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Modular Equation -- from Wolfram MathWorld

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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

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

In terms of and ,

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

In terms of the and defined for modular equations,