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

Since the exponential function is periodic with period ,

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

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

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

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

The image of a horizontal line is an ellipse:

The image of a vertical line is an hyperbola:

LINKS

	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: Sine function By increasing the degree, Taylor polynomial approximates the sine function more and more.
	Inversion 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.