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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]] | |||
:<math>f(z) = \sum^\infty_{k = 0} a_k z^k,</math> | |||
then for ''z'' = ''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–Hill|year = 1979 | isbn = 0-07-085008-9}} | |||
{{DEFAULTSORT:Parseval-Gutzmer formula}} | |||
[[Category:Complex analysis]] | |||
{{mathanalysis-stub}} |
Revision as of 17:23, 26 February 2013
In mathematics, the Parseval–Gutzmer formula states that, if ƒ is an analytic function on a closed disk of radius r with Taylor series
then for z = reiθ on the boundary of the disk,
Proof
The Cauchy Integral Formula for coefficients states that for the above conditions:
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,
We can apply both of these facts to the problem. Using the second fact,
Now, using our Taylor Expansion on the conjugate,
Using the uniform convergence of the Taylor Series and the properties of integrals, we can rearrange this to be
With further rearrangement, we can set it up ready to use the Cauchy Integral Formula statement
Now, applying the Cauchy Integral Formula, we get
Further Applications
Using this formula, it is possible to show that
This is done by using the integral
References
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