Parseval–Gutzmer Formula

In mathematics, the Parseval–Gutzmer formula states that, if $f$ is an analytic function on a closed disk of radius ''r'' with Taylor series

:$f(z) = \sum^\infty_ a_k z^k,$

then for ''z'' = ''re^iθ'' on the boundary of the disk,

:$\int^_0 , f(re^) , ^2 \, \mathrm\theta = 2\pi \sum^\infty_ , a_k, ^2r^,$

which may also be written as

:$\frac\int^_0 , f(re^) , ^2 \, \mathrm\theta = \sum^\infty_ , a_k r^k, ^2.$

Proof

The Cauchy Integral Formula for coefficients states that for the above conditions:

:$a_n = \frac \int^_ \frac \, \mathrm z$

where ''γ'' is defined to be the circular path around origin of radius ''r''. Also for $x \in \Complex,$ we have: $\overline = , x, ^2.$ Applying both of these facts to the problem starting with the second fact:

: \begin
\int^_0 \left , f \left (re^ \right ) \right , ^2 \, \mathrm\theta &= \int^_0 f \left (re^ \right ) \overline \, \mathrm\theta\\ pt&= \int^_0 f \left (re^ \right ) \left (\sum^\infty_ \overline \right ) \, \mathrm\theta && \text \\ pt&= \int^_0 f \left (re^ \right ) \left (\sum^\infty_ \overline \left (re^ \right )^k \right ) \, \mathrm\theta \\ pt&= \sum^\infty_ \int^_0 f \left (re^ \right ) \overline \left (re^ \right )^k \, \mathrm \theta && \text \\ pt&= \sum^\infty_ \left (2\pi \overline r^ \right ) \left (\frac\int^_0 \frac \right ) \mathrm\theta \\
& = \sum^\infty_ \left (2\pi \overline r^ \right ) a_k && \text \\
& = \sum^\infty_
\end

Further Applications

Using this formula, it is possible to show that

:$\sum^\infty_ , a_k, ^2r^ \leqslant M_r^2$

where

:$M_r = \sup\.$

This is done by using the integral

:$\int^_0 \left , f \left (re^ \right ) \right , ^2 \, \mathrm\theta \leqslant 2\pi \left, \max_ \left (f \left (re^ \right ) \right ) \right , ^2 = 2\pi\left , \max_(f(z)) \right , ^2 = 2\pi M_r^2$

References

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Theorems in complex analysis

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