§12.4 Power-Series Expansions

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12.4.1 $U\left(a,z\right)=U\left(a,0\right)u_{1}(a,z)+U'\left(a,0\right)u_{2}(a,z),$

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Symbols:: $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : complex variable, : 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
See also:: Annotations for §12.4 and Ch.12

12.4.2 $V\left(a,z\right)=V\left(a,0\right)u_{1}(a,z)+V'\left(a,0\right)u_{2}(a,z),$

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Symbols:: $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : complex variable, : 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
See also:: Annotations for §12.4 and Ch.12

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

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Defines:: $u_{1}(a,z)$ : solution (locally)
Symbols:: $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), : complex variable and : real or complex parameter
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
See also:: Annotations for §12.4 and Ch.12

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

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Defines:: $u_{2}(a,z)$ : solution (locally)
Symbols:: $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), : complex variable and : real or complex parameter
A&S Ref:: 19.2.3 (modification of)
Permalink:: http://dlmf.nist.gov/12.4.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §12.4 and Ch.12

Equivalently,

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

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Symbols:: $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), : complex variable, : 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
See also:: Annotations for §12.4 and Ch.12

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

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Symbols:: $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), : complex variable, : 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
See also:: Annotations for §12.4 and Ch.12

These series converge for all values of .