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## Complex Cosine

## Contents |

# Complex Cosine

These files show the images of various lines under the complex mapping *f*(*z*) = *c**o**s*(*z*). All of them are, essentially, applications of the complex powers of e. The constructions are based on .

#### Image of a Horizontal Line

The image of a horizontal line is an ellipse whose foci are (1,0) and (-1,0). As z moves along this line, cos(z) makes one complete revolution about the ellipse for every increment of 2π that z moves horizontally.

#### Image of a Vertical Line

The image of a vertical line is one branch of a hyperbola whose foci are (1,0) and (-1,0). As z moves along the vertical line, the hyperbola oscilates back and forth between these two foci, making one complete oscilation for every increment of 2π that z moves along the vertical line.

#### Image of an Arbitrary Line

The image of an arbitrary line is just a very pretty curve!

#### Image of the *x* Axis

When z is restricted to the x-axis, the cos(z) oscilates between (-1,0) and (1,0) along the x-axis. This file shows the relationship between the x-coordinates of z and of cos(z)and how it produces the familiar cosine graph.