proof of complex mean-value theorem

The function h(t)=Ref(a+t(b-a))-f(a)b-a is a function defined on [0,1]. We have h(0)=0 and h(1)=Ref(b)-f(a)b-a. By the ordinary mean-value theorem, there is a number t, 0<t<1, such that h(t)=h(1)-h(0). To evaluate h(t), we use the assumption that f is complex differentiableMathworldPlanetmath (holomorphic). The derivative of f(a+t(b-a))-f(a)b-a is equal to f(a+t(b-a)), then h(t)=Re(f(a+t(b-a))), so u=a+t(b-a) satisfies the required equation. The proof of the second assertion can be deduced from the result just proved by applying it to the function f multiplied by i.

Title proof of complex mean-value theorem
Canonical name ProofOfComplexMeanvalueTheorem
Date of creation 2013-03-22 14:34:39
Last modified on 2013-03-22 14:34:39
Owner Wolfgang (5320)
Last modified by Wolfgang (5320)
Numerical id 21
Author Wolfgang (5320)
Entry type Proof
Classification msc 26A06