The book says that:The function $P_{r}(\theta)$ called the Poisson Kernel, is defined for $\theta \in [-\pi,\pi] $ and $0\leq r < 1$ by the absolutely and uniformly convergent series $$P_{r}(\theta) =\sum_{n = -\infty}^{\infty} r^{|n|}e^{in\theta}.$$
And it added: Note that in calculating the Fourier Coefficients of $P_{r}(\theta)$ we can interchange the order of integration and summation since the sum converges uniformly in $\theta$ for each fixed r, and obtain the Fourier coefficient equals $ r^{|n|}$
My question is that I obtained the nth Fourier Coefficient equals $\sum_{-\infty}^{\infty} r^{|n|} $ and not $r^{|n|}$ only, why should the summation be removed? Could anyone explain to me please?
thanks.
I'm not sure but I think that the property the book is refering to comes form the following:
$\langle P_r(\theta),e^{im\theta}\rangle=\langle \sum_{n=-\infty}^{\infty}r^{|n|}e^{in\theta},e^{im\theta}\rangle=\sum_{n=-\infty}^{\infty}r^{|n|}\langle e^{in\theta},e^{im\theta}\rangle=$ $\sum_{n=-\infty}^{\infty}r^{|n|}\delta_{m,n}=r^{|n|}$.
This is because you can integrate each term separately , and you know that $\int_{-\pi}^{\pi}e^{im\theta}\overline{e^{in\theta}}\neq0$ if and only if $m=n$.