Integral of series with complex exponentials

Question

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$F(z)=\sum_{n=1}^\infty a_nz^n$$ converges in $|z|<1$. How can I evaluate the integral $$\int\int_{|z|<1}\left|\frac{\partial}{\partial r}F(z)\right|^2(1-|z|)dxdy$$ in terms of the coefficients $a_n$?

Answer 1

Related technique: (I), (II). Here is how you advance. We find the partial derivative w.r.t $r$ of $F(r,\theta)$

$$ F(z)=\sum_{n=1}^\infty a_n r^n e^{in\theta} \implies F_r = \sum_{n=1}^\infty n a_n r^{n-1} e^{in\theta}.$$

$$ \Bigg| \sum_{n=1}^\infty n a_n r^{n-1} e^{in\theta} \Bigg|^2 = \sum_{n=1}^\infty n a_n r^{n-1} e^{ni\theta}\sum_{m=1}^\infty m \bar{a_m } r^{m-1} e^{-im\theta}. $$

$$ 1-|z| = 1-|re^{i\theta}|=1-r $$

Gathering the above, the integral becomes

$$ \sum_{n=1}^\infty \sum_{m=1}^\infty n m a_n \bar{a_m}\int_{0}^{2\pi}\int_{0}^{1} r^{n+m-2}(1-r)e^{i(n-m)\theta} r dr d\theta\,. $$

Now, you just need to integrate and simplify. Note that, the

$$ \int_{0}^{2\pi} e^{i(n-m)\theta}d\theta $$

equals $0$ if $n\neq m$ and $2\pi$ if $n=m$.

Note: We used the fact

$$ |w|^2= w\, \bar {w}. $$