
Contents
Next: About this document
Up:
RamanujanModular Equations,
Previous: Sources
![[Annotate]](/organics/icons/sannotate.gif)
![[Shownotes]](../gif/annotate/sshow-171.gif)
References
- 1
- M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,
Dover, New York, 1964.
[1]
- 2
- D.H. Bailey, The computation of
to 29,360,000 decimal digits
using Borweins' quartically convergent algorithm, Math. Comput., 50 (1988)
283--96.
[1] [2] [3] [4]
- 3
- ---, Numerical results on the transcendence of constants involving
, e, and Euler's constant, Math. Comput., 50 (1988) 275--81.
[1] [2]
- 4
- A. Baker, Transcendental Number Theory, Cambridge Univ. Press, London,
1975.
[1] [2]
- 5
- P. Beckmann, A History of Pi, 4th ed., Golem Press, Boulder, CO, 1977.
[1] [2] [3]
- 6
- R. Bellman, A Brief Introduction to Theta Functions, Holt, Reinhart
and Winston, New York, 1961.
[1] [2] [3]
- 7
- B.C. Berndt, Modular Equations of Degrees 3, 5, and 7 and Associated
Theta Functions Identities, chapter 19 of Ramanujan's Second Notebook, Springer ---
Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991.
[1] [2] [3]
- 8
- A. Borodin and I. Munro, The Computational Complexity of Algebraic
and Numeric Problems, American Elsevier, New York, 1975.
[1] [2]
- 9
- J.M. Borwein and P. B. Borwein, The arithmetic-geometric mean and
fast computation of elementary functions, SIAM Rev., 26 (1984)
351--365.
[1] [2] [3] [4]
- 10
- ---, An explicit cubic iteration for
,
BIT, 26 (1986) 123--126.
[1]
- 11
- ---, Pi and the AGM --- A Study in Analytic Number Theory and
Computational Complexity, Wiley, N.Y., 1987.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
- 12
- R.P. Brent, Fast multiple-precision evaluation of elementary
functions, J. ACM 23 (1976) 242--251.
[1] [2]
- 13
- E.O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood
Cliffs, N.J., 1974.
[1] [2]
- 14
- A. Cayley, An elementary Treatise on Elliptic Functions, Bell
and Sons, 1885; reprint Dover, 1961.
[1]
- 15
- ---, A memoir on the transformation of elliptic functions,
Phil. Trans. T., 164 (1874) 397--456.
[1]
- 16
- D.V. Chudnovsky and G.V. Chudnovsky, Padé and Rational
Approximation to Systems of Functions and Their Arithmetic Applications, Lecture Notes
in Mathematics 1052, Springer, Berlin, 1984.
[1] [2]
- 17
- H.R.P. Ferguson and R.W. Forcade,Generalization of the Euclidean
algorithm for real numbers to all dimensions higher than two, Bull. AMS, 1
(1979) 912--914.
[1]
- 18
- C. F. Gauss, Werke, Göttingen 1866--1933, Be 3, pp. 361--403.
[1]
- 19
- G.H. Hardy, Ramanujan, Cambridge Univ. Press, London,1940.
[1] [2]
- 20
- L.V. King, On The Direct Numerical Calculation of Elliptic
Functions and Integrals, Cambridge University Press, 1924.
[1]
- 21
- F. Klein, Development of Mathematics in the 19th Century, 1928,
TRans. Math. Sci. Press, R. Hermann ed., Brookline, MA, 1979.
[1]
- 22
- D. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical
Algorithms, Addison-Wesley, Reading, MA, 1981.
[1] [2]
- 23
- F. Lindemann, Über die Zahl
, Math. Ann., 20
(1882) 213--225.
[1] [2]
- 24
- G. Miel, On calculations past and present: the Archimedean
algorithm, Amer. Math. Monthly, 90 (1983) 17--35.
[1]
- 25
- D.J. Newman, Rational Approximation Versus Fast Computer
Methods, in Lectures on Approximation and Value Distribution, Presses de
l'Université de Montreal, 1982, pp. 149--174.
[1]
- 26
- S. Ramanujan. Modular equations and approximations to
. Quart. J. Math. 45 (1914) 350--72.
[1] [2] [3] [4] [5] [6]
- 27
- E. Salamin. Computation of
using arithmetic-geometric mean.
Math. Comput. 30 (1976) 565--570.
[1] [2]
- 28
- B. Schoenberg, Elliptic Modular Functions, Springer, Berlin, 1976.
[1] [2]
- 29
- A. Schönhage and V. Strassen, Schnelle Multiplikation Grosser
Zahlen, Computing, 7 (1971) 281--292.
[1] [2]
- 30
- D. Shanks, Dihedral quartic approximations and series for
,
J. Number Theory, 14 (1982) 394--423.
[1]
- 31
- D. Shanks and J.W. Wrench, Calculation of
to 100,000
decimals, Math. Comput., 16 (1962) 76--79.
[1] [2]
- 32
- W. Shanks, Contributions to Mathematics Comprising Chiefly of the
Rectification of the Circle to 607 Places of Decimals, G. Bell, London, 1853.
[1]
- 33
- Y. Tamura and Y. Kanada, Calculation of
to 4,196,393,
decimals based on Gauss-Legendre algorithm, preprint (1983).
[1]
- 34
- J. Tannery and J. Molk, Fonctions Elliptiques, vols 1 and 2,
1893; reprint Chelsea, New York, 1972.
[1]
- 35
- S. Wagon, Is
normal?, The Math. Intelligencer,
7 (1985) 65--67.
[1]
- 36
- G.N. Watson, Some singular moduli (1), Quart. J. Math.,
3 (1932) 81--98.
[1] [2]
- 37
- ---, The final problem: an account of the mock theta functions,
J. London Math. Soc., 11 (1936) 55--80.
[1] [2]
- 38
- H. Weber, Lehrbuch der Algebra, Vol. 3, 1908: reprint Chelsea, New
York, 1980.
[1] [2]
- 39
- E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th
ed., Cambridge Univ. Press, London, 1927.
[1] [2]

Contents
Next: About this document
Up:
RamanujanModular Equations,
Previous: Sources