matematicas visuales visual math

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

Complex Taylor polynomials: Exponential function | matematicasVisuales

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:

Complex Taylor polynomials: Exponential function. Taylor's polynomial degree 6 | matematicasVisuales

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

Complex Taylor polynomials: Exponential function. Remainder from polynomial of degree 6| matematicasVisuales

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

Complex Taylor polynomials: Exponential function. Taylor's polynomial degree 40 | matematicasVisuales

This is the remainder:

Complex Taylor polynomials: Remainder from polynomial of degree 40 | matematicasVisuales

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.

REFERENCES

Tristan Needham - Visual Complex Analysis. (pag. 80) - Oxford University Press

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.
Exponentials and Logarithms (7): The exponential as the inverse of the logarithm
After the definition of the natural logarithm function as an integral you can define the exponential function as the inverse function of the logarithm.
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.