§11.9 Lommel Functions

Keywords:: Lommel functions
Permalink:: http://dlmf.nist.gov/11.9

§11.9(i) Definitions
§11.9(ii) Expansions in Series of Bessel Functions
§11.9(iii) Asymptotic Expansion
§11.9(iv) References

§11.9(i) Definitions

Keywords:: Bessel’s equation, Lommel functions, power series
Permalink:: http://dlmf.nist.gov/11.9.i

The inhomogeneous Bessel differential equation

11.9.1 $\frac{{d}^{2}w}{{dz}^{2}}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w=z^{{\mu-1}}$

Symbols:: $\frac{df}{dx}$ : 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
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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 $w=\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)+A\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)+B\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : 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
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where $A$ , $B$ are arbitrary constants, $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ is the Lommel function defined by

11.9.3 $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)=z^{{\mu+1}}\sum _{{k=0}}^{\infty}(-1)^{{k}}\frac{z^{{2k}}}{a_{{k+1}}(\mu,\nu)},$

Defines:: $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : 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
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and

11.9.4 $a_{k}(\mu,\nu)=\prod _{{m=1}}^{k}\left((\mu+2m-1)^{2}-\nu^{2}\right),$ $k=0,1,2,\dots$ .

Symbols:: $\nu$ : real or complex order, $k$ : nonnegative integer and $a_{k}(\mu,\nu)$ : expansion function
Referenced by:: §11.9(iii)
Permalink:: http://dlmf.nist.gov/11.9.E4
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Another solution of (11.9.1) that is defined for all values of $\mu$ and $\nu$ is $\mathop{S_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ , where

11.9.5 $\mathop{S_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)=\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)+2^{{\mu-1}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\*\left(\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}(\mu-\nu)\pi\right)\,\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)-\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}(\mu-\nu)\pi\right)\,\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)\right),$

Defines:: $\mathop{S_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : Lommel function
Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : Lommel function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.9.E5
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the right-hand side being replaced by its limiting form when $\mu\pm\nu$ is an odd negative integer.

¶ Reflection Formulas

Keywords:: Lommel functions

11.9.6

$\mathop{s_{{{\mu},{-\nu}}}\/}\nolimits\!\left(z\right)=\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right),$

$\mathop{S_{{{\mu},{-\nu}}}\/}\nolimits\!\left(z\right)=\mathop{S_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{S_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : Lommel function, $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : Lommel function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.9.E6
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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

Keywords:: Bessel functions, Lommel functions
Permalink:: http://dlmf.nist.gov/11.9.ii

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

11.9.7 $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)=2^{{\mu+1}}\sum _{{k=0}}^{\infty}\*\frac{(2k+\mu+1)\mathop{\Gamma\/}\nolimits\!\left(k+\mu+1\right)}{k!(2k+\mu-\nu+1)(2k+\mu+\nu+1)}\mathop{J_{{2k+\mu+1}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : Lommel function, $!$ : $n!$ : factorial, $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/11.9.E7
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11.9.8 $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)=2^{{(\mu+\nu-1)/2}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)z^{{(\mu+1-\nu)/2}}\*\sum _{{k=0}}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(2k+\mu-\nu+1)}\mathop{J_{{k+\frac{1}{2}(\mu+\nu+1)}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : Lommel function, $!$ : $n!$ : factorial, $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/11.9.E8
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For these and further results see Luke (1969b, §9.4.5).

§11.9(iii) Asymptotic Expansion

Notes:: See Watson (1944, §10.75).
Keywords:: Lommel functions, modified Bessel’s equation
Permalink:: http://dlmf.nist.gov/11.9.iii

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

11.9.9 $\mathop{S_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)\sim z^{{\mu-1}}\sum _{{k=0}}^{\infty}(-1)^{k}a_{k}(-\mu,\nu)z^{{-2k}},$ $z\to\infty$ , $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta(<\pi)$ .

Symbols:: $\mathop{S_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : Lommel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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)
Permalink:: http://dlmf.nist.gov/11.9.E9
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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 $\mathop{S_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ 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).

§11.9(iv) References

Keywords:: Lommel functions, integrals
Permalink:: http://dlmf.nist.gov/11.9.iv

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