I am trying exercises in complex analysis from tutorials of an Institute of which I am not a student.
There is a question in infinite products on which I am struct.
Question is – Prove that infinite product $\prod_{n=1}^{\infty} (1- e^{2πin\tau} )$ , $\tau
$ belongs to Upper Half plane converges absolutely.
I have studied complex analysis from Ponnusamy and silvermann and from that I know that $\prod_{n=1}^{\infty} (1+ a_n ) $ converges absolutely iff $\sum_{n=1}^{\infty} (a_n) $ converges absolutely.
But I don't know how to prove absolute convergence of sum $e^{2πin\tau } $ .
Can someone please explain how to do this
Best Answer
Hint: Write $\tau = a +ib.$ Then $b>0.$ We have
$$e^{2\pi i n\tau} = e^{-2\pi nb}\cdot e^{2\pi i na}.$$
What is the absolute value of the last expression?