In mathematics, the Parseval–Gutzmer formula states that, if is an analytic function on a closed disk of radius r with Taylor series
then for z = reiθ on the boundary of the disk,
which may also be written as
Proof
The Cauchy Integral Formula for coefficients states that for the above conditions:
where γ is defined to be the circular path around origin of radius r. Also for we have: Applying both of these facts to the problem starting with the second fact:
Further Applications Using this formula, it is possible to show that
where
This is done by using the integral
References