Weber Functions
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Anger and Weber Functions." §12.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
30-3-2019
3390
Weber Functions
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).
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.
الاكثر قراءة في التفاضل و التكامل
اخر الاخبار
اخبار العتبة العباسية المقدسة
