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In mathematics, the '''Parseval–Gutzmer formula''' states that, if ''ƒ'' is an [[analytic function]] on a [[closed disk]] of radius ''r'' with [[Taylor series]]
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:<math>f(z) = \sum^\infty_{k = 0} a_k z^k,</math>
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then for ''z''&nbsp;=&nbsp;''re''<sup>''iθ''</sup> on the boundary of the disk,
 
:<math>\int^{2\pi}_0 |f(re^{i\vartheta}) |^2 \, \mathrm{d}\vartheta = 2\pi \sum^\infty_{k = 0} |a_k|^2r^{2k}. </math>
 
== Proof ==
 
The Cauchy Integral Formula for coefficients states that for the above conditions:
:<math>a_n = \frac{1}{2\pi i} \int^{}_{\gamma} \frac{f(z)}{z^{n+1}} \, \mathrm{d}\ z </math>
 
where γ is defined to be the circular path around 0 of radius r. We also have that, for x in the [[complex plane]] C,
 
:<math>\overline{x}{x} = |x|^2</math>
 
We can apply both of these facts to the problem. Using the second fact,
 
:<math>\int^{2\pi}_0 |f(re^{i\vartheta}) |^2 \, \mathrm{d}\vartheta = \int^{2\pi}_0 {f(re^{i\vartheta})}\overline{f(re^{i\vartheta})}  \, \mathrm{d}\vartheta</math>
 
Now, using our Taylor Expansion on the conjugate,
 
:<math> = \int^{2\pi}_0 {f(re^{i\vartheta})}{\sum^\infty_{k = 0} \overline{a_k (re^{i\vartheta})^k}}  \, \mathrm{d}\vartheta</math>
 
Using the uniform convergence of the Taylor Series and the properties of integrals, we can rearrange this to be
 
:<math> = \sum^\infty_{k = 0} \int^{2\pi}_0 \frac{{f(re^{i\vartheta})}\overline{a_k} (r^k)}{(e^{i\vartheta})^k} , \mathrm{d}\vartheta</math>
 
With further rearrangement, we can set it up ready to use the Cauchy Integral Formula statement
 
:<math> = \sum^\infty_{k = 0} ({2\pi}{\overline{a_k} r^{2k}})(\frac{1}{2{\pi}i}\int^{2\pi}_0  \frac{{f(re^{i\vartheta})}}{(r e^{i\vartheta})^{k+1}} {rie^{i\vartheta}}) \mathrm{d}\vartheta</math>
 
Now, applying the Cauchy Integral Formula, we get
 
:<math> = \sum^\infty_{k = 0} ({2\pi}{\overline{a_k} r^{2k}}){a_k} = {2\pi} \sum^\infty_{k = 0} {|a_k|^2 r^{2k}}</math>
 
== Further Applications ==
Using this formula, it is possible to show that
 
:<math>\sum^\infty_{k = 0} |a_k|^2r^{2k} \le {M_r}^2</math>  where <math>M_r = \sup\{|f(z)| : |z| = r\}</math>
 
This is done by using the integral
 
:<math>\int^{2\pi}_0 |f(re^{i\vartheta}) |^2 \, \mathrm{d}\vartheta \le 2\pi |max_{\vartheta \in [0,2\pi)}(f(re^{i\vartheta}))|^2 = 2\pi |max_{|z|=r}(f(z))|^2 = 2\pi(M_r)^2</math>
 
== References ==
*{{cite book|title = Complex Analysis|authorlink = Lars Ahlfors|first = Lars|last = Ahlfors|publisher = McGraw&ndash;Hill|year = 1979 | isbn = 0-07-085008-9}}
 
{{DEFAULTSORT:Parseval-Gutzmer formula}}
[[Category:Complex analysis]]
 
 
{{mathanalysis-stub}}

Latest revision as of 12:38, 21 July 2014

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