# Taylor's theorem for complex valued functions

## Proof of Tayor's theorem for analytic functions

"Figure 1: The circle of convergence C in the complex w plane"
${\displaystyle f(z)={\frac {1}{2\pi i}}\oint _{C}{\frac {f(w)}{w-z}}\;dw}$

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Adding and subtracting the value ${\displaystyle a}$ in the denominator, and rewriting, we have

${\displaystyle f(z)={\frac {1}{2\pi i}}\oint _{C}{\frac {f(w)}{w-a}}\left({\frac {1}{1-{\frac {z-a}{w-a}}}}\right)\;dw}$

We may expand the factor into a geometric series, provided that ${\displaystyle |z-a|<|w-a|}$ meaning that points of ${\displaystyle a}$ and ${\displaystyle z}$ lie inside ${\displaystyle C}$ and points of ${\displaystyle w}$ lie on ${\displaystyle C}$ and that ${\displaystyle C}$ is a disc of radius ${\displaystyle |w-a|}$ called the circle of convergence of the Taylor's series. See Figure 1.

${\displaystyle f(z)={\frac {1}{2\pi i}}\sum _{n=0}^{N-1}\left[\oint _{C}{\frac {f(w)}{(w-a)^{n+1}}}\;dw\right](z-a)^{n}+R_{N}}$

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From Cauchy's integral formulas we recognize

${\displaystyle {\frac {1}{2\pi i}}\oint _{C}{\frac {f(w)}{(w-a)^{n+1}}}\;dw={\frac {f^{(n)}(a)}{n!}}.}$

The only thing that remains is to show that the remainder ${\displaystyle R_{N}}$ vanishes as ${\displaystyle N\rightarrow \infty }$.

${\displaystyle R_{N}={\frac {1}{2\pi i}}\oint _{C}{\frac {f(w)}{(w-z)}}\left({\frac {z-a}{w-a}}\right)^{N}\;dw}$

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${\displaystyle |R_{N}|=\left|{\frac {1}{2\pi i}}\oint _{C}{\frac {f(w)}{(w-z)}}\left({\frac {z-a}{w-a}}\right)^{N}\;dw\right|\leq {\frac {1}{2\pi }}\oint _{C}\left|{\frac {f(w)}{(w-z)}}\left({\frac {z-a}{w-a}}\right)^{N}\right|\;dw}$

We note that

${\displaystyle |w-z|=|w-a-(z-a)|\geq |w-a|-|z-a|}$

${\displaystyle |z-a|<|w-a|.}$

Thus we may define

${\displaystyle \gamma ={\frac {z-a}{w-a}}<1}$

which yields the estimate

${\displaystyle |R_{n}|<{\frac {1}{2\pi }}M(2\pi |w-a|){\frac {\gamma ^{N}}{|w-a|-|z-a|}}={\frac {M\gamma ^{N}|w-a|}{|w-a|-|z-a|}}\rightarrow 0\qquad {\mbox{as}}\qquad N\rightarrow \infty }$

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Here the constant factor ${\displaystyle M}$ follows from the Maximum Modulus Theorem.

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