Elliptic Integral Singular Value--k_3
المؤلف:
Ramanujan, S
المصدر:
"Modular Equations and Approximations to pi." Quart. J. Pure. Appl. Math. 45
الجزء والصفحة:
...
25-4-2019
1730
Elliptic Integral Singular Value--k_3
The third singular value 
, corresponding to
 |
(1)
|
is given by
 |
(2)
|
As shown by Legendre,
 |
(3)
|
(Whittaker and Watson 1990, p. 525). In addition,
![E(k_3)=pi/(4sqrt(3))1/K+(sqrt(3)+1)/(2sqrt(3))K=1/4(pi/(sqrt(3)))^(1/2)[(1+1/(sqrt(3)))(Gamma(1/3))/(Gamma(5/6))+(2Gamma(5/6))/(Gamma(1/3))],](http://mathworld.wolfram.com/images/equations/EllipticIntegralSingularValuek3/NumberedEquation4.gif) |
(4)
|
and
 |
(5)
|
Summarizing,
(Whittaker and Watson 1990).
REFERENCES:
Ramanujan, S. "Modular Equations and Approximations to 
." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.
Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 525-527 and 535, 1990.
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