proof of Gauss’ mean value theorem

We can parameterize the circle by letting $z=z_{0}+re^{i\phi}$. Then $dz=ire^{i\phi}d\phi$. Using the Cauchy integral formula we can express $f(z_{0})$ in the following way:

 $\displaystyle f(z_{0})$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi i}\oint_{C}\frac{f(z)}{z-z_{0}}dz$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi i}\int_{0}^{2\pi}\frac{f(z_{0}+re^{i\phi})}{re^{i% \phi}}ire^{i\phi}d\phi$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}f(z_{0}+re^{i\phi})d\phi.$
Title proof of Gauss’ mean value theorem ProofOfGaussMeanValueTheorem 2013-03-22 13:35:36 2013-03-22 13:35:36 yark (2760) yark (2760) 18 yark (2760) Proof msc 30E20 CauchyIntegralFormula