11 Struve and Related Functions Related Functions 11.8 Analogs to Kelvin Functions 11.10 Anger–Weber Functions

§11.9 Lommel Functions

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Keywords:: Lommel functions, definitions
Permalink:: http://dlmf.nist.gov/11.9
See also:: Annotations for Ch.11

§11.9(i) Definitions

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Keywords:: Bessel’s equation, Lommel functions, definitions, differential equation, inhomogeneous forms, power series, series expansions
Permalink:: http://dlmf.nist.gov/11.9.i
See also:: Annotations for §11.9 and Ch.11

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 of $f$ with respect to $x$ , $z$ : 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		$w=s_{{\mu},{\nu}}\left(z\right)+AJ_{\nu}\left(z\right)+BY_{\nu}\left(z\right),$
ⓘ 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, $s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$ : Lommel function, $z$ : complex variable, $\nu$ : real or complex order, $A$ : arbitrary constant and $B$ : arbitrary constant Permalink: http://dlmf.nist.gov/11.9.E2 Encodings: TeX, pMML, png See also: Annotations for §11.9(i), §11.9 and Ch.11

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

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Keywords:: Lommel functions, reflection formulas
See also:: Annotations for §11.9(i), §11.9 and Ch.11

11.9.6	$\displaystyle s_{{\mu},{-\nu}}\left(z\right)$	$\displaystyle=s_{{\mu},{\nu}}\left(z\right),$
11.9.6	$\displaystyle S_{{\mu},{-\nu}}\left(z\right)$	$\displaystyle=S_{{\mu},{\nu}}\left(z\right).$
ⓘ Symbols: $S_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$ : Lommel function, $s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$ : Lommel function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.9.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.9(i), §11.9(i), §11.9 and Ch.11

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

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Keywords:: Bessel functions, Lommel functions, series expansions
Permalink:: http://dlmf.nist.gov/11.9.ii
See also:: Annotations for §11.9 and Ch.11

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

11.9.7		$s_{{\mu},{\nu}}\left(z\right)=2^{\mu+1}\sum_{k=0}^{\infty}\*\frac{(2k+\mu+1)% \Gamma\left(k+\mu+1\right)}{k!(2k+\mu-\nu+1)(2k+\mu+\nu+1)}J_{2k+\mu+1}\left(z% \right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$ : Lommel function, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/11.9.E7 Encodings: TeX, pMML, png See also: Annotations for §11.9(ii), §11.9 and Ch.11

11.9.8		$s_{{\mu},{\nu}}\left(z\right)=2^{(\mu+\nu-1)/2}\Gamma\left(\tfrac{1}{2}\mu+% \tfrac{1}{2}\nu+\tfrac{1}{2}\right)z^{(\mu+1-\nu)/2}\*\sum_{k=0}^{\infty}\frac% {(\tfrac{1}{2}z)^{k}}{k!(2k+\mu-\nu+1)}J_{k+\frac{1}{2}(\mu+\nu+1)}\left(z% \right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$ : Lommel function, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/11.9.E8 Encodings: TeX, pMML, png See also: Annotations for §11.9(ii), §11.9 and Ch.11

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

§11.9(iii) Asymptotic Expansion

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Keywords:: Lommel functions, asymptotic expansions for large argument, inhomogeneous forms, modified Bessel’s equation
Notes:: See Watson (1944, §10.75).
Referenced by:: §11.9(iii), Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/11.9.iii
Addition (effective with 1.0.11):: A sentence was added at the end of 11.9(iii) including a new reference to Nemes (2015b).
Suggested 2015-10-27 by Gergő Nemes
See also:: Annotations for §11.9 and Ch.11

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

11.9.9		$S_{{\mu},{\nu}}\left(z\right)\sim z^{\mu-1}\sum_{k=0}^{\infty}(-1)^{k}a_{k}(-% \mu,\nu)z^{-2k},$
$z\to\infty$ , $\|\operatorname{ph}z\|\leq\pi-\delta(<\pi)$ .
ⓘ Symbols: $S_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$ : Lommel function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order, $k$ : nonnegative integer, $\delta$ : arbitrary small positive constant and $a_{k}(\mu,\nu)$ : expansion function Referenced by: §11.9(iii), §11.9(iii) Permalink: http://dlmf.nist.gov/11.9.E9 Encodings: TeX, pMML, png See also: Annotations for §11.9(iii), §11.9 and Ch.11

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

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Keywords:: Lommel functions, integral representations, integrals
Permalink:: http://dlmf.nist.gov/11.9.iv
See also:: Annotations for §11.9 and Ch.11

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 (2015, §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).

11.8 Analogs to Kelvin Functions 11.10 Anger–Weber Functions

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