I am seeking a proof for the following statement of the Parseval identity:

If $f$ is a holomorphic function defined on the ball $B(0,r)$, with power series $f(z) = \sum_n a_n z^n$, then $$2 \pi \left( \sum_n |a_n|^2 r^{2n} \right) = \int_0^{2\pi}|f(re^{it})|^2dt.$$

I have seen some proofs of other statements of Parseval identity (specifically using Fourier series), but not of this. I guess I'm looking for one of two possible answers:

1) A proof of the quoted result above (or an outline thereof), or

2) A reference to a standard proof of some statement of the Parseval identity, along with some hints as to how to obtain "my" statement of it from the other (having trouble seeing how to transition).

Thanks in advance for any help!


$$\begin{align}\int_0^{2\pi} |f(re^{it})|^2dt &= \int_0^{2\pi}\overline{f(re^{it})}f(re^{it})dt\\ &=\int_0^{2\pi}\sum_m \overline{a_m}r^me^{-imt}\sum_n {a_n}r^ne^{int}\ dt\\ &=\sum_{m,n}\overline{a_m}a_nr^{m+n}\int_0^{2\pi}e^{it(m-n)}dt\\ &=\sum_{m,n}\overline{a_m}a_nr^{m+n}2\pi\delta_{mn}\\ &=2\pi\sum_n\overline{a_n}a_nr^{n+n}\\ &=2\pi\sum_n |a_n|^2r^{2n}\\ \end{align}$$


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