Ramanujan g- and G-Functions
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:
...
19-8-2019
1963
Ramanujan g- and G-Functions
Following Ramanujan (1913-1914), write
 |
(1)
|
 |
(2)
|
These satisfy the equalities
and 
can be derived using the theory of modular functions and can always be expressed as roots of algebraic equations when 
is rational. They are related to the Weber functions.
For simplicity, Ramanujan tabulated 
for 
even and 
for 
odd. However, (6) allows 
and 
to be solved for in terms of 
and 
, giving
Using (◇) and the above two equations allows 
to be computed in terms of 
or 
 |
(9)
|
In terms of the parameter 
and complementary parameter 
,
Here,
 |
(12)
|
is the elliptic lambda function, which gives the value of 
for which
 |
(13)
|
Solving for 
gives
Solving for 
and 
directly in terms of 
then gives
Analytic values for small values of 
can be found in Ramanujan (1913-1914) and Borwein and Borwein (1987), and have been compiled by Weisstein. Ramanujan (1913-1914) contains a typographical error labeling 
as 
.
REFERENCES:
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.
Ramanujan, S. "Modular Equations and Approximations to 
." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.
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