Although Bessel functions of the second kind are sometimes called Weber functions, Abramowitz and Stegun (1972) define a separate Weber function as
 |
(1) |
These function may also be written as
 |
(2) |
where 
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 
include
where 
is a Struve function.
Letting 
be a root of unity, another set of Weber functions
is defined as
(Weber 1902, Atkin and Morain 1993), where 
is the Dedekind
eta function and 
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 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.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 
and Weber Function 
." §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
Cite this as:
Weisstein, Eric W. "Weber Functions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeberFunctions.html