matematicas visuales visual math

Power Complex Functions with exponent a whole number

have a zero (also called a root) of multiplicity n at 0 (the origin). They are simple examples of polynomial functions.

This representation allow us to see how at a zero of multiplicity n the color cycle goes round the zero n times.

You can change the value of n (the exponent) to see the representation of different power funtions.

The identity function has a zero of multiplicity 1:

Polynimial Complex Functions, power functions: The identity function has a zero of multiplicity 1 | matematicasVisuales

The power function of degre 2 has a zero of multiplicity 2:

Polynimial Complex Functions, power functions: The power function of degre 2 has a zero of multiplicity 2 | matematicasVisuales

The power function of degre 3 has a zero of multiplicity 3:

Polynimial Complex Functions, power functions: The power function of degre 3 has a zero of multiplicity 3 | matematicasVisuales

The power function of degre 4 has a zero of multiplicity 4:

Polynimial Complex Functions, power functions:  The power function of degre 4 has a zero of multiplicity 4 | matematicasVisuales

The power function of degre 5 has a zero of multiplicity 5:

Polynimial Complex Functions, power functions:  The power function of degre 5 has a zero of multiplicity 5 | matematicasVisuales

REFERENCES

Tristan Needham - Visual Complex Analysis. Oxford University Press.

MORE LINKS

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A complex polinomial of degree 3 has three roots or zeros.
Complex Polynomial Functions(4): Polynomial of degree n
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Complex Polynomial Functions(5): Polynomial of degree n (variant)
Every complex polynomial of degree n has n zeros or roots.
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Polynomial Functions (2): Quadratic functions
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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.
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Podemos modificar las multiplicidades del cero y del polo de estas funciones sencillas.
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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
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Inversion
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.