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 = re on the boundary of the disk,

which may also be written as

Proof

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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

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Using this formula, it is possible to show that

 

where

 

This is done by using the integral

 

References

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  • Ahlfors, Lars (1979). Complex Analysis. McGraw–Hill. ISBN 0-07-085008-9.