Re: Riemann Mapping question


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 would like to prove that this is a meromorphic function, and would
also like to find the poles and zeros of this map.

1) I can see that using the Riemann mapping theorem I can get a unique
map from the square to the circle, and that by applying schwarz
reflections (and showing they agree on the overlap) that I can get the
squares to cover the complex plane, which I think obviously (well
visually I can see this!) shows that this set of circles also cover the
complex plane.

I have no idea what you're talking about, sets of circles covering
the plane. I _do_ know what set of squares covering the plane
you're talking about, I think...

I really shouldnt do complex analysis at 4 in the morning! On reviewing
this question this morning, I realised what I said about the circles
makes NO sense at all! (they wouldnt be circles!)

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

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

.

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