“Modular Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Modular Equation

释义

Modular Equation

The modular equation of degree gives an algebraic connection of the form

(1)

between the Transcendental Complete Elliptic Integrals of the FirstKind with moduli

and

. When

and

satisfy a modular equation, arelationship of the form

(2)

exists, and

is called the Modular Function Multiplier. In general, if

is an Odd Prime, then themodular equation is given by

(3)

where

(4)

is a Elliptic Lambda Function, and

(5)

(Borwein and Borwein 1987, p. 126). An Elliptic Integral identity gives

(6)

so the modular equation of degree 2 is

(7)

which can be written as

(8)

A few low order modular equations written in terms of

and

are

	(9)
	(10)

	(11)

In terms of

and

			(12)


			(13)

(14)

where

(15)

and

(16)

Here,

are Theta Functions.

A modular equation of degree for can be obtained by iterating the equation for . Modular equationsfor Prime from 3 to 23 are given in Borwein and Borwein (1987).

Quadratic modular identities include

(17)

Cubic identities include

(18)

(19)

(20)

A seventh-order identity is

(21)

From Ramanujan (1913-1914),

(22)

(23)

See also Schläfli's Modular Form

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 127-132, 1987.

Hanna, M. ``The Modular Equations.'' Proc. London Math. Soc. 28, 46-52, 1928.

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

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