Elliptic Lambda Function
المؤلف:
Borwein, J. M. and Borwein, P. B
المصدر:
Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley
الجزء والصفحة:
...
22-12-2019
1956
Elliptic Lambda Function
The elliptic lambda function 
is a 
-modular function defined on the upper half-plane by
 |
(1)
|
where 
is the half-period ratio, 
is the nome
 |
(2)
|
and 
are Jacobi theta functions.
The elliptic lambda function is essentially the same as the inverse nome, the difference being that elliptic lambda function is a function of the half-period ratio 
, while the inverse nome is a function of the nome 
, where 
is itself a function of 
.
It is implemented as the Wolfram Language function ModularLambda[tau].
The elliptic lambda function 
satisfies the functional equations
has the series expansion
 |
(5)
|
(OEIS A115977), and 
has the series expansion
 |
(6)
|
(OEIS A029845; Conway and Norton 1979; Borwein and Borwein 1987, p. 117).
gives the value of the elliptic modulus 
for which the complementary 
and normal complete elliptic integrals of the first kind 
are related by
 |
(7)
|
i.e., the elliptic integral singular value for 
. It can be computed from
 |
(8)
|
where
 |
(9)
|
and 
is a Jacobi theta function. 
is related to 
by
 |
(10)
|
For all rational 
, 
and 
are known as elliptic integral singular values, and can be expressed in terms of a finite number of gamma functions (Selberg and Chowla 1967). Values of 
for small 
include
where
 |
(28)
|
The algebraic orders of these are given by 2, 2, 4, 2, 8, 4, 4, 4, 8, 4, 12, 4, 8, 8, 8, 4, ... (OEIS A084540).
Some additional exact values are given by
Exact values can also be found for rational 
, including
where 
is a polynomial root.
is related to the Ramanujan g- and G-functions by
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.
Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.
Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.
Sloane, N. J. A. Sequences A029845, A084540, and A115977 in "The On-Line Encyclopedia of Integer Sequences."
Watson, G. N. "Some Singular Moduli (1)." Quart. J. Math. 3, 81-98, 1932.
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