GeoGebra

Complex Natural Logarithm

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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 Ggb.gifImage of a Horizonal Line under the complex natural log map.

Image of a Vertical Line

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

Image of an Arbitrary Line

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

Image of a Circle

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