How coordinate lines transform under ##e^z=\frac{a-w}{a+w}##

[davidbenari]

Homework Statement

Say how coordinate lines of the z plane transform when applied the following transformation

$e^z=\frac{a-w}{a+w}$

Homework Equations

The Attempt at a Solution

This is exactly the way the problem is stated. It is a pretty weird transformation in my opinion and I'm guessing $w$ is the transformation and the coordinate lines in question are cartesian coordinate lines.

That said I've solved for $w$

$w=\frac{a(1-e^z)}{1+e^z}$

And what I've attempted to do is consider the cases $x=C$ $y=C$ where $C$ is some arbitrary constant and find an expression of the form $w=u+iv$. This has proven quite tedious and very little if at all illuminating.

So I was wondering how I could proceed?

At first I thought this was similar to a bilinear transformation but now I've thought not at all.

I don't know this topic too well so I'm probably not seeing something basic here, or failing to identify the class of transformation in questionThanks.

 

[andrewkirk]

I suggest you start with discovering how the x-axis (real axis) transforms. Calculate w for x values of $-\infty,-1,0,+1,+\infty$. You'll find a nice simple result.

Next look at how coordinate lines parallel to the x-axis transform. They will be of the form $a\frac{1-e^{ni+x}}{1+e^{ni+x}} =a\frac{1-e^{ni}e^x}{1+e^{ni}e^x}$ for $n$ an integer. Again calculate w for x values of $-\infty,-1,0,+1,+\infty$.

I think it gets more complicated after the first step. A quicker and easier way to get a feel for it (that would not be available in an exam, so be cautious if this is exam practice) would be to use a spreadsheet or R/matlab-type program to calculate w for the table of coordinate grid points in the square [-10,10] x [-10,10]. I expect a pattern will emerge.

 

[davidbenari]

I just tried plotting the constant x surface on MATLAB and got an oval but strangely its contour is fuzzy (by which I mean its not a perfect line)

 

[davidbenari]

The oval might be a circle I'm adjusting the scale...

edit: Indeed it is a circle.

 

[davidbenari]

It seems the constant x and y surfaces map into circles and rays. I don't see why... Any ideas?

 

[davidbenari]

Nevermind they seem to map into some weird collection of circles