00:01
So we're trying to find a fractional linear transformation.
00:04
So let's zoom in for a thicker pen.
00:08
Which if you recall looks like something like w1 plus w2 times z plus w3 over w3 plus w4 times z as a function of z.
00:27
This is the general form for a fractional linear transformation.
00:29
So fraction of linear functions, right? mapping with z.
00:36
0 goes to 1, i goes to negative 1, and negative i goes to 0.
00:48
So let's see what we got here.
00:51
If you plug 0 into this, we're just going to get w1 over w3, right? this has to equal 1.
01:00
If we plug i into this, so w1 plus i times w2 divided by w3 plus i times w4, this needs to equal negative 1.
01:23
And we need w1 minus i times w2 to equal 0, actually.
01:30
And we don't even need to worry about the denominator here, because in order to get this equal to 0, we need the numerator to equal 0.
01:37
So these are three systems of equations.
01:40
Let's see what we can do here.
01:42
Well if we express everything in terms of w1, actually, to this, let's go ahead and divide both of these.
01:52
We can assume that w1 is not 0, right? because if you divide it by w3, you get 1.
01:57
So let's divide the numerator and the denominator by w1 and represent it with a 1 in the first component there.
02:09
Right? everything, that is totally valid.
02:12
We can do that, multiplying both parts of a fraction by the same thing.
02:18
So now w1 can just be 1 for all three of these.
02:25
What that tells us, first of all now, w3 has to equal 1, because 1 over 1 has to be 1.
02:32
So let's write this out.
02:35
So 1 plus w2z over w3, we've established as 1, plus w4z.
02:46
That's really neat.
02:50
Let's see, what else have we got here? well, if you push around this sign here, 1 has to equal i times w2, right? which means w2 equals negative i.
03:06
So we've just got 1 minus i times z over 1 plus w4 times z.
03:19
And this actually helps us out here.
03:23
So we've got 1 plus i times negative i.
03:27
I times negative i is of course 1.
03:30
So we've got 1 plus 1 divided by, we've established w3 is 1.
03:34
We can do that.
03:35
1 plus i times w4 is equal to negative 1.
03:37
So 2 over 1 plus w times i is negative 1.
03:45
Multiply here.
03:46
2 is equal to negative wi minus 1.
03:51
3 we conclude is equal to negative wi.
03:58
Negative 3 is equal to, or we'll just multiply through by i here.
04:04
And we get 3i is equal to w.
04:09
If you multiply by i, i times i is negative 1.
04:12
Negative negative 1 is 1.
04:13
We get w and 3i there.
04:16
So w4 is equal to 3i.
04:23
So this here is our fractional linear transformation.
04:28
1 minus i times z divided by 1 plus 3i times z.
04:33
We're not done though.
04:34
We want to find the image of the half plane under this fractional linear transformation.
04:38
So let's see what this does.
04:41
It's saying the upper half plane, the real part greater than 0.
04:46
Let's see what this does to the real line.
04:49
That'll give us the boundary.
04:51
So 1 minus i times x.
04:56
X is a real number here.
04:58
Or 1 plus 3i times x.
05:04
Let's see.
05:05
Or we'll call it t.
05:09
The real axis is of course a line.
05:12
So it's either going to map to a line or to a circle.
05:16
We can see what that might be.
05:20
We know that z equals 0 maps to 1.
05:24
So we have 1, 0 on this line.
05:30
Where does it map 1? let's see.
05:33
So 1 minus i over 1 plus 3i is equal to...
05:36
Let's see.
05:38
1 minus i times 1 minus 3i divided by 1 plus 9 is 10.
05:49
Is equal to 1 times 1 plus i.
05:59
I times 3i is negative 3, but we're subtracting that.
06:02
So plus 4 plus i times what i got here.
06:08
1 times negative 3 is negative 3.
06:10
Minus 1 is negative 4.
06:12
So 5 minus 4i on the numerator here divided by 10 is where 1 maps.
06:26
So another point on here is...
06:28
Let's see...
