For the mapping , find the image of
My tutor did this a tricky way, by noticing that:
Therefore, (which is just the open unit circle).
But how do I attempt this if I don't notice such nifty tricks? I keep making a mess of it.
This mapping f(z) is an example of a fractional linear transformation (sometimes called a Möbius transformation). These have the property that they take (lines and circles) to (lines or circles).
Originally Posted by Glitch
The boundary of the domain is the real axis. This is a line, so the mapping f will take it to a line or a circle. A circle is determined by three points on it, so take three points on the real line, say 0, 1 and –1, and see what f does to them. You find that f(0) = –1, f(1) = –i and f(–1) = i. The circle through the three points –1, i and –i is the unit circle.
Therefore f takes the boundary of D to the unit circle. The circle splits the complex plane into two regions, namely the inside and the outside of the circle. So f must take D to one or other of those regions. To find out which, choose a point in D and see where f takes it. If you take the point i, you see that f(i) = 0. From that, you conclude that f takes D to the inside of the unit circle, namely the open unit disc.
That method is not nearly as slick as the trick used by the tutor, but it is a standard method for keeping track of what a fractional linear linear transformation does to a half-plane or a disc.
Re: Mapping question