matematicas visuales visual math

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

Complex Taylor polynomials.Cosine function:  | matematicasVisuales

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):

Complex Taylor polynomials.Cosine function:  Taylor's polynomial degree 5| matematicasVisuales

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

Complex Taylor polynomials.Cosine function:  Remainder from polynomial of degree 5 | matematicasVisuales

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

Complex Taylor polynomials.Cosine function:  Taylor's polynomial degree 10 | matematicasVisuales

The area where the approximation is good is much bigger:

Complex Taylor polynomials.Cosine function: Remainder from polynomial of degree 10 | matematicasVisuales

The Cosine Function is periodic with period .

Complex Taylor polynomials.Cosine function:  is a periodic function with period 2*pi| matematicasVisuales

REFERENCES

Tristan Needham - Visual Complex Analysis. (pags. 84) - Oxford University Press

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.