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Prove Parseval's identity using complex Fourier series

Prove Parseval's identity using complex Fourier series

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Answer #1

Since we are talking about complex series, there can be 2 cases: One, proving Parseval's identity for real valued functions using complex fourier series, and two, proving Parseval's identity for complex valued functions using the same.

We start with the first.

We want to show that :

Proof:

For the second case, we would like to prove:

where is a holomorphic function defined on the ball with power series .

Proof:

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