Parseval–Gutzmer formula

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In mathematics, the Parseval–Gutzmer formula states that, if ƒ is an analytic function on a closed disk of radius r with Taylor series

then for z = re on the boundary of the disk,

Proof[edit]

The Cauchy Integral Formula for coefficients states that for the above conditions:

where γ is defined to be the circular path around 0 of radius r. We also have that, for x in the complex plane C,

We can apply both of these facts to the problem. Using the second fact,

Now, using our Taylor Expansion on the conjugate,

Using the uniform convergence of the Taylor Series and the properties of integrals, we can rearrange this to be

With further rearrangement, we can set it up ready to use the Cauchy Integral Formula statement

Now, applying the Cauchy Integral Formula, we get

Further Applications[edit]

Using this formula, it is possible to show that

where

This is done by using the integral

References[edit]