Problem 1557 (difficulty: 6/10)

Assume that \(\displaystyle f(z)=\displaystyle\sum_{n=1}^\infty a_nz^n\) is convergent on the disc \(\displaystyle |z|<r+\varepsilon\). Prove that

\(\displaystyle \frac1{2\pi r}\int\limits_{|z|=r}|f(z)|^2\cdot|\dz|= \sum_{n=0}^\infty|a_n|^2r^{2n}.\)

(Parseval-formula for power series)

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