Well I have a question that is "identify the two interesting points in the picture for the function $f(z) = z + 1/z$. Explain your answer." where it refers to the tool here:
http://rotormind.com/projects/codework/conformal/
I was told that the 2 interesting points that we must find/prove are:
additionally, there was a hint that the quadratic formula may be useful.
I was also told that I did not need to know anything about conformal mapping and that high school mathematics would enable me to do this question. So my question is, what is the picture showing? and how can I find these points? Thanks!
Let's start with some observations:
The pictured points are the images of the points in a square grid of length and width 2 under the mapping $f$. The original grid (unfortunately hidden) is centered at the mouse cursor, but the image grid may of course lie somewhere else.
Observation 1: The image grid looks mostly like a square, possibly with some bumps and dents.
Observation 2: Angles between curves are mostly preserved, in particular the edges of the image square have mostly 90 degree angles.
Observation 3: the above observations seem to fail and strange things happen, if we move the cursor into the rectangle with vertices $-2-i,~2-i,~2+i,~-2+i$. That is, as soon as some grid points get close to $-1$ or $1$. In this case, some of the image points seem to "cluster" at $-2$ or $2$, respectively.
So, what is wrong about the points $-1$ and $1$? The property that grid points near these, are mapped closer together ("clustering" at $-2$ or $2$), indicates that the derivatives near $-1$ and $1$ are small. Indeed, you can easily calculate that the derivative of $f(z)=z+1/z$ has zeros exactly at these two points and these are mapped to $-2$ and $2$, respectively. We call $-1$ and $1$ critical points, and $-2$ and $2$ critical values.
Now, a conformal mapping is defined to be a locally angle-preserving mapping, which would explain the observations 1 and 2. This, however, is equivalent to being a holomorphic function whose derivative is everywhere non-zero. Our function $f$ thus fails to be conformal at $-1$ and $1$, which explains observation 3.
Finally, it depends on you, whether you want to consider the critical points $-1$ and $1$ or the critical values $-2$ and $2$ to be the "special points".
Edit: Why is there "clustering" near a critical value?
Well, the derivative of a function is by definition the (infinitesimal) ratio of the distance of two image points to their preimage's distance. Now, if you have derivative zero at for instance the point $z=1$, whose image is $2$, this means, that points reasonably close to $1$ must be mapped significantly closer to $2$, otherwise the ratio of those distances could not tend to zero.