# OpenMath Content Dictionary: hypergeo2

Canonical URL:
http://www.openxm.org/...
CD File:
hypergeo2.ocd
CD as XML Encoded OpenMath:
hypergeo2.omcd
Defines:
airyAi, airyBi, besselJ, besselY, hankel1, hankel2, kummer
Date:
2003-11-30
Version:
0 (Revision 3)
Review Date:
2017-12-31
Status:
experimental

  Author: Yasushi Tamura


This CD defines some famous hypergeometric functions such as Bessel functions and Airy functions. These functions are described in the following books. (1) Handbook of Mathematical Functions, Abramowitz, Stegun (2) Higher transcendental functions. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, Erdlyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G.

## kummer

Description:

Kummer's hypergeometric function.

Commented Mathematical property (CMP):
kummer(a,c;z) = hypergeo1.hypergeometric1F1(a,c;z)
Formal Mathematical property (FMP):
$\mathrm{kummer}\left(a,c,z\right)=\mathrm{hypergeometric1F1}\left(a,c,z\right)$
Signatures:
sts

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## besselJ

Description:

The Bessel function. This function is one of the famous two solutions of the Bessel differential equation at z=0.

Commented Mathematical property (CMP):
besselJ(v,z) = (\frac{z}{2})^v \sum_{n=0}^{+\infty} \frac{(-1)^n}{n! \Gamma(v+n+1)} (\frac{z}{2})^2n
Formal Mathematical property (FMP):
$\mathrm{besselJ}\left(v,z\right)={\left(\frac{z}{2}\right)}^{v}\sum _{n=0}^{\mathrm{infty}}\frac{{\left(-1\right)}^{n}}{n!\mathrm{gamma}\left(v+n+1\right)}{\left(\frac{z}{2}\right)}^{\left(2n\right)}$
Example:
${z}^{2}\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{besselJ}\left(v,z\right)\right)\right)+z\frac{d}{dz}\left(\mathrm{besselJ}\left(v,z\right)\right)+\left({z}^{2}-{v}^{2}\right)\mathrm{besselJ}\left(v,z\right)=0$
Signatures:
sts

 [Next: besselY] [Previous: kummer] [Top]

## besselY

Description:

The Bessel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=0.

Commented Mathematical property (CMP):
besselY(v,z) = (\cos(v \pi) besselJ(v,z) - besselJ(-v,z))/\sin(v \pi)
Formal Mathematical property (FMP):
$\mathrm{besselY}\left(v,z\right)=\frac{\mathrm{cos}\left(v\pi \right)\mathrm{besselJ}\left(v,z\right)-\mathrm{besselJ}\left(-v,z\right)}{\mathrm{sin}\left(v\pi \right)}$
Example:
${z}^{2}\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{besselY}\left(v,z\right)\right)\right)+z\frac{d}{dz}\left(\mathrm{besselY}\left(v,z\right)\right)+\left({z}^{2}-{v}^{2}\right)\mathrm{besselY}\left(v,z\right)=0$
Signatures:
sts

 [Next: hankel1] [Previous: besselJ] [Top]

## hankel1

Description:

The first Hankel function. This function is one of the famous two solutions of the Bessel differential equation at z=\infty.

Commented Mathematical property (CMP):
hankel1(v,z) = besselJ(v,z) + i BesselY(v,z)
Formal Mathematical property (FMP):
$\mathrm{hankel1}\left(v,z\right)=\mathrm{besselJ}\left(v,z\right)+i\mathrm{besselY}\left(v,z\right)$
Signatures:
sts

 [Next: hankel2] [Previous: besselY] [Top]

## hankel2

Description:

The second Hankel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=\infty.

Commented Mathematical property (CMP):
hankel2(v,z) = besselJ(v,z) - i BesselY(v,z)
Formal Mathematical property (FMP):
$\mathrm{hankel1}\left(v,z\right)=\mathrm{besselJ}\left(v,z\right)-i\mathrm{besselY}\left(v,z\right)$
Signatures:
sts

 [Next: airyAi] [Previous: hankel1] [Top]

## airyAi

Description:

The first Airy function. This function is one of the famous two solutions of the Airy differential equation, and converges to 0 when z->\infty

Commented Mathematical property (CMP):
(\frac{d^2}{dz^2} - z) airyAi(z) = 0
Formal Mathematical property (FMP):
$\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{airyAi}\left(z\right)\right)\right)-z\mathrm{airyAi}\left(z\right)=0$
Signatures:
sts

 [Next: airyBi] [Previous: hankel2] [Top]

## airyBi

Description:

The second Airy function. This function is the another one of the famous two solutions of the Airy differential equation, and diverges when z->\infty

Commented Mathematical property (CMP):
(\frac{d^2}{dz^2} - z) airyBi(z) = 0
Formal Mathematical property (FMP):
$\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{airyBi}\left(z\right)\right)\right)-z\mathrm{airyBi}\left(z\right)=0$
Signatures:
sts

 [First: kummer] [Previous: airyAi] [Top]