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Multifunctions: Two Branch Points
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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 ![]() 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. ![]() 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. ![]() Similarly, if z travels along a closed loop encircling one of the branch points twice, then f(z) returns to its original value again. ![]() The same happens if the closed loop goes round both branch points. ![]()
REFERENCES
Tristan Needham - Visual Complex Analysis. (pag. 96) - Oxford University Press
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Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.
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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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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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Inversion is a plane transformation that transform straight lines and circles in straight lines and circles.
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Inversion preserves the magnitud of angles but the sense is reversed. Orthogonal circles are mapped into orthogonal circles
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The complex exponential function is periodic. His power series converges everywhere in the complex plane.
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