“Ramanujan g- and G-Functions”的概念、定义、习题解题方法及翻译-数学辞典

单词

Ramanujan g- and G-Functions

释义

Ramanujan g- and G-Functions

Following Ramanujan (1913-14), write

(1)

(2)

These satisfy the equalities

			(3)
			(4)
			(5)
			(6)

and

can be derived using the theory of Modular Functions and can always be expressedas roots of algebraic equations when

is Rational. For simplicity, Ramanujan tabulated

for

Even and

for

Odd. However, (6) allows

and

to be solved for in terms of

and

,giving

			(7)
			(8)

Using (3) and the above two equations allows

to be computed in terms of

(9)

In terms of the Parameter and complementary Parameter ,

			(10)
			(11)

Here,

(12)

is the Elliptic Lambda Function, which gives the value of

for which

(13)

Solving for

gives

			(14)
			(15)

Analytic values for small values of

can be found in Ramanujan

(1913-1914) and Borwein and Borwein (1987),and have been compiled in Weisstein (1996). Ramanujan

(1913-1914) contains a typographical error labeling

See also G-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.

Ramanujan, S. ``Modular Equations and Approximations to .'' Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.

Weisstein, E. W. ``Elliptic Singular Values.'' Mathematica notebook EllipticSingular.m.

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