Anger function

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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[edit]

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[edit]

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[edit]