# Complex analysis

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

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

## References

[1]
Petra Bonfert-Taylor. Introduction to complex analysis. Course on Coursera.