Some modular identities of Ramanujan useful in approximating
Author:
Jon Borwein
Journal:
Proc. Amer. Math. Soc. 95 (1985), 365371
MSC:
Primary 11F03; Secondary 11J72, 41A25
MathSciNet review:
806072
Fulltext PDF Free Access
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Abstract: We show how various modular identities due to Ramanujan may be used to produce simple high order approximations to . Various specializations are considered and the Gaussian arithmetic geometric mean formula for is rederived as a consequence.
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 R. Bellman, A brief introduction to theta functions, Holt, Rinehart and Winston, New York, 1961. MR 0125252 (23:A2556)
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 B. C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan, Rocky Mountain J. Math. 7 (1977), 147189. MR 0429703 (55:2714)
 [3]
 J. M. Borwein and P. B. Borwein, The arithmeticgeometric mean and fast computation of elementary functions, SIAM Rev. 26 (1984), 351366. MR 750454 (86d:65029)
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 A. Cayley, An elementary treatise on elliptic functions, Bell and Sons, 1885, reprinted by Dover, 1961. MR 0124532 (23:A1844)
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 G. H. Hardy, Ramanujan, Chapter 12, Cambridge Univ. Press, 1960.
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 E. Salamin, Computation of using arithmeticgeometric mean, Math. Comp. 135 (1976), 565570. MR 0404124 (53:7928)
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 Y. Tamura and Y. Kanada, Calculation of to 4,196,393 decimals based on Gauss Legendre algorithm, preprint.
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 G. N. Watson, Some singular moduli (1) and (2), Quart. J. Math 3 (1932), 8198 and 189212.
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 E. T. Whittaker and G. N. Watson, A course of modern analysis (4th ed.), Cambridge Univ. Press, 1927. MR 1424469 (97k:01072)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198508060726
PII:
S 00029939(1985)08060726
Article copyright:
© Copyright 1985
American Mathematical Society
