Product of complex numbers
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![Multiplying two complex numbers | matematicasVisuales Multiplying two complex numbers | matematicasVisuales](../../../images/appimg/multiplicacionCompleja.gif) |
We can see it as a dilatative rotation.
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![The product as a complex plane transformation | matematicasVisuales The product as a complex plane transformation | matematicasVisuales](../../../images/appimg/multiplicacion2C.jpg) |
The multiplication by a complex number is a transformation of the complex plane: dilative rotation.
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![Complex Geometric Sequence | matematicasVisuales Complex Geometric Sequence | matematicasVisuales](../../../images/appimg/progresgeom2.gif) |
From a complex number we can obtain a geometric progression obtaining the powers of natural exponent (multiplying successively)
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Complex Functions
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![Complex Polynomial Functions(1): Powers with natural exponent | matematicasVisuales Complex Polynomial Functions(1): Powers with natural exponent | matematicasVisuales](../../../images/appimg/potenciasC.jpg) |
Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.
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![Complex Polynomial Functions(2): Polynomial of degree 2 | matematicasVisuales Complex Polynomial Functions(2): Polynomial of degree 2 | matematicasVisuales](../../../images/appimg/cgrado2.jpg) |
A polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.
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![Complex Polynomial Functions(3): Polynomial of degree 3 | matematicasVisuales Complex Polynomial Functions(3): Polynomial of degree 3 | matematicasVisuales](../../../images/appimg/cubica.jpg) |
A complex polinomial of degree 3 has three roots or zeros.
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![Complex Polynomial Functions(4): Polynomial of degree n | matematicasVisuales Complex Polynomial Functions(4): Polynomial of degree n | matematicasVisuales](../../../images/appimg/polinomicaC.jpg) |
Every complex polynomial of degree n has n zeros or roots.
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![Complex Polynomial Functions(5): Polynomial of degree n (variant) | matematicasVisuales Complex Polynomial Functions(5): Polynomial of degree n (variant) | matematicasVisuales](../../../images/appimg/polinomica2.jpg) |
Every complex polynomial of degree n has n zeros or roots.
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![Cero and polo (Spanish) | matematicasVisuales Cero and polo (Spanish) | matematicasVisuales](../../../images/appimg/ceropolo.jpg) |
Podemos modificar las multiplicidades del cero y del polo de estas funciones sencillas.
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![Cero and polo (variant) (Spanish) | matematicasVisuales Cero and polo (variant) (Spanish) | matematicasVisuales](../../../images/appimg/ceropolov.jpg) |
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![Moebius transformations (Spanish) | matematicasVisuales Moebius transformations (Spanish) | matematicasVisuales](../../../images/appimg/moebius.jpg) |
Una primera aproximación a estas transformaciones. Representación de dos haces coaxiales de circunferencias ortogonales.
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![The Complex Exponential Function | matematicasVisuales The Complex Exponential Function | matematicasVisuales](../../../images/appimg/expC.jpg) |
The Complex Exponential Function extends the Real Exponential Function to the complex plane.
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![The Complex Cosine Function | matematicasVisuales The Complex Cosine Function | matematicasVisuales](../../../images/appimg/cosineC.jpg) |
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.
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![The Complex Cosine Function: mapping an horizontal line | matematicasVisuales The Complex Cosine Function: mapping an horizontal line | matematicasVisuales](../../../images/appimg/cosineCEllipse.jpg) |
The Complex Cosine Function maps horizontal lines to confocal ellipses.
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![Inversion | matematicasVisuales Inversion | matematicasVisuales](../../../images/appimg/inversion.jpg) |
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
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![Inversion: an anticonformal transformation | matematicasVisuales Inversion: an anticonformal transformation | matematicasVisuales](../../../images/appimg/inversion2.jpg) |
Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles
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![Multifunctions: Powers with fractional exponent | matematicasVisuales Multifunctions: Powers with fractional exponent | matematicasVisuales](../../../images/appimg/fractionalpower.jpg) |
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.
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![Multifunctions: Two branch points | matematicasVisuales Multifunctions: Two branch points | matematicasVisuales](../../../images/appimg/multifunctionstwobranch.jpg) |
Multifunctions can have more than one branch point. In this page we can see a two-valued multifunction with two branch points.
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Taylor's Polynomials
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![Taylor polynomials: Rational function with two complex singularities | matematicasVisuales Taylor polynomials: Rational function with two complex singularities | matematicasVisuales](../../../images/appimg/twoComplexSingC.jpg) |
We will see how Taylor polynomials approximate the function inside its circle of convergence.
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![Taylor polynomials: Complex Exponential Function | matematicasVisuales Taylor polynomials: Complex Exponential Function | matematicasVisuales](../../../images/appimg/expCTaylor.jpg) |
The complex exponential function is periodic. His power series converges everywhere in the complex plane.
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![Taylor polynomials: Complex Cosine Function | matematicasVisuales Taylor polynomials: Complex Cosine Function | matematicasVisuales](../../../images/appimg/cosCTaylor.jpg) |
The power series of the Cosine Function converges everywhere in the complex plane.
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