A New Class of Infinite Products Generalizing Viète's Product Formula for π
Tóm tắt
We show how functions F(z) which satisfy an identity of the form F(α z) = g(F(z)) for some complex number α and some function g(z) give rise to infinite product formulas that generalize Viète's product formula for π. Specifically, using elliptic and trigonometric functions we derive closed form expressions for some of these infinite products. By evaluating the expressions at certain points we obtain formulas expressing infinite products involving nested radicals in terms of well-known constants. In particular, simple infinite products for π and the lemniscate constant are obtained.
Tài liệu tham khảo
M. Abramowitz and I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th Printing, Dover, New York, 1972.
P. Beckmann, A History of Pi, 5th edn. The Golem Press, Boulder Colorado, 1982.
L. Berggren, J. Borwein, and P. Borwein, Pi: A Source Book, 2nd edn., Springer Verlag, New York, 2000.
B.C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, New York, 1989.
B.C. Berndt and R.A. Rankin (eds.), Ramanujan: Essays and Surveys, American Mathematical Society, 2001.
J. Borwein and P. Borwein, Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity, Canadian Mathematical Society Series of Monographs and Advanced Texts 4,John Wiley and Sons, Inc., New York, 1998.
B.C. Carlson, “The logarithmic mean,” Amer. Math.Monthly 79 (1972), 615–618.
A. Herschfeld, “On infinite radicals,” Amer. Math.Monthly 42 (1935), 419–429.
K.E. Morrison, “Cosine Products, Fourier Transforms, and Random Sums,” Amer. Math. Monthly 102 (8) (1995),716–724.
T.J. Osler, “The union of Vieta's and Wallis's products forpi,” Amer. Math. Monthly 106 (8) (1999), 774–776.
T.J. Osler, “Interesting finite and infinite products fromsimple algebraic identities,” Math. Gaz. (to appear).
J.F. Ritt, “Periodic functions with a multiplication theorem,”Trans. Amer. Math. Soc. 23(1) (1922), 16–25.
J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, Springer-Verlag, New York, 1994.
K.B. Stolarsky, “Mapping properties, growth, uniqueness of Vieta(infinite cosine) products,” Pacific J. Math. 89(1)(1980), 209–227.
J. Todd, “The lemniscate constants,” Comm. ACM 18 (1975), 14–19; corrigendum, ibid. 18(8), 462.
I.J. Zucker, “The evaluation in terms of Gamma-functions ofthe periods of elliptic curves admitting complexmultiplication,” Math. Proc. Camb. Phil. Soc. 82(1977), 111–118.