Anger function

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

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

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

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

More precisely, the Anger functions satisfy the equation

and the Weber functions satisfy the equation