Basic reference on complex analysis. See background definitions for prerequisites.

Notes taken while taking a course [1].

## Limit

If for all \(\varepsilon > 0\) there is \(\delta > 0\) such that \(|f(z) - c| < \varepsilon\) whenever \(|z-z_0| < \delta\), then

\[ \lim_{z \rightarrow z_0} f(z) = c \]

\(f\) is *continuous* at \(z_0\) if

\[ \lim_{z \rightarrow z_0} f(z) = f(z_0) \]

## Derivative

\[ \frac{d}{dz} f(z_0) = \lim_{z \rightarrow z_0} \frac{f(z) - f(z_0)}{z-z_0} \]

\(f\) is *analytic* in an open set \(U \subset \Complex\) if \(f\) is differentiable for every \(z \in U\).
A function which is analytic in \(\Complex\) is an *entire* function.

### Cauchy-Riemann equations

If \(f(z) = u(x,y) + i v(x,y)\) for \(z = x+iy\) and real functions \(u\) and \(v\),

\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \] \[ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]

Also,

\[ \frac{d f}{d z} = \frac{\partial f}{\partial x} = -i\frac{\partial f}{\partial y} \]

\(f=u+iv\) is analytic in \(D\) if and only if for any \(z \in D\), \(\frac{\partial u}{\partial x}\) and \(\frac{\partial v}{\partial y}\) exist, are continuous, and satisfy the Cauchy-Riemann equations.

## Integral

The path integral over a path \(\gamma: [a,b] \rightarrow \Complex\) is

\[ \int_\gamma f(z) \, dz = \lim_{n \rightarrow \infty} \sum_{k=0}^{n-1} f(z_k) (z_{k+1} - z_k) \]

where \(z_k = \gamma(t_k)\) and \(a = t_0 < t_1 < \ldots < t_n = b\) for any \(n>0\).

If \(\gamma\) is a smooth curve and \(f\) is continuous,

\[ \int_\gamma f(z) \, dz = \int_a^b f(\gamma(t)) \, \gamma'(t) \, dt \]

### By arc length

Integration with respect to arc length is defined as

\[ \int_\gamma f(z) \, |dz| = \int_a^b f(\gamma(t)) \, |\gamma'(t)| \, dt \]

#### ML Estimate

If \(\gamma\) is a curve and \(f\) is continuous on \(\gamma\),

\[ \left| \int_\gamma f(z) \, dz \right| \leq \int_\gamma |f(z)| \, |dz| \]

In particular, if \(|f(z)| \leq M\) on \(\gamma\),

\[ \left| \int_\gamma f(z) \, dz \right| \leq M \cdot \mathrm{length}(\gamma) \]

### Primitives

If \(D \subset \Complex\) and \(f : D \rightarrow \Complex\) is continuous,
a *primitive* of \(f\) is an analytic function \(F : D \rightarrow \Complex\)
such that \(\frac{dF}{dz} = f\) on \(D\).
For any curve \(\gamma : [a,b] \rightarrow D\),

\[ \int_\gamma f(z) \, dz = F(\gamma(b)) - F(\gamma(a)) \]

If \(D\) is a **simply** connected domain in \(\Complex\) and \(f\) is analytic in \(D\),
then \(f\) has a primitive in \(D\).

### Cauchy’s theorem

If \(D\) is a simply connected domain in \(\Complex\), \(f\) is analytic in \(D\), and \(\gamma\) is a piecewise smooth and closed curve in \(D\), then

\[ \int_\gamma f(z) \, dz = 0 \]

If \(\gamma_1\) and \(\gamma_2\) are simply closed curves with the same orientation, \(\gamma_2\) is inside \(\gamma_1\), and \(f\) is analytic in a domain which contains both curves and the region between them, then

\[ \int_{\gamma_1} f(z) \, dz = \int_{\gamma_2} f(z) \, dz \]

### Cauchy integral formula

If \(D\) is a simply connected domain bounded by a piecewise smooth curve \(\gamma\), and \(f\) is analytic in a superset of \(\overline{D}\), then for all \(w \in D\)

\[ f(w) = \frac{1}{2 \pi i} \int_\gamma \frac{f(z)}{z-w} \, dz \]

If \(f\) is analytic in an open set, then \(\frac{df}{dz}\) is also analytic in the same open set. For all \(w \in D\) and \(k \geq 0\),

\[ \frac{d^k f}{dz^k}(w) = \frac{k!}{2 \pi i} \int_\gamma \frac{f(z)}{(z-w)^{k+1}} \, dz \]

#### Cauchy’s estimate

If \(f\) is analytic in an open set which contains \(\overline{B_r(z_0)}\) and \(|f(z)| \leq m\) holds on \(\partial B_r(z_0)\), then for all \(k \geq 0\),

\[ \left| \frac{d^k f}{dz}(z_0) \right| \leq \frac{k! \, m}{r^k} \]

### Liouville’s theorem

If \(f\) is analytic in \(\Complex\) and is bounded, then \(f\) must be constant.

### Maximum principle

If \(f\) is analytic in an open set \(D\) and there exists \(z_0 \in D\) such that \(|f(z)| \leq |f(z_0)|\) for all \(z \in D\), then \(f\) is constant in \(D\).

If \(D \subset \Complex\) is a bounded domain, \(f\) is continuous in \(\overline{D}\), and \(f\) is analytic in \(D\), then \(|f|\) has its maximum on \(\partial D\).

## Fundamental theorem of algebra

If \(a_0\), \(\ldots\), \(a_n\) are complex numbers with \(a_n \neq 0\), then the polynomial

\[ p(z) = \sum_{k=0}^n a_k z^k \]

has \(n\) complex roots \(z_1\), \(\ldots\), \(z_n\), where

\[ p(z) = a_n \prod_{k=1}^n (z - z_k) \]