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Weber Functions

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The name Weber function is used for Bessel functions of the second kind and for the function defined by Abramowitz and Stegun (1972) as

E_nu(z)=1/piint_0^pisin(nutheta-zsintheta)dtheta.
(1)

This function may also be written as

E_nu(z)=sin(1/2pinu)_1F^~_2(1;1/2(2-nu),1/2(2+nu);-1/4z^2)-1/2zcos(1/2pinu)_1F^~_2(1;1/2(3-nu),1/2(nu+3);-1/4z^2),
(2)

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

This function is implemented in the Wolfram Language as WeberE[nu, z] and is an analog of the Anger function.

Special values for real x include

E_0(x)=-H_0(x)
(3)
E_1(x)=H_1(|x|)
(4)
E_2(x)=H_0(x)-(2H_1(x))/x
(5)
E_3(x)=-H_(-1)(x)-(4H_2(x))/x+8/(3pi),
(6)

where H_n(z) is a Struve function.

The Weber modular function is a different function defined using the Dedekind eta function.

See also

Anger Function, Bessel Function of the Second Kind, Modified Struve Function, Struve Function, Weber Modular Function

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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.

Referenced on Wolfram|Alpha

Weber Functions

Cite this as:

Weisstein, Eric W. "Weber Functions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeberFunctions.html

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