The real cosine function can be extended using the exponential function:
Since the exponential function is periodic with period ,
it follows that the cosine function is periodic, but with period .
Dragging the rectangle we can see how the cosine function is periodic.
We can see another point of view of this periodicity in the page about the Taylor's Polynomial of the Cosine Function
The Complex Cosine Function have much in common with his real ancestors, por example:
The image of a horizontal line is an ellipse:
The image of a vertical line is an hyperbola:
Tristan Needham - Visual Complex Analysis. (pag. 88-89) - Oxford University Press