Integral Theorems

If \( f \) is holomorphic on a simply connected domain \( D \), and \( \gamma \) is a closed curve, then:

\[ \oint_\gamma f(z) dz = 0 \]

Let:

1. \( D \) be the closed disk \( D = \{ z : |z - z_0| \le r \} \) centered at \( z_0 \)
2. \( f: D \to \mathbb{C} \) be a holomorphic function
3. \( \gamma \) be the counterclockwise circle forming the boundary of \( D \)

Then:

\[ f(a) = \frac{1}{2\pi i}\oint_\gamma \frac{f(z)}{z - a} dz \]

Holds for all \( a \) in the interior of \( D \).

Let:

1. \( a_n \in U \) be some finite set of points
2. \( D = U \backslash a_n \) be a simply connected domain
3. \( f: D \to \mathbb{C} \) a holomorphic function
4. \( \gamma \) a closed contour in \( D \)

Then:

\[ \oint_\gamma f(z) dz = 2\pi i \sum Res(f, a_k) \]

It is worth noting that CRT is a generalisation of both of the theorems above, the first follows by noting \( a_n \) is empty. For the second:

\begin{align*} \oint_\gamma \frac{f(z)}{z - a} dz &= 2\pi i \ Res\left(\frac{f(z)}{z - a}, a\right) \\ &= 2\pi i \ f(a) \end{align*}