Anger Function

The Anger function is a special function introduced by C. T. Anger in 1855, defined for a complex parameter and a complex variable by an integral of the cosine of a combination of the two over half a period, divided by pi. It is closely related to the Bessel functions, and in fact coincides exactly with the Bessel function of the first kind whenever its parameter is an integer. A closely analogous function, the Weber function or Lommel-Weber function, was introduced by H. F. Weber in 1879 using a sine in place of the cosine in its own defining integral, and relates in the same way to Bessel functions of the second kind. Both the Anger and Weber functions solve inhomogeneous versions of Bessel's differential equation, have their own power series expansions and recurrence relations, and when the parameter is not an integer, each can be written as a linear combination of the other. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/