## 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 Modular Function Multiplier. In general, if is an Odd Prime, then the modular 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 equations for 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.

© 1996-9 Eric W. Weisstein
1999-05-26