#jsDisabledContent { display:none; } My Account | Register | Help

# Anger function

Article Id: WHEBN0016978003
Reproduction Date:

 Title: Anger function Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Anger function

In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as

\mathbf{J}_\nu(z)=\frac{1}{\pi} \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta

and is closely related to Bessel functions.

The Weber function (also known as Lommel-Weber function), introduced by H. F. Weber (1879), is a closely related function defined by

\mathbf{E}_\nu(z)=\frac{1}{\pi} \int_0^\pi \sin (\nu\theta-z\sin\theta) \,d\theta

and is closely related to Bessel functions of the second kind.

## Relation between Weber and Anger functions

The Anger and Weber functions are related by

\sin(\pi \nu)\mathbf{J}_\nu(z) = \cos(\pi\nu)\mathbf{E}_\nu(z)-\mathbf{E}_{-\nu}(z)
-\sin(\pi \nu)\mathbf{E}_\nu(z) = \cos(\pi\nu)\mathbf{J}_\nu(z)-\mathbf{J}_{-\nu}(z)

so in particular if ν is not an integer they can be expressed as linear combinations of each other. If ν is an integer then Anger functions Jν are the same as Bessel functions Jν, and Weber functions can be expressed as finite linear combinations of Struve functions.

## Differential equations

The Anger and Weber functions are solutions of inhomogeneous forms of Bessel's equation z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = 0. More precisely, the Anger functions satisfy the equation

z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = (z-\nu)\sin(\pi z)/\pi

and the Weber functions satisfy the equation

z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = -((z+\nu) + (z-\nu)\cos(\pi z))/\pi.

## References

• .
• C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
• G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952)
• H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.