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: REFERENCES
Tristan Needham  Visual Complex Analysis. Oxford University Press.
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Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.
A polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.
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 xintercept and the yintercept of a linear equation.
Polynomials of degree 2 are quadratic functions. Their graphs are parabolas. To find the xintercepts we have to solve a quadratic equation. The vertex of a parabola is a maximum of minimum of the function.
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the xaxis at least once.
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.
Una primera aproximación a estas transformaciones. Representación de dos haces coaxiales de circunferencias ortogonales.
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.
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles
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 manyvalued function or a multifunction.
Multifunctions can have more than one branch point. In this page we can see a twovalued multifunction with two branch points.
The complex exponential function is periodic. His power series converges everywhere in the complex plane.
