These are some of the images that we saw in class today.

The first set shows a domain and its image as mapped by sin(*z*) and
cos(*z*). Here, the domain runs from -p to
p in the real part and 0 to 2 in the imaginary
part. For a fixed *x*, the numbers *z* = *x* +
*iy* are mapped to an hyperbola, For a fixed *y*, they map
to ellipses. These curves are everywhere perpendicular to each other.

Extending the patch in the real direction, causes the image to continue to
wrap around, over and over again. Extending the image values to higher
positive values causes the ellipses to expand, eventually covering all the
complex plane. Conversely, moving into the negative imaginary parts
causes the image to move through the segment [-1,1] on the real axis and
wraps around the plane from below.

The difference between the sine and cosine is a phase shift of 90
degrees. (You can see this by the coloring of the two images.)

The next set of diagrams show the result of mapping a patch via
tan(*z*). The image is formed by two families of circles, one set
centered on the real axis, and one on the imaginary axis. The latter
shrink down to a point at *i* and -*i*.