Parseval–Gutzmer formula
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
gollark: It's just all hilariously unusably slow.
gollark: Oh hey, matrix→IRC does exist.
gollark: The obvious solution is to implement the system we discussed on esoserver some time ago, and also to actually design that coherently, and to somehow attain arbitrary quantities of motivation and programming skills.
gollark: I don't reaaaally want to figure out how to interact with its APIs.
gollark: Wasn't there an APIONET→MCIR bridge?
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