Re: conformal map regular polygone into unit circle.
- From: Willi Moehring <wmoehri@xxxxxxx>
- Date: 20 Oct 2005 10:15:00 -0400
On 20 Sep 2005 vanesch@xxxxxx wrote:
>
> Hi All,
>
> I'm looking for the conformal mapping (using complex functions) that
> maps the unit circle (or the upper half plane) into a REGULAR polygon
> with n vertices. I know the Schwarz-Christoffel transformation for an
> ARBITRARY polygon, but that doesn't help me because the expression is
> way too complex to be integrated (I'm trying to find the mapping for a
> polygon with 120 vertices). I was hoping that the fact that the polygon
> is REGULAR would simplify the problem. I used the mapping on the unit
> circle in the S-C transform because out of the symmetry of the problem,
> that allowed me (I would guess) to fix the unknown images of the
> vertices: they should also be on a regular polygon. But nevertheless, I
> cannot solve the integral beyond n = 4.
>
> Any hints, papers, books, ... welcome.
>
> thanks,
> Patrick.
>
>
Hi Patrick,
I think your idea with symmetry is excellent. It seem to me, that the
integral is generally a hypergeometric function.
The SC-transformation
(see e.g. Frank, v. Mises Differential-und Integralgleichungen der Physik)
z= G(x) =
int_0^x (1-e^(pi i a) u)^a (1-e^(2 pi i a) u)^a (1-e^(3 pi i a) u)^a
....(1-e^(n pi i a) u) du =
int_0^x (1-u^n)^a du
with a=-2/n
transforms a circle in the x-plane into a regular n-gon in the z-plane. One
has obviously G(e^(pi i a) x) = e^(pi i a) G(x). This means that the image is
unchanged if it is rotated by an angle 2 pi/n.
Substituting v=u^n shows that the function G(x) is an incomplete Beta-function
which can also be expressed in terms of a hypergeometric function (Abramowitz,
Stegun Handbook of Mathematical Functions)
G(x)=1/n B_{x^n}(1/n, 1-2/n)
=x F(1/n,2/n;1+1/n;x^n)
Willi
.
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