The complex function

has two singularities, in z = i and in z = -i.

If we consider complex power series centered at some point, there exists a circle centered at this point such that the series converges everywhere inside the circle and diverges everywhere outside the circle.

It is a very important fact that if this series converges at a point, then its value can be approximated by a partail sum (polynomial), and by choosing a sufficiently large degree we can make the approximation as accurate as we wish. (Tristan Needham)

If we restric our vision to the circle of convergence we can see how these polynomials approximate the function inside the circle of convergence.

The Remainder is the difference between the function and the polynomial. A clear color, almost white, indicates a small module. As we increases the degree of the polynomial the white area approximates the circle of convergence.

This complex function helps us to understand the behavior of the Taylor series of a real function that has no real singularity. This real function coincides with the complex function in the real axis:

LINKS

	Taylor polynomials: Complex Exponential Function The complex exponential function is periodic. His power series converges everywhere in the complex plane.
	Taylor polynomials: Complex Cosine Function The power series of the Cosine Function converges everywhere in the complex plane.
	Taylor polynomials: Rational function without real singularities This is a continuos function and has no real singularities. However, the Taylor series approximates the function only in an interval. To understand this behavior we should consider a complex function.
	Taylor polynomials: Rational function with two real singularities This function has two real singularities at -1 and 1. Taylor polynomials approximate the function in an interval centered at the center of the series. Its radius is the distance to the nearest singularity.
	Taylor polynomials: Rational function 1 The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
	Taylor polynomials: Rational function 2 The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
	Taylor polynomials: Exponential function By increasing the degree, Taylor polynomial approximates the exponential function more and more.
	Taylor polynomials: Sine function By increasing the degree, Taylor polynomial approximates the sine function more and more.
	Taylor polynomials: Square root The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
	More about complex functions Examples of complex functions: polynomials, rationals and Moebius Transformations.