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## Complex Natural Logarithm

## Contents |

# 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 , 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

What we see is that the natural log of a complex number is another complex number with rectangular coordinates .

#### 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.