encodings:: TeX, pMML, png

12.4.4 $u_{2}(a,z)=e^{{-\tfrac{1}{4}z^{2}}}\left(z+(a+\tfrac{3}{2})\frac{z^{3}}{3!}+(a+\tfrac{3}{2})(a+\tfrac{7}{2})\frac{z^{5}}{5!}+\cdots\right).$

Symbols:: $e$ : base of exponential function, $!$ : $n!$ : factorial, $z$ : complex variable, $a$ : real or complex parameter and $u_{2}(a,z)$ : solution
A&S Ref:: 19.2.3 (modification of)
Permalink:: http://dlmf.nist.gov/12.4.E4
Encodings:: TeX, pMML, png

Equivalently,

12.4.5 $u_{1}(a,z)=e^{{\tfrac{1}{4}z^{2}}}\left(1+(a-\tfrac{1}{2})\frac{z^{2}}{2!}+(a-\tfrac{1}{2})(a-\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),$

Symbols:: $e$ : base of exponential function, $!$ : $n!$ : factorial, $z$ : complex variable, $a$ : real or complex parameter and $u_{1}(a,z)$ : solution
A&S Ref:: 19.2.2 (modification of)
Permalink:: http://dlmf.nist.gov/12.4.E5
Encodings:: TeX, pMML, png

12.4.6 $u_{2}(a,z)=e^{{\tfrac{1}{4}z^{2}}}\left(z+(a-\tfrac{3}{2})\frac{z^{3}}{3!}+(a-\tfrac{3}{2})(a-\tfrac{7}{2})\frac{z^{5}}{5!}+\cdots\right).$

Symbols:: $e$ : base of exponential function, $!$ : $n!$ : factorial, $z$ : complex variable, $a$ : real or complex parameter and $u_{2}(a,z)$ : solution
A&S Ref:: 19.2.4 (modification of)
Referenced by:: §12.14(v), §12.7(iv)
Permalink:: http://dlmf.nist.gov/12.4.E6
Encodings:: TeX, pMML, png

These series converge for all values of $z$ .