TOPICS
Search

Elliptic Integral Singular Value--k_3

The third singular value k_3, corresponding to

K^'(k_3)=sqrt(3)K(k_3),
(1)

is given by

k_3=sin(pi/(12))=1/4(sqrt(6)-sqrt(2)).
(2)

As shown by Legendre,

K(k_3)=(sqrt(pi))/(2·3^(3/4))(Gamma(1/6))/(Gamma(2/3))
(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))],
(4)

and

E^'(k_3)=(pisqrt(3))/41/(K^'(k_3))+(sqrt(3)-1)/(2sqrt(3))K^'(k_3).
(5)

Summarizing,

K[1/4(sqrt(6)-sqrt(2))]=(sqrt(pi))/(2·3^(3/4))(Gamma(1/6))/(Gamma(2/3))
(6)
K^'[1/4(sqrt(6)-sqrt(2))]=sqrt(3)K=(sqrt(pi))/(2·3^(1/4))(Gamma(1/6))/(Gamma(2/3))
(7)
E[1/4(sqrt(6)-sqrt(2))]=1/4(pi/(sqrt(3)))^(1/2)[(1+1/(sqrt(3)))(Gamma(1/3))/(Gamma(5/6))+(2Gamma(5/6))/(Gamma(1/3))]
(8)
E^'[1/4(sqrt(6)-sqrt(2))]=(sqrt(pi))/2[3^(3/4)(Gamma(2/3))/(Gamma(1/6))+(sqrt(3)-1)/(2·3^(3/4))(Gamma(1/6))/(Gamma(2/3))].
(9)

(Whittaker and Watson 1990).

See also

Jacobi Theta Functions

Explore with Wolfram|Alpha

References

Ramanujan, S. "Modular Equations and Approximations to pi." 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.

Cite this as:

Weisstein, Eric W. "Elliptic Integral Singular Value--k_3." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticIntegralSingularValuek3.html

Subject classifications