Re: Riemann Mapping question


David C. Ullrich wrote:
On 22 May 2006 07:41:21 -0700, "Fatou" <fatou19@xxxxxxxxxxxxx> wrote:


David C. Ullrich wrote:
On 22 May 2006 06:22:30 -0700, "Fatou" <fatou19@xxxxxxxxxxxxx> wrote:


David C. Ullrich wrote:
On 21 May 2006 20:24:07 -0700, "Fatou" <fatou19@xxxxxxxxxxxxx> wrote:

I'm having trouble getting my teeth into this problem.

Let f be the riemann mapping from the square with vertices 1+i, 1-i,
-1+i, -1-i
to the unit disk, determined by f(0)=0 and f ' (0)>0.

[...]

I suspect you've got something screwed up somewhere. Say Q_0 is the
original square, and Q_1 is the square just to the right. Your
original map takes Q_0 to the unit disk. Now if you use reflection
to extend it to Q_0 union Q_1, can you describe in a few words how
the extended map behaves on the union of those two squares?

the map doesnt act conformally on the union of the squares (2-1 map)

No, it's 1-1 on the union of those two squares! Or maybe it's
2-1 at a few boundary points or something, but I get the idea
that you think that f(Q_1) = f(Q_0). That's definitely not so.
What _is_ f(Q_1)?

f(Q_0) = Q_1,

Huh? At the very start of this we said that f(Q_0) was the
unit disk!

I forgot the notation I was using -apologises
what i meant to say is the schwartz reflection of Q_0 = Q_1! (along the
edge of the square)


I'm not sure whether you're just confused about the notation
or what...

where Q_1 is Q_0 reflected wrt the right hand side of Q_0
so at the intersection of Q_1 and Q_0 it is just the shared boundary
the right hand side of Q_1 is the f(applied to the Q_0 left hand side
boundary)
its basically a piece of paper folded opened out. (I hope you can see
what im trying to say)

so f ' (0) = 0 on the boundaries.

I have no idea what "f'(0) = 0 on the boundaries" means.
But never mind...
at the boundaries the map is no longer conformal (from above they
"overlap" 2 -1 map at the boundaries)

What does that have to do with f'(0) ???

another mistake in my argument.
what i meant to say, is whilst extending the map to the complex plane,
at the vertices (not the boundaries as i previously said) the map sends
2 points to 1,
I thought this just meant it was not conformal at these points => f '
(0) = 0?

I "think" that just tells me the
zeros of the derivative of f.

Intutively (though I cant explain this -would really appreciate if you
could tell me why)
I feel that the reflection of the centre of the square, is a pole.

That's true. That's _immediate_ from the Schwarz reflection principle!

Can you state exactly the version of the reflection principle that
you're using here?


my version of the schwarz reflection principle

Q_0, Q_1 are domains s.t Q_0 n Q_1 =g, where g is an arc of a circle or
a straight line
S_g(Q_0) =Q_1 (will define S_q below)
suppose f:Q_0 u g ---> Q_1 u g cts ly so that it is holomophic in Q_1

definition of the schwartz reflection
suppose g is a straight line or arc of a circle
S_g(z) = h^-1 conjugate (h(z)) where h is a fractional linear
transformation which takes g to a segment of the real line, h(z) c Real
line

ah, I see what you mean by it is immediate from the schwarz reflection
principle that f, will have a pole (as f(0)=0) => meromophic.

I think i've over complicated the problem. Because it all seems rather
simple now.
(i hope), With this function f, i get a checkboard pattern. so above,
below, to the left, to the right, of every square with a zero, they
have poles, and vice versa (hopefully from the diagram below, you can
see what im thinking)

Q_30 ; Q_31 ; Q_32 : Q_33
Q_20 ; Q_21 ; Q_22 : Q_23
Q_10 ; Q_11 ; Q_12 : Q_13
Q_00 ; Q_01 ; Q_02 : Q_03

say Q_00 has a zero, then does Q_02, Q_20, etc
to the right and above Q_00 (Q_01, Q_10) they have poles.

I really appreciate all the time you have put in to helping me
understand this
Thank you
Thank you
The problem is, I'm uncertain whether this is showing that f DOES
actually extend to a meromorphic function

2) With the aid of a diagram I can see that the zeros of the map are
images of 0.
im pretty sure that not all the maps of 0 (2a+2b*i where a,b element of
the integers) are zeros, but im confused to how to progress with this
question.

any help would be much appreciated


************************

David C. Ullrich


************************

David C. Ullrich


************************

David C. Ullrich

.

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