In mathematics, the Parseval–Gutzmer formula states that, if
is an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
on a
closed disk of radius ''r'' with
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
:
then for ''z'' = ''re
iθ'' 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
*
Theorems in complex analysis
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