Parseval–Gutzmer formula explained

In mathematics, the Parseval - Gutzmer formula states that, if

f is an analytic function on a closed disk of radius r with Taylor seriesf(z) = \sum ^{inf ty} _{k =0} a _{k} z ^{k,}

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

\int ^{2 \pi} _{0} |f(re ^{i\theta} ) | ^{2} d \theta = 2 \pi \sum ^{inf ty} _{k =0} |a ^{2 r} _{k|} ^{2k} ,

which may also be written as

(1)/(2 \pi) \int ^{2 \pi} _{0} |f(re ^{i\theta} ) | ^{2} d \theta = \sum ^{inf ty} _{k =0} |a _{k} r ^{k|} ^{2.}

Proof

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

a _{n} = (1)/(2 \pi i) \int _{\gamma} (f(z))/(z ^{n+1}) d z

where γ is defined to be the circular path around origin of radius r. Also for

x \in \Complex, we have: \overline{x}{x} = |x| ^{2.} Applying both of these facts to the problem starting with the second fact:\begin{align} \int ^{2 \pi} _{0} \left |f \left (re ^{i\theta} \right ) \right | ^{2} d \theta & = \int ^{2 \pi} _{0} f \left (re ^{i\theta} \right ) \overline{f \left (re ^{i\theta} \right )} d \theta\\[ 6 pt] & = \int ^{2 \pi} _{0} f \left (re ^{i\theta} \right ) \left (\sum ^{inf ty} _{k =0} \overline{a _{k} \left (re ^{i\theta} \right ) ^{k}} \right ) d \theta && UsingTaylorexpansionontheconjugate \\[ 6 pt] & = \int ^{2 \pi} _{0} f \left (re ^{i\theta} \right ) \left (\sum ^{inf ty} _{k =0} \overline{a _{k}} \left (re ^{-i\theta} \right ) ^{k} \right ) d \theta \\[ 6 pt] & = \sum ^{inf ty} _{k =0} \int ^{2 \pi} _{0} f \left (re ^{i\theta} \right ) \overline{a _{k}} \left (re ^{-i\theta} \right ) ^{k} d \theta && UniformconvergenceofTaylorseries \\[ 6 pt] & = \sum ^{inf ty} _{k =0} \left ( 2 \pi \overline{a _{k}} r ^{2k} \right ) \left ( (1)/(2 {\pi) i}\int ^{2 \pi} _{0} (f \left (re ^{i\theta} \right ))/((r e ^{i\theta} ) ^{k+1}) {rie ^{i\theta} } \right) \mathrm\theta \\& = \sum^\infty_ \left (2\pi \overline r^ \right) a_k && \text \\& = \sum^\infty_

^2 r^

\end

Further Applications

Using this formula, it is possible to show that

\sum ^{inf ty} _{k =0} |a ^{2 r} _{k|} ^{2k} \leqslant M ^{2} _{r}

where

M _{r} = \sup\{|f(z)| : |z| = r\}.

This is done by using the integral

\int ^{2 \pi} _{0} \left |f \left (re ^{i\theta} \right ) \right | ^{2} d \theta \leqslant 2 \pi \left| max _{\theta} \left (f \left (re ^{i\theta} \right ) \right ) \right | ^{2} = 2 \pi\left | max _{|z|=r} (f(z)) \right | ^{2} = 2 \pi M ^{2} _{r}

References

• Book: Ahlfors, Lars. Lars Ahlfors

. Complex Analysis. Lars Ahlfors. McGraw - Hill. 1979 . 0-07-085008-9.