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

The first set shows a domain and its image as mapped by
*e*^{z}. Here, the domain runs from -1 to 1 in the real
part and -p to p. For
a fixed *x*, the numbers *z* = *x* + *iy*
are mapped to a circle of radius *e*^{x}; you can see
this by looking at the equation
*e*^{x}(cos(*y*) + *i*
sin(*y*)), where the *y* gives the argument of the
result, and *e*^{x} gives the radius.

Conversely, if we hold *y* fixed and let *x* vary, *z* will
map to the ray at angle *y* from (but not including) the origin, to
infinity (since the limit of *e*^{x} is zero as
*x* goes to negative infinity and is infinity as *x* goes to
infinity). In this way, the horizontal strip from negative infinity to
infinity in the real part and from -p to p covers all of the complex plane (other than 0).

Because of the periodic nature of cos(*y*) + *i*
sin(*y*), if we look at horizontal strips at higher or lover
values in the imaginary part, we cover the complex plane over and over
again.

The next set of diagrams show the result of mapping a square patch (minus
the origin) via the map 1/*z*. We noted that the images of the
horizontal and vertical lines appear to be circles. You will analyse this
observation more carefully on the next problem set.