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Anger Function

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

J_nu(z)=1/piint_0^picos(nutheta-zsintheta)dtheta.

Anger's original function had an upper limit of 2pi, but the current notation was standardized by Watson (1966).

The Anger function may also be written as

J_nu(z)=1/2zsin(1/2pinu)_1F^~_2(1;1/2(3-nu),1/2(3+nu);-1/4z^2)+cos(1/2pinu)_1F^~_2(1;1-1/2nu,1+1/2nu;-1/4z^2),

where _1F^~_2(a;b,c;z) is a regularized hypergeometric function.

If nu is an integer n, then J_n(z)=J_n(z), where J_n(z) is a Bessel function of the first kind.

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

See also

Anger Differential Equation, Bessel Function, Modified Struve Function, Parabolic Cylinder Function, Struve Function, Weber Functions

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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 J_nu(x) and Weber Function E_nu(x)." §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.

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Anger Function

Cite this as:

Weisstein, Eric W. "Anger Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AngerFunction.html

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