Taylor Series of Holomorphic Function

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Theorem

Let $a \in \C$ be a complex number.

Let $r > 0$ be a real number.

Let $f$ be a function holomorphic on some open ball, $D = B \paren {a, r}$.


Then:

$\displaystyle \map f z = \sum_{n \mathop = 0}^\infty \frac {\map {f^n} a} {n!} \paren {z - a}^n$

for all $z \in D$.


Proof

In Holomorphic Function is Analytic, it is shown that:

$\displaystyle \map f z = \sum_{n \mathop = 0}^\infty \paren {\frac 1 {2 \pi i} \int_{\partial D} \frac {f \left({t}\right)} {\left({t - a}\right)^{n + 1} } \mathrm dt} \paren {z - a}^n$

for all $z \in D$.

From Cauchy's Integral Formula: General Result, we have:

$\displaystyle \frac 1 {2 \pi i} \int_{\partial D} \frac {f \left({t}\right)} {\left({t - a}\right)^{n + 1} } \mathrm dt = \frac {\map {f^n} a} {n!}$

Hence the result.

$\blacksquare$