proof of Gauss’ mean value theorem

We can parameterize the circle by letting z=z0+reiϕ. Then dz=ireiϕdϕ. Using the Cauchy integral formulaPlanetmathPlanetmath we can express f(z0) in the following way:

f(z0) = 12πiCf(z)z-z0𝑑z
= 12πi02πf(z0+reiϕ)reiϕireiϕ𝑑ϕ
= 12π02πf(z0+reiϕ)𝑑ϕ.
Title proof of Gauss’ mean value theorem
Canonical name ProofOfGaussMeanValueTheorem
Date of creation 2013-03-22 13:35:36
Last modified on 2013-03-22 13:35:36
Owner yark (2760)
Last modified by yark (2760)
Numerical id 18
Author yark (2760)
Entry type Proof
Classification msc 30E20
Related topic CauchyIntegralFormula