Conformal Mapping: Find Points of Z Plane for f(z)=-(1-z)/(1+z)

[KleZMeR]

Homework Statement

With The map f(z) = -(1-z)/(1+z)

where z=x+iy,

and f(z) maps z onto w = u + iv plane.

show for which points of the z plane this map is conformal.

Homework Equations

The Attempt at a Solution

I have read a lot about this subject, and I think I understand the theory, but I have no idea how to start. I can get f(z) into w = u + iv form. I just don't understand how to find these points. Any hints would be appreciated. I want to know how to do this and am not asking for a solution, just a push in the right direction.

 

[D_Tr]

It will be conformal in points where it is analytic and the derivative is non-zero. If any of the above are not satisfied at a point, the function is not conformal there.

 

[KleZMeR]

So, D_Tr, my function is analytic, but it is not safe to just assume that at all points it is analytic? It seems that there may be some maxima and minima in my map where the derivative is zero, yes? Given my f(z), I could find some zero point on the map, as well as a singularity, but can I assume that these two points are the only non-conformal points?. By inspection I see z=1, z=-1, as both being points in my map where it is non-conformal, but there may be other points which are I think found by use of my partial derivatives?

 

[D_Tr]

At first, when you say that a function is analytic, you specify at which points is it analytic. If and only if, at a point z0, a function is analytic and its derivative is not zero, then it is conformal at z0. The function you have is analytic everywhere except -1 because its derivative does not exist there. This is the only point the function is non conformal because there is no point where the derivative becomes zero.
Even if you examine the function at infinity by looking at the behavior of f(1/z) at z=0, you will find out the function is also conformal at the point at infinity.
You look at where the derivative is zero, not the map. You are interested in the derivative because of the property of a conformal map of preserving shapes (rotation and size change are allowed). You do not have to look to any partial derivatives. A function is not analytic if its derivative is dependent on whether you differentiate with respect to the real or the imaginary part of z (or with respect to any other combination).

 

[KleZMeR]

Thank you SO much D_Tr! I have been trying to get to this point of understanding for days. I am using "Mathematical Methods" by Arfken, 2nd edition (1965) from my Uni library. There is only 1.5 pages on conformal mapping in there. If convenient can you suggest any literature/website that may have more information? I have googled but found nothing close to your explanation. Thank you again.

 

[D_Tr]

I read about conformal maps from a chapter 24 in "Mathematical Methods for Physics and Engineering" (Riley, Hobson, Bence, 3rd edition).
They prove the shape-preserving property of these maps, but this book is, like the one you are using, a bit hard if you are studying these topics for the first time (Arfken is more advanced, as far as I know). There is simply not enough space in 1200 pages to cover every subject in detail.
I use this book and also online sources like Wikipedia or pdf papers freely available in various university websites, to clear things up. I had to look online to see that for complex functions, holomorphic implies analytic. Not knowing this led me to be confused when the authors used the word "analytic", when they should have used "holomorphic" in proving Cauchy's theorem.

 

[D_Tr]

I just wanted to add that I plan to study "Visual Complex Analysis" by Tristan Needham some time in the near future. I have only read positive opinions on this book, especially about the intuitive explanations it gives. You may want to check it out!

 

[KleZMeR]

Thank you! Yes I will definitely look for those at my library. I'm sure I can find the second edition of the Math Methods book. Our course is spending a lot of time on this but because the homework is quite challenging we spend much time solving one homework problem and not much time solving shorter problems that demonstrate the theory in a more concise manner.