Jump to content

Parseval–Gutzmer formula

From Wikipedia, the free encyclopedia

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

[edit]

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

[edit]

Using this formula, it is possible to show that

where

This is done by using the integral

References

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