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Elliptic Integral Singular Value

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When the elliptic modulus k has a singular value, the complete elliptic integrals may be computed in analytic form in terms of gamma functions. Abel (quoted in Whittaker and Watson 1990, p. 525) proved that whenever

(K^'(k))/(K(k))=(a+bsqrt(n))/(c+dsqrt(n)),
(1)

where a, b, c, d, and n are integers, K(k) is a complete elliptic integral of the first kind, and K^'(k)=K(sqrt(1-k^2)) is the complementary complete elliptic integral of the first kind, then the elliptic modulus k is the root of an algebraic equation with integer coefficients.

A elliptic modulus k_r such that

(K^'(k_r))/(K(k_r))=sqrt(r),
(2)

is called a singular value of the elliptic integral. The elliptic lambda function lambda^*(r) gives the value of k_r.

Selberg and Chowla (1967) showed that K(lambda^*(r)) and E(lambda^*(r)) are expressible in terms of a finite number of gamma functions. The complete elliptic integrals of the second kind E(k_r) and E^'(k_r) can be expressed in terms of K(k_r) and K^'(k_r) with the aid of the elliptic alpha function alpha(r).

Values of K(k_r) for small integer r in terms of gamma functions Gamma(z) are summarized below.

K(k_1)=(Gamma^2(1/4))/(4sqrt(pi))
(3)
K(k_2)=(sqrt(sqrt(2)+1)Gamma(1/8)Gamma(3/8))/(2^(13/4)sqrt(pi))
(4)
K(k_3)=(3^(1/4)Gamma^3(1/3))/(2^(7/3)pi)
(5)
K(k_4)=((sqrt(2)+1)Gamma^2(1/4))/(2^(7/2)sqrt(pi))
(6)
K(k_5)=(sqrt(5)+2)^(1/4)sqrt((Gamma(1/(20))Gamma(3/(20))Gamma(7/(20))Gamma(9/(20)))/(160pi))
(7)
K(k_6)=sqrt((sqrt(2)-1)(sqrt(3)+sqrt(2))(2+sqrt(3)))sqrt((Gamma(1/(24))Gamma(5/(24))Gamma(7/(24))Gamma((11)/(24)))/(384pi))
(8)
K(k_7)=(Gamma(1/7)Gamma(2/7)Gamma(4/7))/(7^(1/4)·4pi)
(9)
K(k_8)=sqrt((2sqrt(2)+sqrt(1+5sqrt(2)))/(4sqrt(2)))((sqrt(2)+1)^(1/4)Gamma(1/8)Gamma(3/8))/(8sqrt(pi))
(10)
K(k_9)=(3^(1/4)sqrt(2+sqrt(3))Gamma^2(1/4))/(12sqrt(pi))
(11)
K(k_(10))=sqrt((2+3sqrt(2)+sqrt(5)))sqrt((Gamma(1/(40))Gamma(7/(40))Gamma(9/(40))Gamma((11)/(40))Gamma((13)/(40))Gamma((19)/(40))Gamma((23)/(40))Gamma((37)/(40)))/(2560pi^3))
(12)
K(k_(11))=[2+(17+3sqrt(33))^(1/3)-(3sqrt(33)-17)^(1/3)]^2(Gamma(1/(11))Gamma(3/(11))Gamma(4/(11))Gamma(5/(11))Gamma(9/(11)))/(11^(1/4)144pi^2)
(13)
K(k_(12))=(3^(1/4)(sqrt(2)+1)(sqrt(3)+sqrt(2))sqrt(2-sqrt(3))Gamma^3(1/3))/(2^(13/3)pi)
(14)
K(k_(13))=((18+5sqrt(13))^(1/4))/(sqrt(6656pi^5))sqrt(Gamma(1/(52))Gamma(7/(52))Gamma(9/(52))Gamma((11)/(52))Gamma((15)/(52))Gamma((17)/(52))Gamma((19)/(52))Gamma((25)/(52))Gamma((29)/(52))Gamma((31)/(52))Gamma((47)/(52))Gamma((49)/(52)))
(15)
K(k_(14))=-11-8sqrt(2)-4sqrt(5+4sqrt(2))-2sqrt(2(5+4sqrt(2)))+2sqrt(11+8sqrt(2))+2sqrt(2(11+8sqrt(2)))+sqrt(2(5+4sqrt(2))(11+8sqrt(2)))
(16)
K(k_(15))=sqrt(((sqrt(5)+1)Gamma(1/(15))Gamma(2/(15))Gamma(4/(15))Gamma(8/(15)))/(240pi))
(17)
K(k_(16))=((2^(1/4)+1)^2Gamma^2(1/4))/(2^(9/2)sqrt(pi))
(18)
K(k_(17))=C_1[(Gamma(1/(68))Gamma(3/(68))Gamma(7/(68))Gamma((11)/(68))Gamma((13)/(68)))/(Gamma(5/(68))Gamma((15)/(68))Gamma((19)/(68))Gamma((29)/(68)))]^(1/4)[Gamma((21)/(68))Gamma((25)/(68))Gamma((27)/(68))Gamma((31)/(68))Gamma((33)/(68))]^(1/4)
(19)
K(k_(25))=(sqrt(5)+2)/(20)(Gamma^2(1/4))/(sqrt(pi)),
(20)

where Gamma(z) is the gamma function and C_1 is an algebraic number (Borwein and Borwein 1987, p. 298).

Borwein and Zucker (1992) give amazing expressions for singular values of complete elliptic integrals in terms of central beta functions

beta(p)=B(p,p).
(21)

Furthermore, they show that K(k_n) is always expressible in terms of these functions for n=1,2 (mod 4). In such cases, the Gamma(z) functions appearing in the expression are of the form Gamma(t/4n) where 1<=t<=(2n-1) and (t,4n)=1. The terms in the numerator depend on the sign of the Kronecker symbol {t/4n}. Values for the first few n are

K(k_1)=2^(-2)beta(1/4)
(22)
K(k_2)=2^(-13/4)beta(1/8)
(23)
K(k_3)=2^(-4/3)3^(-1/4)beta(1/3)
(24)
=2^(-5/3)3^(-3/4)beta(1/6)
(25)
K(k_5)=2^(-33/20)5^(-5/8)(11+5sqrt(5))^(1/4)sin(1/(20)pi)beta(1/2)
(26)
=2^(-29/20)5^(-3/8)(1+sqrt(5))^(1/4)sin(3/(20)pi)beta(3/(20))
(27)
K(k_6)=2^(-47/12)3^(-3/4)(sqrt(2)-1)(sqrt(3)+1)beta(1/(24))
(28)
=2^(-43/12)3^(-1/4)(sqrt(3)-1)beta(5/(24))
(29)
K(k_7)=2·7^(-3/4)sin(1/7pi)sin(2/7pi)B(1/7,2/7)
(30)
=2^(-2/7)7^(-1/4)(beta(1/7)beta(2/7))/(beta(1/(14)))
(31)
K(k_(10))=2^(-61/20)5^(-1/4)(sqrt(5)-2)^(1/2)(sqrt(10)+3)(beta(1/8)beta(7/(40)))/(beta(1/340))
(32)
=2^(-15/4)5^(-3/4)(sqrt(5)-2)^(1/2)(beta(1/(40))beta(1/940))/(beta(3/8))
(33)
K(k_(11))=R·2^(-7/11)sin(1/(11)pi)sin(3/(11)pi)B(1/(22),3/(22))
(34)
K(k_(13))=2^(-3)13^(-5/8)(5sqrt(13)+18)^(1/4)[tan(1/(52)pi)tan(3/(52)pi)tan(9/(52)pi)]^(1/2)(beta(1/(52))beta(9/(52)))/(beta((23)/(52)))
(35)
K(k_(14))=sqrt(sqrt(4sqrt(2)+2)+sqrt(2)+sqrt(2sqrt(2)-1))·2^(-13/4)7^(-3/8)×[(tan(5/(56)pi)tan((13)/(56)pi))/(tan((11)/(56)pi))]^(1/4)sqrt((beta(5/(56))beta((13)/(56))beta(1/8))/(beta((11)/(56))))
(36)
K(k_(15))=2^(-1)3^(-3/4)5^(-7/12)B(1/(15),4/(15))
(37)
=(2^(-2)3^(-3/4)5^(-3/4)(sqrt(5)-1)beta(1/(15))beta(4/(15)))/(beta(1/3))
(38)
K(k_(17))=C_2[(beta(1/(68))beta(3/(68))beta(7/(68))beta(9/(68))beta((11)/(68))beta((13)/(68)))/(beta(5/(68))beta((15)/(68)))]^(1/4),
(39)

where R is the real root of

x^3-4x=4=0
(40)

and C_2 is an algebraic number (Borwein and Zucker 1992). Note that K(k_(11)) is the only value in the above list which cannot be expressed in terms of central beta functions.

Using the elliptic alpha function, the elliptic integrals of the second kind can also be found from

E=pi/(4sqrt(r)K)+[1-(alpha(r))/(sqrt(r))]K
(41)
E^'=pi/(4K)+alpha(r)K,
(42)

and by definition,

K^'=Ksqrt(r).
(43)

See also

Central Beta Function, Elliptic Alpha Function, Elliptic Delta Function, Elliptic Integral of the First Kind, Elliptic Integral of the Second Kind, Elliptic Lambda Function, Elliptic Modulus, Gamma Function

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References

Abel, N. H. "Recherches sur les fonctions elliptiques." J. reine angew. Math. 3, 160-190, 1828. Reprinted in Abel, N. H. Oeuvres Completes (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., p. 377, 1988.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.Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evaluation of the Gamma Function at Rational Values of Small Denominator." IMA J. Numerical Analysis 12, 519-526, 1992.Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, pp. 75, 95, and 98, 1961.Glasser, M. L. and Wood, V. E. "A Closed Form Evaluation of the Elliptic Integral." Math. Comput. 22, 535-536, 1971.Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 524-528, 1990.Wrigge, S. "An Elliptic Integral Identity." Math. Comput. 27, 837-840, 1973.Zucker, I. J. "The Evaluation in Terms of Gamma-Functions of the Periods of Elliptic Curves Admitting Complex Multiplication." Math. Proc. Cambridge Philos. Soc. 82, 111-118, 1977.Zucker, I. J. and Joyce, G. S. "Special Values of the Hypergeometric Series II." Math. Proc. Cambridge Philos. Soc. 131, 309-319, 2001.

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Elliptic Integral Singular Value

Cite this as:

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

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