In mathematics, the Anger function, introduced by , is a function defined as

and is closely related to Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

s.

The Weber function, introduced by , is a closely related function defined by

and is closely related to Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

s of the second kind.

Relation between Weber and Anger functions

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.

Differential equations

The Anger and Weber functions are solutions of inhomogenous forms of Bessel's equation . More precisely,
the Anger functions satisfy the equation


and the Weber functions satisfy the equation