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"

By Cauchy's integral formula

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

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

$ 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 $ |z-a|<|w-a| $ meaning that points of $ a $ and $ z $ lie inside Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C and points of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): w lie on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C and that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C is a disc of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): | w - a | called the circle of convergence of the Taylor's series. See Figure 1.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) = \frac{1}{2\pi i } \sum_{n=0}^{N-1} \left[ \oint_C \frac{ f(w) }{ ( w - a)^{n+1} } \; d w \right] (z - a)^n + R_N

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From Cauchy's integral formulas we recognize

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_N vanishes as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N \rightarrow \infty .

$ 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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |R_N| = \left| \frac{1}{2\pi i } \oint_C \frac{ f(w) }{ ( w - z )} \left( \frac{ z - a }{ w - a} \right)^N \; d w \right| \le \frac{1}{2\pi } \oint_C \left| \frac{ f(w) }{ ( w - z )} \left( \frac{ z - a }{ w - a} \right)^N \right| \; d w

We note that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): | w - z | = | w - a - (z - a) | \ge | w - a | - | z - a |

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |z - a | < | w - a |.

Thus we may define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \gamma = \frac{z -a }{ w -a } < 1

which yields the estimate

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): |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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M follows from the Maximum Modulus Theorem.

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