The complex Exponential Function is the complex function that we can define as a power series and that extends the real Exponential function to complex values

The Exponential Function is periodic with period .

This series converges everywhere in the complex plane. Then the value of the function can be approximated by a partial sum (polynomial), and by choosing a sufficiently large degree we can make the approximation as accurate as we wish.

We can see it if we look at the Remainder (the difference between the function and the polynomial).

For example, this is one polynomial approximation of degree 6:

And this is a representation of the remainder. The approximation is good but only near the center:

We can see how the approximation improves when the polynomial is of degree 40:

This is the remainder:

The behavior of the power series of the complex exponential function is very good because the series converges everywhere. We can see a similar behavior studying the complex cosine function.

LINKS

	The Complex Exponential Function The Complex Exponential Function extends the Real Exponential Function to the complex plane.
	Taylor polynomials (1): Exponential function By increasing the degree, Taylor polynomial approximates the exponential function more and more.
	The exponential as the inverse of the logarithm (Spanish)
	Taylor polynomials: Complex Cosine Function The power series of the Cosine Function converges everywhere in the complex plane.
	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.