# OpenMath Content Dictionary: bessel_gen

Canonical URL:
http://www.openmath.org/CDs/bessel_gen.ocd
CD File:
bessel_gen.ocd
CD as XML Encoded OpenMath:
bessel_gen.omcd
Defines:
AngerJ, LommelS1, LommelS2, StruveH, StruveL, WeberE
Date:
23/8/2001
Version:
(Revision )
Review Date:
Status:
private

1/1/5000

This content dictionary contains symbols to describe the functions generated by the bessel functions.

## StruveH

The symbol StruveH takes two arguments. It represents the Struve function, defined as follows. The Struve function StruveH(v,x) solves the inhomogeneous Bessel equation:

$$x^2y^{\prime\prime}+xy^\prime+(x^2-v^2)y= 4\frac{((1/2)~x)^{v+1}}{\pi^{1/2}\Gamma(v+1/2)}$$

Signatures:
sts

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## StruveL

The symbol StruveL takes two arguments. It represents the modified Struve function, defined as follows. The modified Struve function StruveL(v,x) solves the inhomogeneous Bessel equation:

$$x^2y^{\prime\prime}+xy^\prime-(x^2+v^2)y= 4\frac{((1/2)~x)^{v+1}}{\pi^{1/2}\Gamma(v+1/2)}$$

Signatures:
sts

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## AngerJ

The symbol AngerJ takes two arguments. It represents the Anger function, defined as follows. The Anger function AngerJ(v,x) solves the inhomogeneous Bessel equation:

$$x^2y^{\prime\prime}+xy^\prime+(x^2-v^2)y=\frac{(x-v){\rm sin}(v\pi)}{\pi}$$

Signatures:
sts

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## WeberE

The symbol WeberE takes two arguments. It represents the Weber function, defined as follows. The Weber function WeberE(v,x) solves the inhomogeneous Bessel equation:

$$x^2y^{\prime\prime}+xy^\prime+(x^2-v^2)y=\frac{(v-x){\rm cos}(v\pi)-(v+x)}{\pi}$$

Signatures:
sts

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## LommelS1

The LommelS1 symbol represents the Lommel function of the first kind. The Lommel functions are solutions to the differential function:

$$z^2y^{\prime\prime}+zy^\prime+(z^2-\nu^2)y=z^{\mu+1}$$

The Lommel function of the first kind is specifically:

$$U_1(u,v)=\sum_{s=0}^\infty {(-1)}^s {\left ( \frac{u}{v} \right )}^{1+2s}J_{1+2s}(\pi x y)$$

Signatures:
sts

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## LommelS2

The LommelS1 symbol represents the Lommel function of the second kind.

The Lommel functions are solutions to the differential function:

$$z^2y^{\prime\prime}+zy^\prime+(z^2-\nu^2)y=z^{\mu+1}$$

The Lommel function of the second kind is specifically:

$$U_2(u,v)=\sum_{s=0}^\infty {(-1)}^s {\left ( \frac{u}{v} \right )}^{2+2s}J_{2+2s}(\pi x y)$$

Signatures:
sts

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