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

As a power series the equivalent definition is:

This series converges everywhere in the complex plane.

If we increase the degree of the Taylor polynomial, this polynomial approaches the function more and more. We can see it if we look at the Remainder (the difference between the function and the polynomial).

For example, this is a representation of a Taylor's polynomial of degree 5 (In the mathlet we can change the center):

And this is the remainder. We can see how the approximation is much better near the center:

If we increase the degree of the polynomial the approximation is better (degree 10):

The area where the approximation is good is much bigger:

The Cosine Function is periodic with period .

LINKS

	The Complex Cosine Function The Complex Cosine Function extends the Real Cosine Function to the complex plane. It is a periodic function that shares several properties with his real ancestor.
	The Complex Cosine Function: mapping an horizontal line The Complex Cosine Function maps horizontal lines to confocal ellipses.
	Taylor polynomials (2): Sine function By increasing the degree, Taylor polynomial approximates the sine function more and more.
	Taylor polynomials: Complex Exponential Function The complex exponential function is periodic. His power series converges everywhere in the complex plane.
	Taylor polynomials (1): Exponential function By increasing the degree, Taylor polynomial approximates the exponential function more and more.
	Taylor polynomials: Rational function with two complex singularities We will see how Taylor polynomials approximate the function inside its circle of convergence.
	More about complex functions Examples of complex functions: polynomials, rationals and Moebius Transformations.