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

[Annotate][Shownotes]


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]


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