matematicas visuales visual math
Complex Analysis

Multiplying two complex numbers | matematicasVisuales The product as a complex plane transformation | matematicasVisuales Geometric sequence | matematicasVisuales Complex Polynomial Functions(1): Powers with natural exponent | matematicasVisuales Complex Polynomial Functions(2): Polynomial of degree 2 | matematicasVisuales Complex Polynomial Functions(3): Polynomial of degree 3 | matematicasVisuales Complex Polynomial Functions(4): Polynomial of degree n | matematicasVisuales
Complex Polynomial Functions(5): Polynomial of degree n (variant) | matematicasVisuales Cero and polo (Spanish) | matematicasVisuales Cero and polo (variant) (Spanish) | matematicasVisuales Moebius transformations (Spanish) | matematicasVisuales The Complex Exponential Function | matematicasVisuales The Complex Cosine Function | matematicasVisuales The Complex Cosine Function: mapping an horizontal line | matematicasVisuales
Inversion | matematicasVisuales Inversion: an anticonformal transformation | matematicasVisuales Multifunctions: Powers with fractional exponent | matematicasVisuales Multifunctions: Two branch points | matematicasVisuales Taylor polynomials: Rational function with two complex singularities | matematicasVisuales Taylor polynomials: Complex Exponential Function | matematicasVisuales Taylor polynomials: Complex Cosine Function | matematicasVisuales


Product of complex numbers
Multiplying two complex numbers | matematicasVisuales
We can see it as a dilatative rotation.
The product as a complex plane transformation | matematicasVisuales
The multiplication by a complex number is a transformation of the complex plane: dilative rotation.
Geometric sequence | matematicasVisuales
From a complex number we can obtain a geometric progression obtaining the powers of natural exponent (multiplying successively)

Complex Functions
Complex Polynomial Functions(1): Powers with natural exponent | matematicasVisuales
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 | matematicasVisuales
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 | matematicasVisuales
A complex polinomial of degree 3 has three roots or zeros.
Complex Polynomial Functions(4): Polynomial of degree n | matematicasVisuales
Every complex polynomial of degree n has n zeros or roots.
Complex Polynomial Functions(5): Polynomial of degree n (variant) | matematicasVisuales
Every complex polynomial of degree n has n zeros or roots.
Cero and polo (Spanish) | matematicasVisuales
Podemos modificar las multiplicidades del cero y del polo de estas funciones sencillas.
Cero and polo (variant) (Spanish) | matematicasVisuales
Moebius transformations (Spanish) | matematicasVisuales
Una primera aproximación a estas transformaciones. Representación de dos haces coaxiales de circunferencias ortogonales.
The Complex Exponential Function | matematicasVisuales
The Complex Exponential Function extends the Real Exponential Function to the complex plane.
The Complex Cosine Function | matematicasVisuales
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 | matematicasVisuales
The Complex Cosine Function maps horizontal lines to confocal ellipses.
Inversion | matematicasVisuales
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
Inversion: an anticonformal transformation | matematicasVisuales
Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles
Multifunctions: Powers with fractional exponent | matematicasVisuales
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 | matematicasVisuales
Multifunctions can have more than one branch point. In this page we can see a two-valued multifunction with two branch points.

Taylor's Polynomials
Taylor polynomials: Rational function with two complex singularities | matematicasVisuales
We will see how Taylor polynomials approximate the function inside its circle of convergence.
Taylor polynomials: Complex Exponential Function | matematicasVisuales
The complex exponential function is periodic. His power series converges everywhere in the complex plane.
Taylor polynomials: Complex Cosine Function | matematicasVisuales
The power series of the Cosine Function converges everywhere in the complex plane.