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Modular Equation

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

(K^'(l))/(K(l))=n(K^'(k))/(K(k))
(1)

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

(M(l,k)dy)/(sqrt((1-y^2)(1-l^2y^2)))=(dx)/(sqrt((1-x^2)(1-k^2x^2)))
(2)

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

Omega_p(u,v)=(v-u_0)(v-u_1)...(v-u_p),
(3)

where

u_p=(-1)^((p^2-1)/8)[lambda(q^p)]^(1/8)
(4)
=(-1)^((p^2-1)/8)u(q^p),
(5)

lambda is a elliptic lambda function, and

q=e^(ipitau)
(6)

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

(K^'(k))/(K(k))=2(K^'((2sqrt(k))/(1+k)))/(K((2sqrt(k))/(1+k))),
(7)

so the modular equation of degree 2 is

l=(2sqrt(k))/(1+k),
(8)

which can be written as

l^2(1+k)^2=4k.
(9)

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

Omega_2=l^2(1+k)^2-4k=0
(10)
Omega_7=(kl)^(1/4)+(k^'l^')^(1/4)-1=0
(11)
Omega_(23)=(kl)^(1/4)+(k^'l^')^(1/4)+2^(2/3)(klk^'l^')^(1/12)-1=0.
(12)

In terms of u and v,

Omega_3(u,v)=u^4-v^4+2uv(1-u^2v^2)=0
(13)
Omega_5(u,v)=v^6-u^6+5u^2v^2(v^2-u^2)+4uv(u^4v^4-1)
(14)
=(u/v)^3+(v/u)^3=2(u^2v^2-1/(u^2v^2))=0
(15)
Omega_7(u,v)=(1-u^8)(1-v^8)-(1-uv)^8=0,
(16)

where

u^2=sqrt(k)=(theta_2(q))/(theta_3(q))
(17)

and

v^2=sqrt(l)=(theta_2(q^p))/(theta_3(q^p)).
(18)

Here, theta_i are Jacobi theta functions.

A modular equation of degree 2^r for r>=2 can be obtained by iterating the equation for 2^(r-1). Modular equations for prime p from 3 to 23 are given in Borwein and Borwein (1987).

Quadratic modular identities include

(theta_3(q))/(theta_3(q^4))-1=[(theta_3^2(q^2))/(theta_3^2(q^4))-1]^(1/2).
(19)

Cubic identities include

[3(theta_2(q^9))/(theta_2(q))-1]^3=9(theta_2^4(q^3))/(theta_2^4(q))-1
(20)
[3(theta_3(q^9))/(theta_3(q))-1]^3=9(theta_3^4(q^3))/(theta_3^4(q))-1
(21)
[3(theta_4(q^9))/(theta_4(q))-1]^3=9(theta_4^4(q^3))/(theta_4^4(q))-1.
(22)

A seventh-order identity is

sqrt(theta_3(q)theta_3(q^7))-sqrt(theta_4(q)theta_4(q^7))=sqrt(theta_2(q)theta_2(q^7)).
(23)

From Ramanujan (1913-1914),

(1+q)(1+q^3)(1+q^5)...=2^(1/6)q^(1/24)(kk^')^(-1/12)
(24)
(1-q)(1-q^3)(1-q^5)...=2^(1/6)q^(1/24)k^(-1/12)k^('1/6).
(25)

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

(M(l,k)dy)/(sqrt((1-y^2)(1-l^2y^2)))=(dx)/(sqrt((1-x^2)(1-k^2x^2)))
(26)

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

M_n(l,k)=(theta_3^2(q))/(theta_3^2(q^(1/p)))=(K(k))/(K(l)),
(27)

where theta_i is a Jacobi theta function and K(k) is a complete elliptic integral of the first kind.

The first few multipliers in terms of l and k are

M_2(l,k)=1/(1+k)=(1+l^')/2
(28)
M_3(l,k)=(1-sqrt((l^3)/k))/(1-sqrt((k^3)/l)).
(29)

In terms of the u and v defined for modular equations,

M_3=v/(v+2u^3)=(2v^3-u)/(3u)
(30)
M_5=(v(1-uv^3))/(v-u^5)=(u+v^5)/(5u(1+u^3v))
(31)
M_7=(v(1-uv)[1-uv+(uv)^2])/(v-u^7)
(32)
=(v^7-u)/(7u(1-uv)[1-uv+(uv)^2]).
(33)

See also

Modular Form, Modular Function, Schläfli's Modular Form

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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 pi." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.

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Modular Equation

Cite this as:

Weisstein, Eric W. "Modular Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ModularEquation.html

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