Lommel–Weber function

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.

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

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

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