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


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 .


We note that

Thus we may define

which yields the estimate


Here the constant factor follows from the Maximum Modulus Theorem.

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