Anger function

In mathematics, the Anger function, introduced in harv|Anger|1855, is a function defined by

: $mathbf\{J\}\_\; u(z)=frac\{1\}\{pi\}\; int\_0^pi\; cos\; (\; u\; heta-zsin\; heta)\; ,d\; heta.$

and is closely related to Bessel functions.

The Weber function, introduced by harvs|txt|authorlink=Heinrich Friedrich Weber|first=H. F.|last=Weber|year=1879, is a closely related function defined by

: $mathbf\{E\}\_\; u(z)=frac\{1\}\{pi\}\; int\_0^pi\; sin\; (\; u\; heta-zsin\; heta)\; ,d\; heta.$

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

:$sin(pi\; u)mathbf\{J\}\_\; u(z)\; =\; cos(pi\; u)mathbf\{E\}\_\; u(z)-mathbf\{E\}\_\{-\; u\}(z)$

:$-sin(pi\; u)mathbf\{E\}\_\; u(z)\; =\; cos(pi\; u)mathbf\{J\}\_\; u(z)-mathbf\{J\}\_\{-\; u\}(z)$

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 J_ν are the same as Bessel functions "J"_ν, and Weber functions can be expressed as finite linear combinations of Struve functions.

Differential equations

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

:$z^2y^\{primeprime\}\; +\; zy^prime\; +(z^2-\; u^2)y\; =\; (z-\; u)sin(pi\; z)/pi$

and the Weber functions satisfy the equation

:$z^2y^\{primeprime\}\; +\; zy^prime\; +(z^2-\; u^2)y\; =\; -((z+\; u)\; +\; (z-\; u)cos(pi\; z))/pi.$

References

*AS ref|12|498
*C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig , 5 (1855) pp. 1–29
*springer|id=A/a012490|title=Anger function|first=A.P.|last= Prudnikov
*springer|id=W/w097320|title=Weber function|first=A.P.|last= Prudnikov
*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