MATH529: L17 Complex analysis

Overview

• Complex numbers

• Powers and roots

• Sets in the complex plane

• Functions of a complex variable

Powers and roots

• Modulus and argument: $z=r{e}^{i\theta }$, $mod\left(z\right)=\left|z\right|=r$, $\arg \left(z\right)=\theta$

• $z^n$, $n\in \mathbb{Z}$, $z^n={(r{e}^{i\theta })}^n=r^n{e}^{in\theta }$

• Roots, $w$ is $n^{\mathrm{th}}$ root of $z$, $w=\sqrt[n]{z}=z^{1/n}$ if $w^n=z$

  ${\displaystyle w=\rho e^{i\phi },z=r{e}^{i\theta }\Rightarrow {\rho }^n(\cos n\phi +i\sin n\phi )=r(\cos \theta +i\sin \theta )\Rightarrow \rho =r^{1/n}}$ ${\displaystyle \cos n\phi =\cos \theta ,\sin n\phi =\sin \theta \Rightarrow {\phi }_k=\frac{\theta +2k\pi }{n},k=0,1,\dots ,n-1}$ ${\displaystyle w_k=r^{1/n}[\cos \left(\frac{\theta +2k\pi }{n}\right)+i\sin \left(\frac{\theta +2k\pi }{n}\right)],k=0,1,\dots ,n-1}$

• Example $w=\sqrt[6]{1}\Rightarrow w_k=\cos \left(\frac{\theta +2k\pi }{6}\right)+i\sin \left(\frac{\theta +2k\pi }{6}\right)$, $k=0,1,\dots ,5$

Sets in complex plane

• $|z-z_0|=R$, circle of radius $R$ centered at $z_0$

• $\mathrm{Im}\left(z\right)<0$, lower half plane

• $-1<\mathrm{Re}\left(z\right)<1$, centered infinite vertical strip of width 2

• $1<\left|z\right|<2$, circular ring

A neighborhood of $z_0$, $|z-z_0|<\epsilon$, $\epsilon \in {ℝ}_{+}$

Types of points in sets

• $z_0$ is an interior point of set $S$, if all points of a neighborhood of $z_0$ are inside $S$

  ${\displaystyle \exists \epsilon >0,\forall z\in N(z_0,\epsilon )=\{z:|z-z_0|<\epsilon \},z\in S\Rightarrow z_0\mathrm{interior}\mathrm{point}}$

• $S$ is open if it only contains interior points

• $z_0$ is an exterior point of set $S$, if all points of a neighborhood of $z$ are outside $S$

  ${\displaystyle \exists \epsilon >0,\forall z\in N(z_0,\epsilon )=\{z:|z-z_0|<\epsilon \},z\notin S\Rightarrow z_0\mathrm{exterior}\mathrm{point}}$

• $z_0$ is a boundary point of set $S$, every neighborhood of $z\in S$ contains a point in $S$ and a point outside $S$

  ${\displaystyle \forall \epsilon >0,\exists z_1\in N(z_0,\epsilon ),\exists z_2\in N(z_0,\epsilon )=\{z:|z-z_0|<\epsilon \},z_1\in S,z_2\notin S\Rightarrow z_0\mathrm{boundary}\mathrm{point}}$

Types of sets

• $S$ open is connected if all points can be connected by a polygonal line completely within $S$

• $S$ is a domain if it is open and connected

Functions of complex variable

• $w=f\left(z\right)=u(x,y)+iv(x,y)$, $f:ℂ\to ℂ$

• Expect some differences w.r.t. real functions, e.g.

  • $g:{ℝ}_{+}\cup \left\{0\right\}\to ℝ$, $g\left(x\right)=\sqrt{x}$

  • $f:ℂ\to ℂ$, $f\left(z\right)=\sqrt{z}$, careful about possibility of multiple outputs

• Examples:

  • polynomials, e.g., $f\left(z\right)=z^4-3z$

  • rational functions, e.g. $f\left(z\right)=(z^3-2)/(z^3+2)$

  • $f\left(z\right)=z+\mathrm{Re}\left(z\right)$