Complex Natural Logarithm

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1 Complex Natural Logarithm

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 $\left(ln(r),\theta\right)$ .