These are some of the images that we saw in class today.
The first set shows a domain and its image as mapped by ez. Here, the domain runs from -1 to 1 in the real part and -p to p. For a fixed x, the numbers
z = x + iyare mapped to a circle of radius ex; you can see this by looking at the equation ex(cos(y) + i sin(y)), where the y gives the argument of the result, and ex 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 ex 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.