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12 Parabolic Cylinder FunctionsProperties

§12.4 Power-Series Expansions

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

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

12.4.3u_{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:
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
12.4.4u_{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:
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,

12.4.5u_{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),
12.4.6u_{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).

These series converge for all values of z.

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