An entire function which is a generalization of the Bessel function of the first kind defined by

Anger's original function had an upper limit of , but the current notation was standardized by Watson (1966).

The Anger function may also be written as

where  is a regularized hypergeometric function.

If  is an integer , then , where  is a Bessel function of the first kind.

The Anger function is implemented in the Wolfram Language as AngerJ[nu, z].

REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Anger Function  and Weber Function ." §1.5 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, p. 28, 1990.

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