matematicas visuales visual math

We are going to study the complex function

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:

The real function has no singularities but the complex one has two singularities, in z = i and in z = -i.

Complex Taylor polynomials: Rational function with two complex singularities  | matematicasVisuales

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. We are going to see that the radius of convergence is the distance from the center to the nearest singularity.

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)

For example, this is a representation of the Taylor's polynomial of degree 6:

Complex Taylor polynomials:  Rational function with two complex singularities. Taylor's polynomial of degree 6  | matematicasVisuales

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 is a representation of the remainder when we consider the Taylor's polynomial of degree 6. The approximation is very accurate near the center:

Complex Taylor polynomials:  Rational function with two complex singularities. Remainder from polynomial of degree 6 | matematicasVisuales

The approximation is better when we consider the polynomial of degree 50:

Complex Taylor polynomials:  Rational function with two complex singularities. Taylor's polynomial of degree 50 | matematicasVisuales
Complex Taylor polynomials:  Rational function with two complex singularities. Remainder from polynomial of degree 50 | matematicasVisuales

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

This is a representation of the Taylor's polynomial of degree 50 only inside the circle of convergence:

Complex Taylor polynomials: Rational function with two complex singularities. Taylor's polynomial of degree 50 restricted inside the circle of convergence | matematicasVisuales

And this is a representation of the function only inside the same circle. We can see that the image below is very similar to this.

Complex Taylor polynomials: Rational function with two complex singularities. Function restricted inside the circle of convergence | matematicasVisuales

Another example, when the Taylor's polynomial has only degree 15:

Complex Taylor polynomials: Rational function with two complex singularities. Taylor's polynomial of degree 15 | matematicasVisuales

The same polynomial but only inside the circle of convergence:

Complex Taylor polynomials: Rational function with two complex singularities. Taylor's polynomial of degree 15 restricted inside the circle of convergence | matematicasVisuales

And the function restricted to the same circle:

Complex Taylor polynomials: Rational function with two complex singularities. Function restricted inside the circle of convergence | matematicasVisuales

Inside the circle of convergence the function and the approximation are similar but outside they are very different.

REFERENCES

Tristan Needham - Visual Complex Analysis. (pags. 64-77) - Oxford University Press

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 (7): 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 (1): Exponential function
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
Taylor polynomials (2): Sine function
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Taylor polynomials (3): Square root
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (4): Rational function 1
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (5): Rational function 2
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (6): 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.
More about complex functions
Examples of complex functions: polynomials, rationals and Moebius Transformations.