In mathematics, the Parseval–Gutzmer formula states that, if
is an analytic function on a closed disk of radius r with Taylor series
![{\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}z^{k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/372f324efd09e4b5152c355949d7fdfb31419178)
then for z = reiθ on the boundary of the disk,
![{\displaystyle \int _{0}^{2\pi }|f(re^{i\theta })|^{2}\,\mathrm {d} \theta =2\pi \sum _{k=0}^{\infty }|a_{k}|^{2}r^{2k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3336cf64781193d57a670cfc7845668b26fc9d5a)
which may also be written as
![{\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }|f(re^{i\theta })|^{2}\,\mathrm {d} \theta =\sum _{k=0}^{\infty }|a_{k}r^{k}|^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29ce141ae62a904a30fe5da00c1e78b84445a85d)
Proof
The Cauchy Integral Formula for coefficients states that for the above conditions:
![{\displaystyle a_{n}={\frac {1}{2\pi i}}\int _{\gamma }^{}{\frac {f(z)}{z^{n+1}}}\,\mathrm {d} z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/600470bf379b545e20de8eeb978030358a157206)
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:
![{\displaystyle {\begin{aligned}\int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{2}\,\mathrm {d} \theta &=\int _{0}^{2\pi }f\left(re^{i\theta }\right){\overline {f\left(re^{i\theta }\right)}}\,\mathrm {d} \theta \\[6pt]&=\int _{0}^{2\pi }f\left(re^{i\theta }\right)\left(\sum _{k=0}^{\infty }{\overline {a_{k}\left(re^{i\theta }\right)^{k}}}\right)\,\mathrm {d} \theta &&{\text{Using Taylor expansion on the conjugate}}\\[6pt]&=\int _{0}^{2\pi }f\left(re^{i\theta }\right)\left(\sum _{k=0}^{\infty }{\overline {a_{k}}}\left(re^{-i\theta }\right)^{k}\right)\,\mathrm {d} \theta \\[6pt]&=\sum _{k=0}^{\infty }\int _{0}^{2\pi }f\left(re^{i\theta }\right){\overline {a_{k}}}\left(re^{-i\theta }\right)^{k}\,\mathrm {d} \theta &&{\text{Uniform convergence of Taylor series}}\\[6pt]&=\sum _{k=0}^{\infty }\left(2\pi {\overline {a_{k}}}r^{2k}\right)\left({\frac {1}{2{\pi }i}}\int _{0}^{2\pi }{\frac {f\left(re^{i\theta }\right)}{(re^{i\theta })^{k+1}}}{rie^{i\theta }}\right)\mathrm {d} \theta \\&=\sum _{k=0}^{\infty }\left(2\pi {\overline {a_{k}}}r^{2k}\right)a_{k}&&{\text{Applying Cauchy Integral Formula}}\\&={2\pi }\sum _{k=0}^{\infty }{|a_{k}|^{2}r^{2k}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4226b99415ef1d1ce74760cd96c184e0ddd44b91)
Further Applications
Using this formula, it is possible to show that
![{\displaystyle \sum _{k=0}^{\infty }|a_{k}|^{2}r^{2k}\leqslant M_{r}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/976957e3a62387d4c1e7a6ec1b6e9b96ff493076)
where
![{\displaystyle M_{r}=\sup\{|f(z)|:|z|=r\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68aa8d684701007c01221d4a701d3e6a6a81a3ff)
This is done by using the integral
![{\displaystyle \int _{0}^{2\pi }\left|f\left(re^{i\theta }\right)\right|^{2}\,\mathrm {d} \theta \leqslant 2\pi \left|\max _{\theta \in [0,2\pi )}\left(f\left(re^{i\theta }\right)\right)\right|^{2}=2\pi \left|\max _{|z|=r}(f(z))\right|^{2}=2\pi M_{r}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61d7493e06f35cf83c0cd8f98af73c3db67275f8)
References