Complex functions or functions of complex numbers are functions that their argument is complex number. In other words, Complex function is are functions whose domain and codomain are subsets of the set of complex numbers C.

While it is easy to visualize real functions — functions whose domain and codomain are subsets of the set of complex numbers C R, It is more difficult to visualize Complex functions. In the following movies we see nice method to visualize Complex functions.

## Visualize Complex Functions

My idea behind the visualization of Complex functions is displaying the graph of the domain and codomain as graph where the x axis display is real number and the y axis is the imaginary part. Yes, this a simple task to do. But you may ask how can we see the mapping between the complex number in the domain maps to the complex number in the codomain? We add a moving red dot that will display this mapping – In other words, the when the dot is displayed in complex number in the domain graph it will also displayed in complex number in the codomain graph.

By doing this, We add another time to visualization.

Let’s see some examples:

## Narrowing

The following demo shows how can we narrow complex domain by k by applying complex function. The domains in the movie are a hexagon and a circle. They are being narrowed by 2,3,4.

## Expansion of Complex Functions

The following demo shows how can we expand complex domain by k by applying complex function. The domains in the movie are a hexagon and a circle. They are being expanded by 2,3,4.

## Rotation

The following demos shows how can we rotate complex domain by angle k by applying complex function.

In the first demo, the domain is a arc(0°,270°) of circle with radius 3. It is being rotated by 60° steps (60°, 120°, 180°, 240°, 300°, 0°).

In the first demo, The domain is the rectangle (0,4,4+4i,4i,0) and it is being rotated by 45° steps (45°, 90°, 135°, 180°, 225°, 270°, 315° and 0°).

## Translation

The following demo show how can we move complex domain by c by applying complex function. The domains in the movie are a helmensicata and a circle. They are being moved by ( 5, 5+5i, 5i, -5+5i, -5. -5-5i, -5i . -5i+5)

## Absolute value and argument

Absolute value and argument are basic complex numbers operators. Let’s see them in action, we see squares,hexagons and circles.

## Summery

What do you think about this method visualization of Complex functions? Do you have more complex function you want to visualize?