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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

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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?

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  • $\begingroup$ It's always less than <1 hence absolutely convergent. $\endgroup$
    – user775699
    Commented Jan 11, 2020 at 20:46

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