Taylor's theorem for complex valued functions

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Proof of Tayor's theorem for analytic functions

"Figure 1: The circle of convergence C in the complex w plane"

By Cauchy's integral formula

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Adding and subtracting the value in the denominator, and rewriting, we have



We may expand the factor into a geometric series, provided that meaning that points of and lie inside and points of lie on and that is a disc of radius called the circle of convergence of the Taylor's series. See Figure 1.

.


From Cauchy's integral formulas we recognize


The only thing that remains is to show that the remainder vanishes as .

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We note that



Thus we may define


which yields the estimate


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Here the constant factor follows from the Maximum Modulus Theorem.





Named for Brook Taylor (1685–1731), English mathematician.