Taylor Series of Holomorphic Function

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:

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

for all $z \in D$.

Proof

In Holomorphic Function is Analytic, it is shown that:

$\ds \map f z = \sum_{n \mathop = 0}^\infty \paren {\frac 1 {2 \pi i} \int_{\partial D} \frac {\map f t} {\paren {t - a}^{n + 1} } \rd t} \paren {z - a}^n$

for all $z \in D$.

From Cauchy's Integral Formula for Derivatives, we have:

$\ds \frac 1 {2 \pi i} \int_{\partial D} \frac {\map f t} {\paren {t - a}^{n + 1} } \rd t = \frac {\map {f^n} a} {n!}$

Hence the result.

$\blacksquare$