Complex Natural Logarithm

These sketches illustrate the complex natural logarithm and the images of simple lines and the circle. Each file has a Complex Natural Log tool that takes z, the origin, and the unit point as inputs.

The construction is based on the polar form of a complex number $z=r \cdot e^{i \theta}$, as well as other unstated assumptions about complex powers of e and the complex logarithm (one being that they work the same way for complex numbers as they do for real numbers).

We start with $f(z)=ln(z) =ln \left(re^{i\theta}\right) =ln(r)+ln\left(e^{i\theta}\right) =ln(r)+i\theta$

What we see is that the natural log of a complex number $z=r \cdot e^{i \theta}$ is another complex number with rectangular coordinates $\left(ln(r),\theta\right)$.

Image of a Horizontal Line

This file shows the Image of a Horizonal Line under the complex natural log map.

Image of a Vertical Line

This file shows the Image of a Vertical Line under the complex natural log map.

Image of an Arbitrary Line

This file shows the Image of an Arbitrary Line under the complex natural log map.

Image of a Circle

Nothing would be complete without the Image of a Circle under the complex natural log map.