# Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi

@article{Borwein1989RamanujanME, title={Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi}, author={Jonathan M. Borwein and Peter B. Borwein and David H. Bailey}, journal={American Mathematical Monthly}, year={1989}, volume={96}, pages={201-219} }

The year 1987 was the centenary of Ramanujan’s birth. He died in 1920 Had he not died so young, his presence in modern mathematics might be more immediately felt. Had he lived to have access to powerful algebraic manipulation software. such as MACSYMA, who knows how much more spectacular his already astonishing career might have been.

#### 129 Citations

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Rational analogues of Ramanujan's series for 1/π†

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#### References

SHOWING 1-10 OF 64 REFERENCES

The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm

- Mathematics
- 1988

Paper 7: David H. Bailey, “The computation of pi to 29,360,000 decimal digits using Borweins’ quartically convergent algorithm,” Mathematics of Computation, vol. 50 (1988), p. 283–296. Reprinted by… Expand

Transcendental Number Theory

- Mathematics
- 1975

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having… Expand

Numerical results on the transcendence of constants involving pi, e, and Euler's constant

- Mathematics
- 1988

The existence of simple polynomial equations (integer relations) for the constants e/pi, e + pi, log pi, gamma (Euler's constant), e exp gamma, gamma/e, gamma/pi, and log gamma is investigated by… Expand

Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity

- Mathematics
- 1988

Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration. Theta Functions and the Arithmetic-Geometric Mean Iteration. Jacobi's Triple-Product and Some Number Theoretic Applications.… Expand

Lehrbuch der Algebra

- Nature
- 1896

A “Treatise on Algebra” is rarely found to fulfil the promise of its title. It is too often a mere collection of problems and examples, thrown together without much regard to order or method; such… Expand

Computation of π Using Arithmetic-Geometric Mean

- Mathematics
- 1976

A new formula for π is derived. It is a direct consequence of Gauss’ arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre’s relation for elliptic… Expand

OF CALCULATIONS PAST AND PRESENT: THE ARCHIMEDEAN ALGORITHM

- Mathematics
- 1983

was a forerunner of a certain type of algorithms used on today's computers. Stimulated by an article of G. M. Phillips [49], which recently appeared in this monthly, we trace the evolution of… Expand

The computational complexity of algebraic and numeric problems

- Computer Science
- Elsevier computer science library
- 1975

Thank you for downloading the computational complexity of algebraic and numeric problems elsevier computer science library theory of computation series 1. As you may know, people have search numerous… Expand

Dihedral quartic approximations and series for π

- Mathematics
- 1982

Abstract Imaginary quadratic fields with class groups that have C(4) as a subgroup are analyzed in depth, and the units of associated dihedral quartic fields are thereby evaluated using Epstein zeta… Expand

III. A memoir on the transformation of elliptic functions

- Mathematics
- Proceedings of the Royal Society of London
- 1874

The theory of Transformation in Elliptic Functions was established by Jacobi in the ‘Fundamenta Nova’ (1829); and he has there developed, transcendentally, with an approach to completeness, the… Expand