Conformal Mappings

From: Julien Santini (santini.julien_at_wanadoo.fr)
Date: 07/15/04 

Date: Thu, 15 Jul 2004 08:41:39 +0200

Hello,

In the definition of a conformal mapping, Serge Lang [Complex Analysis] says
that "they conserve [oriented] angles", and taking a holomorphic map f: U->C
(C the complex and U an open subset of C) and two curves g: [a,b]->U and h:
[a,b]->U which are differentiable, he shows that at any point z = g(t) =
h(t) such that f '(z) <>0, g '(t) <>0 and h'(t) <>0, we must have
Angle(vector(g(t)),vector(h(t))) = Angle(vector(f(g(t)),vector(f(h(t))),
where it is understood that I identified complex numbers with elements in
R^2.
The point is: we're dealing only with regular points of the curves f and g
here. Hopefully, when f is holomorphic (whence infinitely differentiable,
which the reader is not supposed to know at this point of the book) angles
are preserved also at singular points where the tangent lines exist.
But what if f were not holomorphic ?? In this case, the tangent lines of g
and h may exist but f may not be sufficiently differentiable in order to
insure that the tangent lines of fog and foh exist.
That raises the question: what's the exact definition of a conformal mapping
? Shall it also conserve angles at singular points (or maybe it is a direct
consequence of the conservation of angles at regular points ? -not clear)
Everywhere I looked on the web nothing really precise was said about that
(mathworld, planetmath ...) 

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