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
The oval hits the real axis at this point:
To calculate where this oval hits the imaginary axis, we consider
The oval hits the imaginary axis at this point:
The oval is a perfect ellipse. If we calculate
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.
Tristan Needham - Visual Complex Analysis. (pag. 88-89) - Oxford University Press