Modular Equation
المؤلف:
Borwein, J. M. and Borwein, P. B
المصدر:
Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley
الجزء والصفحة:
...
24-12-2019
1048
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
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
![(theta_3(q))/(theta_3(q^4))-1=[(theta_3^2(q^2))/(theta_3^2(q^4))-1]^(1/2).](http://mathworld.wolfram.com/images/equations/ModularEquation/NumberedEquation10.gif) |
(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](http://mathworld.wolfram.com/images/equations/ModularEquation/NumberedEquation11.gif) |
(20)
|
![[3(theta_3(q^9))/(theta_3(q))-1]^3=9(theta_3^4(q^3))/(theta_3^4(q))-1](http://mathworld.wolfram.com/images/equations/ModularEquation/NumberedEquation12.gif) |
(21)
|
![[3(theta_4(q^9))/(theta_4(q))-1]^3=9(theta_4^4(q^3))/(theta_4^4(q))-1.](http://mathworld.wolfram.com/images/equations/ModularEquation/NumberedEquation13.gif) |
(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,
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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