I am self studying analytic number theory from Tom M Apostol Modular functions and Dirichlet series in number theory and I am unable to think about a step in a proof which uses complex analysis but I am not able to recall how to think about it
Can somebody please give some hint. I am not writing here whole proof, just the step which I can't understand but I will make sure it is absolutely clear to reader.
Step is - Apostol mentions a function f(x) and proves it analytic in D= { x : 0 < |x| <1 } which I can understand. Now since f is analytic, so it must have a Laurent expansion about 0 . So, f(x) = $\sum_{n= -\infty}^\infty a(n)x^n $ where assume x=$ exp(2πit) $ ( it is mentioned in proof why x is equal to this particular value)
Now, I can't understand this statement which appears next in text - ** f(x) is absolutely convergent for each x in D ** .
I tried looking for some related theorem in my complex analysis textbook but couldn't find any.
Can somebody please explain this step.
The Laurent series of $f$ has the form
$$f(z)=\sum_{n=-\infty}^{\infty} a_nz^n,\,\,0<|z|<1.$$
You can write $z=re^{it},0<r=|z|<1,t\in \mathbb R.$ In any decent treatment of Laurent expansions it is shown that such a series converges absolutely at each such $z.$ Also shown: the series converges uniformly on compact subsets of $\{0<|z|<1\}.$