Complex Analysis

Complex analysis investigates functions of complex numbers.

Functions

The polynomial map

z\mapsto z^3

cubes the distance from the origin and triples the argument.

A complex polynomial is a function of the form

\begin{equation*}p(z)=a_0+a_1z+\cdots+a_nz^n,\end{equation*}

where

a_0,\ldots,a_n\in\mathbb{C}

. Even the simple monomial

z^3

has a clear geometric effect: if

z=re^{i\theta}

, then

\begin{equation*}z^3=r^3e^{i3\theta}.\end{equation*}

The exponential map

z=x+iy\mapsto e^z=e^xe^{iy}

sends vertical strips to annuli and horizontal strips to sectors.

For

z=x+iy

, the complex exponential separates into a radial and angular part:

\begin{equation*}e^z=e^xe^{iy}.\end{equation*}

Thus the real part

x

controls the radius, while the imaginary part

y

controls the argument.

The logarithm map

z=re^{i\theta}\mapsto \operatorname{Log}z=\log r+i\theta

sends annular sectors to rectangles.

The logarithm reverses the geometric effect of the exponential map. In polar coordinates,

\begin{equation*}\operatorname{Log}(re^{i\theta})=\log r+i\theta,\end{equation*}

so radii become vertical lines in the logarithm plane, and angles become horizontal lines.

[Needham2023]Needham, Tristan. Visual Complex Analysis. 25th anniversary edition. Oxford University Press, 2023.