Anger Function

A generalization of the Bessel Function of the First Kind defined by

$${\mathcal J}_\nu(z) \equiv {1\over\pi} \int_0^\pi \cos(\nu\theta-z\sin\theta)\,d\theta.$$

If $\nu$ is an Integer $n$, then ${\mathcal J}_n(z) = J_n(z)$, where $J_n(z)$ is a Bessel Function of the First Kind. Anger's original function had an upper limit of $2\pi$, but the current Notation was standardized by Watson (1966).

See also Bessel Function, Modified Struve Function, Parabolic Cylinder Function, Struve Function, Weber Functions

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Anger and Weber Functions.'' §12.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 498-499, 1972.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.