Cauchy's Integral Formula/General Result

Theorem
Let $D = \set {z \in \C: \left\lvert{z}\right\rvert \le r}$ be the closed disk of radius $r$ in $\C$.

Let $f: U \to \C$ be holomorphic on some open set $U$ such that $D \subseteq U$. Let $n \in \N$ be a natural number.

Then for each $a$ in the interior of $D$:


 * $\displaystyle f^{\paren n} \paren a = \dfrac {n!} {2 \pi i} \int_{\partial D} \frac {\map f z} {\paren {z - a}^{n + 1} } \rd z$

where $\partial D$ is the boundary of $D$, and is traversed anticlockwise.

Proof
Proof by induction:

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:


 * $\displaystyle f^{\left({n}\right)} \left({a}\right) = \frac {n!} {2 \pi i} \int_{\partial D} \frac {f \left({z}\right)} {\left({z - a}\right)^{n + 1} } \rd z$

Basis for the Induction
$P \left({0}\right)$ holds, as this is:


 * $\displaystyle f \left({a}\right) = \frac 1 {2 \pi i} \int_{\partial D} \frac {f \left({z}\right)} {\left({z - a}\right)} \rd z$

which is Cauchy's Integral Formula.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 0$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:


 * $\displaystyle f^{\left({k}\right)} \left({a}\right) = \frac {k!} {2 \pi i} \int_{\partial D} \frac {f \left({z}\right)} {\left({z - a}\right)^{k + 1} } \rd z$

Then we need to show:


 * $\displaystyle f^{\left({k + 1}\right)} \left({a}\right) = \frac {\left({k + 1}\right)!} {2 \pi i} \int_{\partial D} \frac {f \left({z}\right)} {\left({z - a}\right)^{k + 2} } \rd z$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall n \in \N: f^{\left({n}\right)} \left({a}\right) = \frac {n!} {2 \pi i} \int_{\partial D} \frac {f \left({z}\right)} {\left({z - a}\right)^{n + 1} } \rd z$

Also known as
This result can also be referred to as Cauchy's Integral Formula for Derivatives.