Editing Riemann mapping theorem (section)

=== Simple connectivity ===

'''Theorem.''' For an open domain <math>G\subset\mathbb{C}</math> the following conditions are equivalent:<ref>See
*{{harvnb|Ahlfors|1978}}
*{{harvnb|Beardon|1979}}
*{{harvnb|Conway|1978}}
*{{harvnb|Gamelin|2001}}</ref>

# <math>G</math> is simply connected;
# the integral of every holomorphic function <math>f</math> around a closed piecewise smooth curve in <math>G</math> vanishes;
# every holomorphic function in <math>G</math> is the derivative of a holomorphic function; 
# every nowhere-vanishing holomorphic function <math>f</math> on <math>G</math> has a holomorphic logarithm;
# every nowhere-vanishing holomorphic function <math>g</math> on <math>G</math> has a holomorphic square root;
# for any <math>w\notin G</math>, the [[winding number]] of <math>w</math> for any piecewise smooth closed curve in <math>G</math> is <math>0</math>;
# the complement of <math>G</math> in the extended complex plane <math>\mathbb{C}\cup\{\infty\}</math> is connected.

(1) ⇒ (2) because any continuous closed curve, with base point <math>a\in G</math>, can be continuously deformed to the constant curve <math>a</math>.  So the line integral of <math>f\,\mathrm{d}z</math> over the curve is <math>0</math>.

(2) ⇒ (3) because the integral over any piecewise smooth path <math>\gamma</math> from <math>a</math> to <math>z</math> can be used to define a primitive.

(3) ⇒ (4) by integrating <math>f^{-1}\,\mathrm{d}f/\mathrm{d}z</math> along <math>\gamma</math> from <math>a</math> to <math>x</math> to give a branch of the logarithm.

(4) ⇒ (5) by taking the square root as <math>g(z)=\exp(f(x)/2)</math> where <math>f</math> is a holomorphic choice of logarithm.

(5) ⇒ (6) because if <math>\gamma</math> is a piecewise closed curve and <math>f_n</math> are successive square roots of <math>z-w</math> for <math>w</math> outside <math>G</math>, then the winding number of <math>\gamma</math> about <math>w</math> is <math>2^n</math> times the winding number of <math>f_n\circ\gamma</math> about <math>0</math>. Hence the winding number of <math>\gamma</math> about <math>w</math> must be divisible by <math>2^n</math> for all <math>n</math>, so it must equal <math>0</math>.

(6) ⇒ (7) for otherwise the extended plane <math>\mathbb{C}\cup\{\infty\}\setminus G</math> can be written as the disjoint union of two open and closed sets <math>A</math> and <math>B</math> with <math>\infty\in B</math> and <math>A</math> bounded. Let <math>\delta>0</math> be the shortest Euclidean distance between <math>A</math> and <math>B</math> and build a square grid on <math>\mathbb{C}</math> with length <math>\delta/4</math> with a point <math>a</math> of <math>A</math> at the centre of a square. Let <math>C</math> be the compact set of the union of all squares with distance <math>\leq\delta/4</math> from <math>A</math>. Then <math>C\cap B=\varnothing</math> and <math>\partial C</math> does not meet <math>A</math> or <math>B</math>: it consists of finitely many horizontal and vertical segments in <math>G</math> forming a finite number of closed rectangular paths <math>\gamma_j\in G</math>. Taking <math>C_i</math> to be all the squares covering <math>A</math>, then <math>\frac{1}{2\pi}\int_{\partial C}\mathrm{d}\mathrm{arg}(z-a)</math> equals the sum of the winding numbers of <math>C_i</math> 
over <math>a</math>, thus giving <math>1</math>. On the other hand the sum of the winding numbers of <math>\gamma_j</math> about <math>a</math> equals <math>1</math>. Hence the winding number of at least one of the <math>\gamma_j</math> about <math>a</math> is non-zero.

(7) ⇒ (1) This is a purely topological argument. Let <math>\gamma</math> be a piecewise smooth closed curve based at <math>z_0\in G</math>. By approximation γ is in the same [[homotopy]] class as a rectangular path on the square grid of length <math>\delta>0</math> based at <math>z_0</math>; such a rectangular path is determined by a succession of <math>N</math> consecutive directed vertical and horizontal sides. By induction on <math>N</math>, such a path can be deformed to a constant path at a corner of the grid. If the path intersects at a point <math>z_1</math>, then it breaks up into two rectangular paths of length <math><N</math>, and thus can be deformed to the constant path at <math>z_1</math> by the induction hypothesis and elementary properties of the [[fundamental group]]. The reasoning follows a "northeast argument":<ref>{{harvnb|Gamelin|2001|pages=256–257}}, elementary proof</ref><ref>{{harvnb|Berenstein|Gay|1991|pages=86–87}}</ref> in the non self-intersecting path there will be a corner <math>z_0</math> with largest real part (easterly) and then amongst those one with largest imaginary part (northerly). Reversing direction if need be, the path go from <math>z_0-\delta</math> to <math>z_0</math> and then to <math>w_0=z_0-in\delta</math> for <math>n\geq1</math> and then goes leftwards to <math>w_0-\delta</math>. Let <math>R</math> be the open rectangle with these vertices. The winding number of the path is <math>0</math> for points to the right of the vertical segment from <math>z_0</math> to <math>w_0</math> and <math>-1</math> for points to the right; and hence inside <math>R</math>. Since the winding number is <math>0</math> off <math>G</math>, <math>R</math> lies in <math>G</math>. If <math>z</math> is a point of the path, it must lie in <math>G</math>; if <math>z</math> is on <math>\partial R</math> but not on the path, by continuity the winding number of the path about <math>z</math> is <math>-1</math>, so <math>z</math> must also lie in <math>G</math>. Hence <math>R\cup\partial R\subset G</math>. But in this case the path can be deformed by replacing the three sides of the rectangle by the fourth, resulting in two less sides (with self-intersections permitted).