On a double series representation of $\pi$

E. N. Galushina

The Bulletin of Irkutsk State University. Series Mathematics, Tome 17 (2016), pp. 3-11

Résumé

This paper proposes a new representation of $\pi$ as a double series. This representation follows from the relation between the Weierstrass $\wp$-function and the Jacobi theta-function. In the beginning of the paper we give definitions of the classical Weierstrass $\wp$-function and Jacobi theta-function. In the beginning of 1980s Italian mathematician P.Zappa attempted to generalize $\wp$-function to multidimensional spaces using methods of multidimensional complex analysis. Using the Bochner-Martinelli kernel he found a generalizationof the $\wp$-function with properties similar to the classical one-dimensional $\wp$-function, and and analog of the identity that connects the $\wp$-function and a certain theta-function of several variables. This identity involves a constant given by an integral representation that also holds in the one-dimensional case. Computing this constant in one-dimensional case by two different methods, namely, using the integral representation and using known series whose sums involve the digamma function, we obtain a representation of $\pi$ as an absolutely convergent double series. We have performed computational experiments to estimate the rate of convergence of this series. Although it is not fast, hopefully, the proposed representation will be useful in fundamental studies in the field of mathematical analysis and number theory.

Détail

Keywords: Weierstrass $\wp$-function, Jacobi theta-function, $\pi$.