Some modular identities of Ramanujan useful in approximating

Author:
Jon Borwein

Journal:
Proc. Amer. Math. Soc. **95** (1985), 365-371

MSC:
Primary 11F03; Secondary 11J72, 41A25

MathSciNet review:
806072

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Richard Bellman,*A brief introduction to theta functions*, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, 1961. MR**0125252****[2]**Bruce C. Berndt,*Modular transformations and generalizations of several formulae of Ramanujan*, Rocky Mountain J. Math.**7**(1977), no. 1, 147–189. MR**0429703****[3]**J. M. Borwein and P. B. Borwein,*The arithmetic-geometric mean and fast computation of elementary functions*, SIAM Rev.**26**(1984), no. 3, 351–366. MR**750454**, 10.1137/1026073**[4]**-,*Elliptic integrals and approximations to pi*, Dalhousie Research Report, 1984.**[5]**-,*The arithmetic-geometric mean and its relatives with applications in number theory, Analysis and complexity theory*(in preparation).**[6]**Richard P. Brent,*Fast multiple-precision evaluation of elementary functions*, J. Assoc. Comput. Mach.**23**(1976), no. 2, 242–251. MR**0395314****[7]**Arthur Cayley,*An elementary treatise on elliptic functions*, 2nd ed. Dover Publications, Inc., New York, 1961. MR**0124532****[8]**G. H. Hardy,*Ramanujan*, Chapter 12, Cambridge Univ. Press, 1960.**[9]**Ramanujan,*Modular equations and approximations to*, Quart. J. Math.**45**(1914), 350-372.**[10]**Eugene Salamin,*Computation of 𝜋 using arithmetic-geometric mean*, Math. Comp.**30**(1976), no. 135, 565–570. MR**0404124**, 10.1090/S0025-5718-1976-0404124-9**[11]**Y. Tamura and Y. Kanada,*Calculation of**to*4,196,393*decimals based on Gauss Legendre algorithm*, preprint.**[12]**G. N. Watson,*Some singular moduli*(1)*and*(2), Quart. J. Math**3**(1932), 81-98 and 189-212.**[13]**E. T. Whittaker and G. N. Watson,*A course of modern analysis*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR**1424469**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
11F03,
11J72,
41A25

Retrieve articles in all journals with MSC: 11F03, 11J72, 41A25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0806072-6

Article copyright:
© Copyright 1985
American Mathematical Society