400
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
where obviously lim wn(k) = w(k) since one is passing to the limit in a
product of a fixed number k of terms. To show that the infinite product Pn
tends to the infinite product P when n --+ +00, i.e. that the sum of the series
L Wn (k) tends to that of the series L w( k), it thus suffices (Chap. III, nO 13,
Theorem 17) to establish that there exists a convergent series L v(k) with
positive terms such that
for any k and n.
By (10) we must therefore first estimate the
for 1 ::; k ::; m, n = 2m + 1. Now we have sin t 2: 2t/7r for 0 ::; t ::; 7r /2
(examine the graphs of the two sides), whence
sin 2 (7rk/n) 2: 4k 2 /n 2 •
On the other hand, I sinzl ::; Izl[1 + IzI 2 /3! + ... J, whence
I sin(7rz/n)I ::; l7rz/nl[1 + l7rz/nI 2 /3! + ... J ::; M(z)/n
where M(z) = 7rlzl[1 + l7rzI 2 /3! + ... J does not depend on n. It follows that
(18.14)
the general term of a convergent series independent of n.
It follows that, for any nand k, we have
IPn(k)1 ::; (1 + IUl(k)l) ... (1 + lun(k)l) ::; il[1 + a(k)J = p'
the final result since the last product converges absolutely; whence, from (13')
and (14), a bound
(18.15)
IWn(k)1 ::; p'a(k) = v(k)
by the general term of a convergent series independent of n, which, at long
last, justifies the expansion of the sine function as an infinite product.
This proof, directly inspired by Euler in his Introductio (apart, of course,
from convergence ... ), is not, by far, the simplest 30 , but it is surely the most
spectacular. One may clearly generalise the argument:
30 See other proofs in Remmert, Funktionentheorie 2, pp. 10-16.