“Elliptic Lambda Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Elliptic Lambda Function

释义

Elliptic Lambda Function

The Group is the Subgroup of the Gamma Group with and Odd; and Even. The function

(1)

where

(2)

is a

-Modular Function and

are Theta Functions.

gives the value of the Modulus for which the complementary and normal complete Elliptic Integrals ofthe First Kind are related by

(3)

It can be computed from

(4)

where

(5)

and

is a Theta Function.

From the definition of the lambda function,

(6)

For all rational

and

are expressible in terms of a finite number of GammaFunctions (Selberg and Chowla 1967).

is related to the G-Functions">Ramanujan g- and G-Functions by

			(7)
			(8)

Special values are

where

In addition,

See also Elliptic Alpha Function, Elliptic Integral of the First Kind, Modulus (Elliptic Integral),G-Functions">Ramanujan g- and G-Functions, Theta Function

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.

Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, pp. 75, 95, and 98, 1961.

Selberg, A. and Chowla, S. ``On Epstein's Zeta-Function.'' J. Reine. Angew. Math. 227, 86-110, 1967.

Watson, G. N. ``Some Singular Moduli (1).'' Quart. J. Math. 3, 81-98, 1932.

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