“Elliptic Integral Singular Value”的概念、定义、习题解题方法及翻译-数学辞典

单词

Elliptic Integral Singular Value

释义

Elliptic Integral Singular Value

When the Modulus has a singular value, the complete elliptic integrals may becomputed in analytic form in terms of Abel (quoted in Whittaker andWatson 1990, p. 525) proved that whenever

(1)

where

, and

are Integers,

is a complete Elliptic Integral of theFirst Kind, and

is the complementary complete Elliptic Integral of the First Kind,then the Modulus

is the Root of an algebraic equation with IntegerCoefficients.

A 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 finitenumber of Gamma Functions. The complete Elliptic Integrals of the Second Kind

and

can be expressed in terms of

and

with the aid of theElliptic Alpha Function

The following table gives the values of for small integral in terms ofGamma Functions.

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

(3)

Furthermore, they show that

is always expressible in terms of these functions for

. In suchcases, 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

where

is the Real Root of

(4)

and

is an algebraic number (Borwein and Zucker 1992). Note that

is the only value in the above list which cannot beexpressed in terms of Central Beta Functions.

Using the Elliptic Alpha Function, the Elliptic Integrals of the Second Kind can also be found from

			(5)
			(6)

and by definition,

(7)

See also Central Beta Function, Elliptic Alpha Function, Elliptic Delta Function, Elliptic Integral ofthe First Kind, Elliptic Integral of the Second Kind, Elliptic Lambda Function, Gamma Function,Modulus (Elliptic Integral)

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.

Weisstein, E. W. ``Elliptic Singular Values.'' Mathematica notebook EllipticSingular.m.

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 Phil. Soc. 82, 111-118, 1977.

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