matematicas visuales visual math
The Complex Cosine Function: a geometric approach

The real cosine function can be extended using the exponential function:

The Complex Cosine Function formula  | matematicasvisuales

Since the exponential function is periodic with period ,

The Complex Exponential Function is periodic | matematicasvisuales

it follows that the cosine function is periodic, but with period .

Dragging the rectangle we can see how the cosine function is periodic.

The cosine function is periodic | matematicasvisuales

We can see another point of view of this periodicity in the page about the Taylor's Polynomial of the Cosine Function

Taylor's Polynomial of the Cosine Function | Cosine Periodicity | matematicasvisuales

The Complex Cosine Function have much in common with his real ancestors, por example:

Complex Cosine Function is an even function | matematicasvisuales

The image of a horizontal line is an ellipse:

Complex Cosine Function | ellipse | matematicasvisuales

The image of a vertical line is an hyperbola:

Complex Cosine Function |hyperbola | matematicasvisuales


Tristan Needham - Visual Complex Analysis. (pag. 88-89) - Oxford University Press


The Complex Cosine Function: mapping an horizontal line
The Complex Cosine Function maps horizontal lines to confocal ellipses.
Taylor polynomials: Complex Cosine Function
The power series of the Cosine Function converges everywhere in the complex plane.
The Complex Exponential Function
The Complex Exponential Function extends the Real Exponential Function to the complex plane.
Taylor polynomials (2): Sine function
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
Multifunctions: Powers with fractional exponent
The usual definition of a function is restrictive. We may broaden the definition of a function to allow f(z) to have many differente values for a single value of z. In this case f is called a many-valued function or a multifunction.
More about complex functions
Examples of complex functions: polynomials, rationals and Moebius Transformations.