# Functions of complex variable

The field $\mathbb C$ is naturally extending the field $\mathbb R$, which means that all arithmetic operations on $\mathbb R$ extend naturally as operations on $\mathbb C$. In particular, any polynomial $p(x)=a_0x^n+a_1 x^{n-1}+\cdots+a_n$ can be interpreted as a map $p:\mathbb C\to\mathbb C$. Geometrically, this can be visualized as a map of the 2-plane to the 2-plane.

We discussed the maps $p(x)=x^2$ and $p(x)=1/x$.

Smooth functions are those which can be accurately approximated by the $\mathbb R$-affine maps $x\mapsto c+\lambda(x-a)$ near each point $a\in\mathbb R$. Functions that can be accurately approximated by $\mathbb C$-affine maps (the same formula, but over the complex numbers), are called holomorphic (or complex analytic). Such maps are characterized by the property that small circles are mapped into small almost-circles, that is,

1. angles are preserved, and
2. lengths are scaled

Sometimes these local conditions become global. Examples: the affine maps $x\mapsto \lambda (x-a)+b$ send lines to lines and circles to circles. The map $x\mapsto 1/x$ maps circles and lines into circles or lines (depending on whether the circles/lines pass through the origin $x=0$).

## Complex integration

Integration is carried over smooth (or piecewise smooth) paths in $\mathbb C$, using Riemann-like sums. It depends linearly on the function which we integrate, but in contrast with the real case we have much more freedom in choosing the paths.

1. If $f(x)=c$ is a constant, and $\gamma =[p_0,p_1]+[p_1,p_2]+[p_2,p_0]$ is a closed triangle, then the integral is zero as the sum $c(p_1-p_0)+c(p_2-p_1)+c(p_0-p_2)=c\cdot0=0$.
2. If $f(x)$ is non-constant, then the integral identity is valid only for “very small triangles” near a point $a\in\mathbb C$ with $c=f(a)$.
3. This implies that $\displaystyle \int_\gamma f(x)\,\mathrm dx=\int_{\gamma'} f(x)\,\mathrm dx$ as long as the paths $\gamma,\gamma'$ share the common endpoints and can be continuously deformed one into the other.

Integrals over closed loops are zeros, unless there are singular points (where the function is non-holomoprhic) inside.

Example: $f(x)=\frac 1x$ is non-holomorphic at $x=0$, and $\displaystyle \oint_{|x|=1}\tfrac 1x\,\mathrm dx=2\pi i$.

## Cauchy integral formula

If $f$ is holomorphic inside a domain $U$ bounded by a closed curve $\gamma$ and $a\in U$, then $\displaystyle f(a)=\frac 1{2\pi i}\oint_\gamma \frac{f(x)\,\mathrm dx}{x-a}$.

In other words, the value of $f$ on the boundary uniquely determine its values inside the domain. This is in wild contrast with functions of real variable!

## Taylor series

As a function of $a$, the expression $\frac1{x-a}$ admits a converging Taylor expansion (in fact, the same old geometric progression series) in powers of $a-a_0$ for any $a_0\ne x$. Thus if we choose $a_0\in U$ off the path $\gamma$, then the series will converge for any $x\in\gamma$ (warning! note that the “variable” and the “parameter” exchanged their roles!!!), the Cauchy integral can be expanded in the converging series of powers $(a-a_0)^n, \ n=0,1,2,\dots$, hence the function $f(x)$ gets expanded in the series $f(a)=c_0+c_1(a-a_0)+c_2(a-a_0)^2+\cdots$.

Conclusion: functions that are $\mathbb C$-differentiable can be expanded in the convergent Taylor series (hence are “polynomials of infinite degree”) and vice versa, “polynomials of infinite degree” are infinitely $\mathbb C$-differentiable. This is a miracle that so many functions around us are actually holomorphic!