matematicas visuales visual math

We have already seen a multifunction. It is the case of the cubic root that has three values and the point z=0 is called a branch point. Fractional powers are multifunctions.

Now we are considering the two-valued multifunction

Two-valued multifunction with two branch points  | matematicasvisuales

We can draw a path on the left panel and see how this path is transformed on the right panel.

If z travels along a closed loop like in this picture, not encircling any of the two branch points, its image f(z) travels along a closed loop and returns to its original value.

Two branch points: a closed loop not encircling any of the two branch points  | matematicasvisuales

However, if z instead travels along a closed loop which goes round only one of the branch points, then f(z) does not return to its original value but instead ends up at a different value of the multifunction.

Two branch points: a closed loop going round one of the branch points once | matematicasvisuales

Similarly, if z travels along a closed loop encircling one of the branch points twice, then f(z) returns to its original value again.

Two branch points: a closed loop going round one of the branch points twice | matematicasvisuales

The same happens if the closed loop goes round both branch points.

Two branch points: a closed loop encircling both branch points | matematicasvisuales

REFERENCES

Tristan Needham - Visual Complex Analysis. (pag. 96) - Oxford University Press

LINKS

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.
Inversion
Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
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.
More about complex functions
Examples of complex functions: polynomials, rationals and Moebius Transformations.