We already know that cos(z) is periodic with period . In this page we are going to explore how an horizontal line is transformed by the function cos(z), following Tristan Needham's Visual Complex Analysis. Playing the animation we can see that the image of is "some kind of symmetrical oval" (is just the sum of two circular motions). To calculate where this oval hits the real axis, we consider Then The oval hits the real axis at this point: To calculate where this oval hits the imaginary axis, we consider Then The oval hits the imaginary axis at this point:
The oval is a perfect ellipse. If we calculate Using Considering the real and imaginary parts We can write Which is the familiar representation of the ellipse, that we can write (implicit formula): To calculate the foci of the ellipse we can use The shapes of the ellipse change as we vary c but all these ellipses are confocal. The image under cos(z) of a vertical line is an hyperbola with the same foci as the ellipse. Ellipses and hyperbolas meet at right angles.
REFERENCES
Tristan Needham  Visual Complex Analysis. (pag. 8889)  Oxford University Press
LINKS
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.
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.
