When the elliptic modulus 
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
 |
(1) |
where 
,
,
,
,
and 
are integers, 
is a complete
elliptic integral of the first kind, and 
is the complementary complete
elliptic integral of the first kind, then the elliptic
modulus 
is the root of an algebraic equation
with integer coefficients.
A elliptic modulus 
such that
 |
(2) |
is called a singular value of the elliptic integral. The elliptic lambda function 
gives the value of 
.
Selberg and Chowla (1967) showed that 
and 
are expressible in terms of a finite number of
gamma functions. The complete elliptic
integrals of the second kind 
and 
can be expressed in terms of 
and 
with the aid of the elliptic
alpha function 
.
Values of 
for small integer 
in terms of gamma functions 
are summarized below.
where 
is the gamma function and 
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
 |
(21) |
Furthermore, they show that 
is always expressible in terms of these functions
for 
.
In such cases, the 
functions appearing in the expression are of
the form 
where 
and 
. The terms in the numerator depend on the sign of the
Kronecker symbol 
. Values for the first few 
are
is the  |
(40) |
and 
is an algebraic number (Borwein and Zucker 1992). Note that 
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
and by definition,
 |
(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
Explore with Wolfram|Alpha
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 
-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.
Referenced on Wolfram|Alpha
Elliptic Integral Singular Value
Cite this as:
Weisstein, Eric W. "Elliptic Integral Singular Value." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipticIntegralSingularValue.html