# §33.20 Expansions for Small $|\epsilon|$

## §33.20(i) Case $\epsilon=0$

 33.20.1 $\displaystyle f\left(0,\ell;r\right)$ $\displaystyle=(2r)^{1/2}J_{2\ell+1}\left(\sqrt{8r}\right),$ $\displaystyle h\left(0,\ell;r\right)$ $\displaystyle=-(2r)^{1/2}Y_{2\ell+1}\left(\sqrt{8r}\right)$, $r>0$,
 33.20.2 $\displaystyle f\left(0,\ell;r\right)$ $\displaystyle=(-1)^{\ell+1}(2|r|)^{1/2}I_{2\ell+1}\left(\sqrt{8|r|}\right),$ $\displaystyle h\left(0,\ell;r\right)$ $\displaystyle=(-1)^{\ell}(2/\pi)(2|r|)^{1/2}K_{2\ell+1}\left(\sqrt{8|r|}\right)$, $r<0$.

For the functions $J$, $Y$, $I$, and $K$ see §§10.2(ii), 10.25(ii).

## §33.20(ii) Power-Series in $\epsilon$ for the Regular Solution

 33.20.3 $f\left(\epsilon,\ell;r\right)=\sum_{k=0}^{\infty}\epsilon^{k}{\sf F}_{k}(\ell;% r),$

where

 33.20.4 ${\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}J_{2\ell+1+p}\left(% \sqrt{8r}\right),$ $r>0$,
 33.20.5 ${\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(-1)^{\ell+1+p}(2|r|)^{(p+1)/2}C_{k,p}I_{2% \ell+1+p}\left(\sqrt{8|r|}\right),$ $r<0$.

The functions $J$ and $I$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by $C_{0,0}=1$, $C_{1,0}=0$, and

 33.20.6 $\displaystyle C_{k,p}$ $\displaystyle=0,$ $p<2k$ or $p>3k$, $\displaystyle C_{k,p}$ $\displaystyle=\left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}\right)/(4p),$ $k>0$, $2k\leq p\leq 3k$. ⓘ Symbols: $k$: nonnegative integer, $\ell$: nonnegative integer and $C_{k,p}$: coefficients Referenced by: §33.20(iii) Permalink: http://dlmf.nist.gov/33.20.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.20(ii), §33.20 and Ch.33

The series (33.20.3) converges for all $r$ and $\epsilon$.

## §33.20(iii) Asymptotic Expansion for the Irregular Solution

As $\epsilon\to 0$ with $\ell$ and $r$ fixed,

 33.20.7 $h\left(\epsilon,\ell;r\right)\sim-A(\epsilon,\ell)\sum_{k=0}^{\infty}\epsilon^% {k}{\sf H}_{k}(\ell;r),$

where $A(\epsilon,\ell)$ is given by (33.14.11), (33.14.12), and

 33.20.8 ${\sf H}_{k}(\ell;r)=\sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}Y_{2\ell+1+p}\left(% \sqrt{8r}\right),$ $r>0$,
 33.20.9 ${\sf H}_{k}(\ell;r)=(-1)^{\ell+1}\frac{2}{\pi}\sum_{p=2k}^{3k}(2|r|)^{(p+1)/2}% C_{k,p}K_{2\ell+1+p}\left(\sqrt{8|r|}\right),$ $r<0$.

The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6).

## §33.20(iv) Uniform Asymptotic Expansions

For a comprehensive collection of asymptotic expansions that cover $f\left(\epsilon,\ell;r\right)$ and $h\left(\epsilon,\ell;r\right)$ as $\epsilon\to 0\pm$ and are uniform in $r$, including unbounded values, see Curtis (1964a, §7). These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\ell+1$ and $2\ell+2$.