Weber Functions
Although Bessel functions of the second kind are sometimes called Weber functions, Abramowitz and Stegun (1972) define a separate Weber function as
These 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
where H_n(z) is a Struve function.
Letting zeta_n=e^(2pii/n) be a root of unity, another set of Weber functions is defined as
![画像:([f^(24)(tau)+8][f_1^8(tau)-f_2^8(tau)])/(f^8(tau))](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/WeberFunctions/Inline31.svg){kind=link}
(Weber 1902, Atkin and Morain 1993), where eta(tau) is the Dedekind eta function and tau is the half-period ratio. These functions are related to the Ramanujan g- and G-functions and the elliptic lambda function.
The Weber functions satisfy the identities
(Weber 1902, Atkin and Morain 1993).
See also
Anger Function, Bessel Function of the Second Kind, Dedekind Eta Function, Elliptic Lambda Function, j-Function, Jacobi Identities, Jacobi Triple Product, Klein's Absolute Invariant, Modified Struve Function, Ramanujan g- and G-Functions, Q-Function, Struve FunctionExplore with Wolfram|Alpha
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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.Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primality Proving." Math. Comput. 61, 29-68, 1993.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 68-69, 1987.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.Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, pp. 113-114, 1902.Weng, A. "Class Polynomials of CM-Fields. http://www.exp-math.uni-essen.de/zahlentheorie/classpol/class.html.Referenced on Wolfram|Alpha
Weber FunctionsCite this as:
Weisstein, Eric W. "Weber Functions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeberFunctions.html