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Ramanujan's explicit values for the classical theta-function

Part of: Other special functions

Published online by Cambridge University Press:  26 February 2010

Bruce C. Berndt  and
Heng Huat Chan
Bruce C. Berndt
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, U. S. A.
Heng Huat Chan
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, U.S.A.
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Extract

Let

where φ is the notation used by Ramanujan in his notebooks [15], and is the familiar notation of Whittaker and Watson [20, p. 464]. It is well known that [1, p. 102] (with a misprint corrected)

where denotes the ordinary or Gaussian hypergeometric function; k, 0 < k < 1, is the modulus; K is the complete elliptic integral of the first kind; and

where K′=K(k′) and is the complementary modulus. Thus, an evaluation of any one of the functions φ, , or K yields an evaluation of the other two functions. However, such evaluations may not be very explicit. For example, if K(k) is known for a certain value of k, it may be difficult or impossible to explicitly determine K′, and so q cannot be explicitly determined. Conversely, it may be possible to evaluate φ(q) for a certain value of q, but it may be impossible to determine the corresponding value of k. (Recall that [1, p. 102].)

MSC classification

Secondary: 33E05: Elliptic functions and integrals

Information

Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 278 - 294
Copyright
Copyright © University College London 1995

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