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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