Inversion: an anticonformal mapping


Inversion in a circle is a transformation that preserves the magnitude of angles but each angle is mapped to an angle of opposite sign. The sense or direction of each angle is reversed. Inversion in a circle is an anticonformal mapping (like a reflection in a line). In particular, inversion is a mapping that preserves the angle of intersection of two circles. Orthogonal circles (or lines) invert into orthogonal circles (or lines). A circle orthogonal to the circle of inversion is mapped into itself. All the lines through a point and all the circles concentric with the same point as a center can be seen as two particular cases of coaxal sistem of circles. These two coaxal sistems are orthogonal. The lines form a coaxal system of intersecting type and the concentric circles are a nonintersecting coaxal system. These coaxal systems are mapped into two orthogonal coaxal systems: Every "small" rectangle is transformed into an "small" rectangle: In this picture we can see how the order of colors is reversed: REFERENCES Hilber and CohnVossen  Geometry and the Imagination (pag. 253)  Chelsea Publishing Company
Coxeter  Introduction to Geometry  Wiley and Sons.
Pedoe  Circles, a Mathematical View  Dover
Tristan Needham  Visual Complex Analysis. (pag. 124)  Oxford University Press
Rademacher and Toeplitz  The Enjoyment of Mathematics
Ogilvy  Excursions in Geometry (pag. 24) 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.
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.
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.
