In mathematics, the Anger function, introduced by , is a function defined as
with complex parameter and complex variable . It is closely related to the Bessel functions.
The Weber function (also known as LommelâÂÂWeber function), introduced by , 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 is not an integer they can be expressed as linear combinations of each other. If is an integer then Anger functions are the same as Bessel functions , and Weber functions can be expressed as finite linear combinations of Struve functions.
The Anger function has the power series expansion
While the Weber function has the power series expansion
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
The Anger function satisfies this inhomogeneous form of recurrence relation
While the Weber function satisfies this inhomogeneous form of recurrence relation
The Anger and Weber functions satisfy these homogeneous forms of delay differential equations
The Anger and Weber functions also satisfy these inhomogeneous forms of delay differential equations