Infinite Products of Holomorphic Functions

Abstract

Infinite products first appeared in 1579 in the work of F. Vieta (Opera, p. 400, Leyden, 1646); he gave the formula $$\frac{2}{\pi } = \sqrt {\frac{1}{2}} \cdot \sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2}} } \cdot \sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2}} } } \cdot ...$$ for π (cf. [Z], p. 104 and p. 118). In 1655 J. Wallis discovered the famous product $$\frac{\pi }{2} = \frac{{2\cdot 2}}{{1\cdot 3}}\cdot \frac{{4\cdot 4}}{{3\cdot 5}}\cdot \frac{{6\cdot 6}}{{5\cdot 7}}\cdot ...\cdot \frac{{2n\cdot 2n}}{{(2n - 1)\cdot (2n - 1)}}\cdot ...,$$ which appears in “Arithmetica infinitorum,” Opera I, p. 468 (cf. [Z], p. 104 and p. 119). But L. Euler was the first to work systematically with infinite products and to formulate important product expansions; cf. Chapter 9 of his Introductio. The first convergence criterion is due to Cauchy, Cours d’analyse, p. 562 ff. Infinite products had found their permanent place in analysis by 1854 at the latest, through Weierstrass ([Wei], p. 172 ff.).1