# §11.9 Lommel Functions

## §11.9(i) Definitions

The inhomogeneous Bessel differential equation

 11.9.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\frac{1}{z}\frac{\mathrm{d}w}{% \mathrm{d}z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w=z^{\mu-1}$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $z$: complex variable and $\nu$: real or complex order Referenced by: §11.9(i), §11.9(i), §11.9(iii) Permalink: http://dlmf.nist.gov/11.9.E1 Encodings: TeX, pMML, png See also: Annotations for §11.9(i), §11.9 and Ch.11

can be regarded as a generalization of (11.2.7). Provided that $\mu\pm\nu\neq-1,-3,-5,\dots$, (11.9.1) has the general solution

 11.9.2 $w=s_{{\mu},{\nu}}\left(z\right)+AJ_{\nu}\left(z\right)+BY_{\nu}\left(z\right),$

where $A$, $B$ are arbitrary constants, $s_{{\mu},{\nu}}\left(z\right)$ is the Lommel function defined by

 11.9.3 $s_{{\mu},{\nu}}\left(z\right)=z^{\mu+1}\sum_{k=0}^{\infty}(-1)^{k}\frac{z^{2k}% }{a_{k+1}(\mu,\nu)},$ ⓘ Defines: $s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$: Lommel function Symbols: $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $a_{k}(\mu,\nu)$: expansion function Permalink: http://dlmf.nist.gov/11.9.E3 Encodings: TeX, pMML, png See also: Annotations for §11.9(i), §11.9 and Ch.11

and

 11.9.4 $a_{k}(\mu,\nu)=\prod_{m=1}^{k}\left((\mu+2m-1)^{2}-\nu^{2}\right)=4^{k}{\left(% \frac{\mu-\nu+1}{2}\right)_{k}}{\left(\frac{\mu+\nu+1}{2}\right)_{k}},$ $k=0,1,2,\dots$. ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\nu$: real or complex order, $k$: nonnegative integer and $a_{k}(\mu,\nu)$: expansion function Referenced by: §11.9(iii), Erratum (V1.1.3) for Additions Permalink: http://dlmf.nist.gov/11.9.E4 Encodings: TeX, pMML, png Addition (effective with 1.1.3): An alternative Pochhammer symbol representation was added. See also: Annotations for §11.9(i), §11.9 and Ch.11

Another solution of (11.9.1) that is defined for all values of $\mu$ and $\nu$ is $S_{{\mu},{\nu}}\left(z\right)$, where

 11.9.5 $S_{{\mu},{\nu}}\left(z\right)=s_{{\mu},{\nu}}\left(z\right)+2^{\mu-1}\Gamma% \left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}% {2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\*\left(\sin\left(\tfrac{1}{2}(\mu-% \nu)\pi\right)\,J_{\nu}\left(z\right)-\cos\left(\tfrac{1}{2}(\mu-\nu)\pi\right% )\,Y_{\nu}\left(z\right)\right),$ ⓘ Defines: $S_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$: Lommel function Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$: gamma function, $s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$: Lommel function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $z$: complex variable and $\nu$: real or complex order Permalink: http://dlmf.nist.gov/11.9.E5 Encodings: TeX, pMML, png See also: Annotations for §11.9(i), §11.9 and Ch.11

the right-hand side being replaced by its limiting form when $\mu\pm\nu$ is an odd negative integer.

### Reflection Formulas

 11.9.6 $\displaystyle s_{{\mu},{-\nu}}\left(z\right)$ $\displaystyle=s_{{\mu},{\nu}}\left(z\right),$ $\displaystyle S_{{\mu},{-\nu}}\left(z\right)$ $\displaystyle=S_{{\mu},{\nu}}\left(z\right).$

For the foregoing results and further information see Watson (1944, §§10.7–10.73) and Babister (1967, §3.16).

## §11.9(ii) Expansions in Series of Bessel Functions

When $\mu\pm\nu\neq-1,-2,-3,\dots$,

 11.9.7 $s_{{\mu},{\nu}}\left(z\right)=2^{\mu+1}\sum_{k=0}^{\infty}\*\frac{(2k+\mu+1)% \Gamma\left(k+\mu+1\right)}{k!(2k+\mu-\nu+1)(2k+\mu+\nu+1)}J_{2k+\mu+1}\left(z% \right),$
 11.9.8 $s_{{\mu},{\nu}}\left(z\right)=2^{(\mu+\nu-1)/2}\Gamma\left(\tfrac{1}{2}\mu+% \tfrac{1}{2}\nu+\tfrac{1}{2}\right)z^{(\mu+1-\nu)/2}\*\sum_{k=0}^{\infty}\frac% {(\tfrac{1}{2}z)^{k}}{k!(2k+\mu-\nu+1)}J_{k+\frac{1}{2}(\mu+\nu+1)}\left(z% \right).$

For these and further results see Luke (1969b, §9.4.5).

## §11.9(iii) Asymptotic Expansion

For fixed $\mu$ and $\nu$,

 11.9.9 $S_{{\mu},{\nu}}\left(z\right)\sim z^{\mu-1}\sum_{k=0}^{\infty}(-1)^{k}a_{k}(-% \mu,\nu)z^{-2k},$ $z\to\infty$, $|\operatorname{ph}z|\leq\pi-\delta(<\pi)$.

For $a_{k}(\mu,\nu)$ see (11.9.4). If either of $\mu\pm\nu$ equals an odd positive integer, then the right-hand side of (11.9.9) terminates and represents $S_{{\mu},{\nu}}\left(z\right)$ exactly.

For uniform asymptotic expansions, for large $\nu$ and fixed $\mu=-1,0,1,2,\dots$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). For an error bound for (11.9.9) and an exponentially-improved extension see Nemes (2015b).

## §11.9(iv) References

For further information on Lommel functions see Watson (1944, §§10.7–10.75) and Babister (1967, Chapter 3). For descriptive properties of $s_{{\mu},{\nu}}\left(x\right)$ see Steinig (1972).

For collections of integral representations and integrals see Apelblat (1983, §12.17), Babister (1967, p. 85), Erdélyi et al. (1954a, §§4.19 and 5.17), Gradshteyn and Ryzhik (2000, §6.86), Marichev (1983, p. 193), Oberhettinger (1972, pp. 127–128, 168–169, and 188–189), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105–106 and 191–192), Oberhettinger and Badii (1973, §2.14), Prudnikov et al. (1990, §§1.6 and 2.9), Prudnikov et al. (1992a, §3.34), and Prudnikov et al. (1992b, §3.32).