PlanetMath: proof of Gauss' mean value theorem

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proof of Gauss' mean value theorem

(Proof)

We can parameterize the circle by letting $z=z_0 + r e^{i\phi}$ . Then $dz=ir e^{i\phi}d\phi$ . Using the Cauchy integral formula we can express in the following way:

$\displaystyle f(z_0)$	$\displaystyle =$	$\displaystyle \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{z-z_0}dz$
	$\displaystyle =$	$\displaystyle \frac{1}{2\pi i} \int_{0}^{2\pi} \frac{f(z_0 + r e^{i\phi})}{r e^{i\phi}} ir e^{i\phi} d\phi$
	$\displaystyle =$	$\displaystyle \frac{1}{2\pi} \int_{0}^{2\pi} f(z_0 + r e^{i\phi}) d\phi .$

"proof of Gauss' mean value theorem" is owned by yark. [ full author list (2) | owner history (1) ]

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See Also: Cauchy integral formula

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Cross-references: Cauchy integral formula, circle

This is version 15 of proof of Gauss' mean value theorem, born on 2003-04-28, modified 2006-10-01.
Object id is 4217, canonical name is ProofOfGaussMeanValueTheorem.
Accessed 3299 times total.

Classification:

AMS MSC:

30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)

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