Complex Analysis Midterm Review

Basics

Differentiable / Holomorphic

\(f: \Omega \to \mathbb{C}\) is differentiable at \(z_0 \in \Omega\) if the limit exists: (\(h \in \mathbb{C}\))

\[ f'(z) := \lim_{h \to 0} \frac{f(z + h) - f(z)}{h} = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0} \]

Domain

Open & Connected.

Connected

Can't split into disjoint open sets.

Simply Connected

Connected & no holes.

Tips and Tricks

Norm

\[ N(z) := |z|^2 = z \cdot \bar{z} = x^2 + y^2 \]

Facts

Cauchy-Riemann

Write \(f(z) = u + iv\).

\(f\) is differentiable at \(z\)
\(\iff\)
\(\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y}\) & \(\dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x}\)

Identity Theorem

Suppose \(f\) & \(g\) are holomorphic on a domain \(D\). \(S \subset D\) has an accumulation point of \(D\).

\(f = g\) on \(S\)
\(\Rightarrow\)
\(f = g\) on \(D\)

Cauchy Integral Formula

\(\Omega\) is a bounded, simply connected Domain, & \(f\) is holomorphic on \(\overline{\Omega}\). \(\forall a \in \Omega\):

\[ \Rightarrow \quad f(a) = \frac{1}{2\pi i} \cdot \oint_{\partial \Omega} \frac{f(z)}{z - a}\, dz \]

&

\[ f^{(n)}(a) = \frac{n!}{2\pi i} \cdot \oint_{\partial \Omega} \frac{f(z)}{(z - a)^{n+1}}\, dz \]

Cauchy's Theorem

\(\Omega\) is a bounded, simply connected Domain, & \(f\) is holomorphic on \(\overline{\Omega}\).

\[ \Rightarrow \quad 0 = \oint_{\partial \Omega} f(z)\, dz \]

Liouville's Theorem

\(f\) is entire & bounded
\(\Rightarrow\)
\(f\) is constant

Cauchy Bounds

\(f\) is holomorphic on \(\overline{\Omega}\), where \(B_R(a) \subset \Omega\).

\[ \Rightarrow \quad |f^{(n)}(a)| \leq \frac{n!}{R^n} \cdot \max_{|z - a| = R} |f(z)| \]

M-L Inequality

\(f\) is holomorphic on \(\overline{\Omega}\). Simply connected.

\[ \left| \oint_{\partial \Omega} f(z)\, dz \right| \leq M \cdot L \]

where \(M = \displaystyle\max_{z \in \partial \Omega} |f(z)|\) & \(L = \text{length}(\partial \Omega)\).

Residue Theorem

\(f\) holomorphic on \(\overline{\Omega}\), simply connected, except for \(z_1, \ldots, z_n\).

\[ \Rightarrow \quad \oint_{\partial \Omega} f(z)\, dz = 2\pi i \cdot \sum_{i=1}^{n} \underset{z_i}{\operatorname{Res}}[f(z)] \]

@ multiplicity \(m\)...

\[ \underset{z_i}{\operatorname{res}}(f) = \frac{1}{(m-1)!} \cdot \lim_{z \to z_i} \left( \frac{d^{m-1}}{dz^{m-1}} \left[ (z - z_i)^m \cdot f(z) \right] \right) \]

@ multiplicity \(m = 1\)...

\[ \underset{z_i}{\operatorname{res}}(f) = \lim_{z \to z_i} \left[ (z - z_i) \cdot f(z) \right] \]