Mobius transformation

**thespian** · October 26th 2009, 10:23 PM

Sketch:

w=((i+z)/(i-z))
for:C\{z=x+iy,x≤0,y=0}

So basically for all z, ecxluding :x is less than or equal to zero and y=0.

I have solved for z which gives me:
z=((wi-i)/(w+1))

What is the easiest way to graph this? the lecturer graphed the answer but didnt really explain how to get the answer.

**Bruno J.** · October 26th 2009, 10:35 PM

Hint : evaluate $\frac{1+it}{1-it}$ , where $t=\tan \theta$ .

**thespian** · October 26th 2009, 10:49 PM

i would like to learn how to approach this type of question. how did you arrive at that trigonometric substitution? say as opposed to cosx, or cosecx? Can you please explain the logic to me, because this is what i find most difficult.

**Bruno J.** · October 27th 2009, 08:39 AM

Originally Posted by thespian

i would like to learn how to approach this type of question. how did you arrive at that trigonometric substitution? say as opposed to cosx, or cosecx? Can you please explain the logic to me, because this is what i find most difficult.

I know it because I've done it before! It's as simple as that.

In fact, sending the real number $t$ to $\frac{1+it}{1-it}$ is just like doing a stereographic projection of the real line to the unit circle. It's essentially the complex form of the tangent half-angle formula. You should have gotten $e^{2i\theta}=\frac{1+it}{1-it}=\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}i$ .

Practice, my friend! I had almost the exact same question two weeks ago - I had to find the image of the first quadrant under that transformation. I could see that the real line was going to the unit circle, but not much more than that. It was the first time I ever asked for help on MHF (and I got it

).

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