Modular Equation

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

(1)

between the transcendental complete elliptic integrals of the first kind with moduli  and . When  and  satisfy a modular equation, a relationship of the form

(2)

exists, and  is called the multiplier. In general, if  is an odd prime, then the modular equation is given by

(3)

where

			(4)
			(5)

 is a elliptic lambda function, and

(6)

(Borwein and Borwein 1987, p. 126), where  is the half-period ratio. An elliptic integral identity gives

(7)

so the modular equation of degree 2 is

(8)

which can be written as

(9)

A few low order modular equations written in terms of  and  are

			(10)
			(11)
			(12)

In terms of  and ,

			(13)
			(14)
			(15)
			(16)

where

(17)

and

(18)

Here,  are Jacobi theta functions.

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

Quadratic modular identities include

(19)

Cubic identities include

(20)

(21)

(22)

A seventh-order identity is

(23)

From Ramanujan (1913-1914),

(24)

(25)

When  and  satisfy a modular equation, a relationship of the form

(26)

exists, and  is called the multiplier. The multiplier of degree  can be given by

(27)

where  is a Jacobi theta function and  is a complete elliptic integral of the first kind.

The first few multipliers in terms of  and  are

			(28)
			(29)

In terms of the  and  defined for modular equations,

			(30)
			(31)
			(32)
			(33)

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.