proof of complex mean-value theorem

The function $h(t)=\\operatorname{Re}\\frac{f(a+t(b-a))-f(a)}{b-a}$ is a function defined on [0,1].We have $h(0)=0$ and $h(1)=\\operatorname{Re}\\frac{f(b)-f(a)}{b-a}$ .By the ordinary mean-value theorem, there is a number $t$ , $0<t<1$ , such that $h^{\\prime}(t)=h(1)-h(0)$ .To evaluate $h^{\\prime}(t)$ , we use the assumption that $f$ is complex differentiable (holomorphic). The derivative of $\\frac{f(a+t(b-a))-f(a)}{b-a}$ is equal to $f^{\\prime}(a+t(b-a))$ , then $h^{\\prime}(t)=\\operatorname{Re}(f^{\\prime}(a+t(b-a)))$ , 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.

单词	ProofOfComplexMeanvalueTheorem
释义	proof of complex mean-value theorem The function $h(t)=\\operatorname{Re}\\frac{f(a+t(b-a))-f(a)}{b-a}$ is a function defined on [0,1].We have $h(0)=0$ and $h(1)=\\operatorname{Re}\\frac{f(b)-f(a)}{b-a}$ .By the ordinary mean-value theorem, there is a number $t$ , $0<t<1$ , such that $h^{\\prime}(t)=h(1)-h(0)$ .To evaluate $h^{\\prime}(t)$ , we use the assumption that $f$ is complex differentiable (holomorphic). The derivative of $\\frac{f(a+t(b-a))-f(a)}{b-a}$ is equal to $f^{\\prime}(a+t(b-a))$ , then $h^{\\prime}(t)=\\operatorname{Re}(f^{\\prime}(a+t(b-a)))$ , 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.
随便看	Ackermann function AckermannFunctionIsNotPrimitiveRecursive Ackermann function is not primitive recursive AckermannFunctionIsTotalRecursive Ackermann function is total recursive AClosedSubsetOfACompleteMetricSpaceIsComplete a closed subset of a complete metric space is complete ACohnsIrreducibilityCriterion A. Cohn’s irreducibility criterion ACompactMetricSpaceIsSecondCountable a compact metric space is second countable ACompactSetInAHausdorffSpaceIsClosed a compact set in a Hausdorff space is closed ACompleteSubspaceOfAMetricSpaceIsClosed a complete subspace of a metric space is closed AConditionOfAlgebraicExtension a condition of algebraic extension AConnectedAndLocallyPathConnectedSpaceIsPathConnected a connected and locally path connected space is path connected AConnectedNormalSpaceWithMoreThanOnePointIsUncountable a connected normal space with more than one point is uncountable ActaMathematica Acta Mathematica AcyclicGraph acyclic graph