# §12.4 Power-Series Expansions

 12.4.1 $\mathop{U\/}\nolimits\!\left(a,z\right)=\mathop{U\/}\nolimits\!\left(a,0\right% )u_{1}(a,z)+{\mathop{U\/}\nolimits^{\prime}}\!\left(a,0\right)u_{2}(a,z),$
 12.4.2 $\mathop{V\/}\nolimits\!\left(a,z\right)=\mathop{V\/}\nolimits\!\left(a,0\right% )u_{1}(a,z)+{\mathop{V\/}\nolimits^{\prime}}\!\left(a,0\right)u_{2}(a,z),$

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