matematicas visuales visual math
Complex Polynomial Functions(5): Degree n (variant)

In the applet of the page Complex Polynomial Functions I have added control about what points of the Complex Plane are depicted with colors.

The functions transform points in the Complex Plane. The transformed points have a modulus. In the applet you can see colored points with modulus in a range.

For example, in this representation of a polynomial of degree 7 with two simple roots, one double and one triple root the only points with colors are points whose transformed points have modulus between 1 and 10:

Funciones polinómicas complejas de grado n: polinomio de grado 7 | matematicasVisuales


Tristan Needham - Visual Complex Analysis. Oxford University Press.


Complex Polynomial Functions(1): Powers with natural exponent
Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.
Complex Polynomial Functions(2): Polynomial of degree 2
A polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.
Complex Polynomial Functions(3): Polynomial of degree 3
A complex polinomial of degree 3 has three roots or zeros.
Polynomial Functions (1): Linear functions
Two points determine a stright line. As a function we call it a linear function. We can see the slope of a line and how we can get the equation of a line through two points. We study also the x-intercept and the y-intercept of a linear equation.
Polynomial Functions (2): Quadratic functions
Polynomials of degree 2 are quadratic functions. Their graphs are parabolas. To find the x-intercepts we have to solve a quadratic equation. The vertex of a parabola is a maximum of minimum of the function.
Polynomial Functions (3): Cubic functions
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the x-axis at least once.
Polynomial Functions (4): Lagrange interpolating polynomial
We can consider the polynomial function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.
Cero and polo (Spanish)
Podemos modificar las multiplicidades del cero y del polo de estas funciones sencillas.
Moebius transformations (Spanish)
Una primera aproximación a estas transformaciones. Representación de dos haces coaxiales de circunferencias ortogonales.
The Complex Exponential Function
The Complex Exponential Function extends the Real Exponential Function to the complex plane.
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.
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
Inversion: an anticonformal transformation
Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles
Multifunctions: Powers with fractional exponent
The usual definition of a function is restrictive. We may broaden the definition of a function to allow f(z) to have many differente values for a single value of z. In this case f is called a many-valued function or a multifunction.
Multifunctions: Two branch points
Multifunctions can have more than one branch point. In this page we can see a two-valued multifunction with two branch points.
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 with two complex singularities
We will see how Taylor polynomials approximate the function inside its circle of convergence.