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Associated Anger Function


The associated Anger function is the two-parameter extension of the Anger function defined for Re(mu)>-1 by

 J_(nu,mu)(z)=1/piint_0^pi(2sintheta)^mucos(nutheta-zsintheta)dtheta.

The power uses the real natural logarithm of 2sintheta>0 for 0<theta<pi. For each such mu, this is an entire function of nu and z, and J_(nu,0)(z)=J_nu(z).

Its regularized hypergeometric function representation is

 J_(nu,mu)(z)=sqrt(pi)cos(1/2pinu)Gamma(1+mu)_2F^~_3(1/2(mu+1),1/2(mu+2);1/2,1+1/2(mu+nu),1+1/2(mu-nu);-1/4z^2)+1/4sqrt(pi)sin(1/2pinu)Gamma(2+mu)z_2F^~_3(1/2(mu+2),1/2(mu+3);3/2,1/2(3+mu+nu),1/2(3+mu-nu);-1/4z^2).

The associated Anger function is implemented in the Wolfram Language as AngerJ[nu, mu, z] (Wolfram Research).


See also

Anger Function

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References

Wolfram Research. "AngerJ." https://reference.wolfram.com/language/ref/AngerJ.html.

Cite this as:

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

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