Editing Riemann mapping theorem (section)

=== Riemann mapping theorem ===

*'''Weierstrass' convergence theorem.''' The uniform limit on compacta of a sequence of holomorphic functions is holomorphic; similarly for derivatives.
::This is an immediate consequence of [[Morera's theorem]] for the first statement. [[Cauchy's integral formula]] gives a formula for the derivatives which can be used to check that the derivatives also converge uniformly on compacta.<ref>{{harvnb|Gamelin|2001}}</ref>

*'''[[Hurwitz's theorem (complex analysis)|Hurwitz's theorem]].''' If a sequence of nowhere-vanishing holomorphic functions on an open domain has a uniform limit on compacta, then either the limit is identically zero or the limit is nowhere-vanishing. If a sequence of univalent holomorphic functions on an open domain has a uniform limit on compacta, then either the limit is constant or the limit is univalent.
::If the limit function is non-zero, then its zeros have to be isolated. Zeros with multiplicities can be counted by the winding number <math>\frac{1}{2\pi i}\int_Cg^{-1}(z)g'(z)\mathrm{d}z</math> for a holomorphic function <math>g</math>. Hence winding numbers are continuous under uniform limits, so that if each function in the sequence has no zeros nor can the limit. For the second statement suppose that <math>f(a)=f(b)</math> and set <math>g_n(z)=f_n(z)-f_n(a)</math>. These are nowhere-vanishing on a disk but <math>g(z)=f(z)-f(a)</math> vanishes at <math>b</math>, so <math>g</math> must vanish identically.<ref>{{harvnb|Gamelin|2001}}</ref>

'''Definitions.''' A family <math>{\cal F}</math> of holomorphic functions on an open domain is said to be ''normal'' if any sequence of functions in <math>{\cal F}</math> has a subsequence that converges to a holomorphic function uniformly on compacta. 
A family <math>{\cal F}</math> is ''compact'' if whenever a sequence <math>f_n</math> lies in <math>{\cal F}</math> and converges uniformly to <math>f</math> on compacta, then <math>f</math> also lies in <math>{\cal F}</math>. A family <math>{\cal F}</math> is said to be ''locally bounded'' if their functions are uniformly bounded on each compact disk. Differentiating the  [[Cauchy integral formula]], it follows that the derivatives of a locally bounded family are also locally bounded.<ref>{{harvnb|Duren|1983}}</ref><ref>{{harvnb|Jänich|1993}}</ref>

*'''[[Montel's theorem]].''' Every locally bounded family of holomorphic functions in a domain <math>G</math> is normal.  
::Let <math>f_n</math> be a totally bounded sequence and chose a countable dense subset <math>w_m</math> of <math>G</math>. By locally boundedness and a "[[Diagonal argument (proof technique)|diagonal argument]]", a subsequence can be chosen so that <math>g_n</math> is convergent at each point <math>w_m</math>. It must be verified that this sequence of holomorphic functions converges on <math>G</math> uniformly on each compactum <math>K</math>. Take <math>E</math> open with <math>K\subset E</math> such that the closure of <math>E</math> is compact and contains <math>G</math>. Since the sequence <math>\{g_n'\}</math> is locally bounded, <math>|g_n'|\leq M</math> on <math>E</math>. By compactness, if <math>\delta>0</math> is taken small enough, finitely many open disks <math>D_k</math> of radius <math>\delta>0</math> are required to cover <math>K</math> while remaining in <math>E</math>. Since 
:::<math>g_n(b) - g_n(a)= \int_a^b g_n^\prime(z)\, dz</math>,
::we have that <math>|g_n(a)-g_n(b)|\leq M|a-b|\leq2\delta M</math>. Now for each <math>k</math> choose some <math>w_i</math> in <math>D_k</math> where <math>g_n(w_i)</math> converges, take <math>n</math> and <math>m</math> so large to be within <math>\delta</math> of its limit. Then for <math>z\in D_k</math>,
:::<math>|g_n(z) - g_m(z)| \leq |g_n(z) - g_n(w_i)| + |g_n(w_i) - g_m(w_i)| + |g_m(w_1) - g_m(z)|\leq  4M\delta + 2\delta.</math>
::Hence the sequence <math>\{g_n\}</math> forms a Cauchy sequence in the uniform norm on <math>K</math> as required.<ref>{{harvnb|Duren|1983}}</ref><ref>{{harvnb|Jänich|1993}}</ref>

*'''Riemann mapping theorem.''' If <math>G\neq\mathbb{C}</math> is a simply connected domain and <math>a\in G</math>, there is a unique conformal mapping <math>f</math> of <math>G</math> onto the unit disk <math>D</math> normalized such that <math>f(a)=0</math> and <math>f'(a)>0</math>.
::Uniqueness follows because if <math>f</math> and <math>g</math> satisfied the same conditions, <math>h=f\circ g^{-1}</math> would be a univalent holomorphic map of the unit disk with <math>h(0)=0</math> and <math>h'(0)>0</math>. But by the [[Schwarz lemma]], the univalent holomorphic maps of the unit disk onto itself are given by the [[Möbius transformation]]s
:::<math>k(z)=e^{i\theta}(z-\alpha)/(1-\overline{\alpha} z)</math>
::with <math>|\alpha|<1</math>. So <math>h</math> must be the identity map and <math>f=g</math>.
::To prove existence, take <math>{\cal F}</math> to be the family of holomorphic univalent mappings <math>f</math> of <math>G</math> into the open unit disk <math>D</math> with <math>f(a)=0</math> and <math>f'(a)>0</math>. It is a normal family by Montel's theorem. By the characterization of simple-connectivity, for <math>b\in\mathbb{C}\setminus G</math> there is a holomorphic branch of the square root <math>h(z)=\sqrt{z -b}</math> in <math>G</math>. It is univalent and <math>h(z_1)\neq-h(z_2)</math> for <math>z_1,z_2\in G</math>. By the [[open mapping theorem (complex analysis)| open mapping theorem]], <math>h(G)</math> contains a closed disk <math>\Delta</math>; say with centre <math>h(a)</math> and radius <math>r>0</math>.  Thus no points of <math>-\Delta</math> can lie in <math>h(G)</math>. Let <math>F</math> be the unique Möbius transformation taking <math>\mathbb{C}\setminus-\Delta</math> onto <math>D</math> with the normalization <math>F(h(a))=0</math> and <math>(F \circ h)'(a)=F'(h(a))\cdot h'(a)>0</math>. By construction <math>F\circ h</math> is in <math>{\cal F}</math>, so that <math>{\cal F}</math> is ''non-empty''. The method of [[Paul Koebe|Koebe]] is to use an ''extremal function'' to produce a conformal mapping solving the problem: in this situation it is often called the ''Ahlfors function'' of {{math|''G''}}, after [[Lars Ahlfors|Ahlfors]].<ref>{{harvnb|Gamelin|2001|page=309}}</ref> Let <math>0<M\leq\infty</math> be the supremum of <math>f'(a)</math> for <math>f\in{\cal F}</math>. Pick <math>f_n\in{\cal F}</math> with <math>f_n'(a)</math> tending to <math>M</math>. By Montel's theorem, passing to a subsequence if necessary, <math>f_n</math> tends to a holomorphic function <math>f</math> uniformly on compacta. By Hurwitz's theorem, <math>f</math> is either univalent or constant. But <math>f</math> has <math>f(a)=0</math> and <math>f'(a)>0</math>. So <math>M</math> is finite, equal to <math>f'(a)>0</math> and <math>{f\in\cal F}</math>. It remains to check that the conformal mapping <math>f</math> takes <math>G</math> ''onto'' <math>D</math>. If not, take <math>c\neq0</math> in <math>D\setminus f(G)</math> and let <math>H</math> be a holomorphic square root of <math>(f(z)-c)/(1-\overline{c}f(z))</math> on <math>G</math>. The function <math>H</math> is univalent and maps <math>G</math> into <math>D</math>. Let
:::<math>F(z)=\frac{e^{i\theta}(H(z)-H(a))}{1-\overline{H(a)}H(z)},</math>
::where <math>H'(a)/|H'(a)|=e^{-i\theta}</math>. Then <math>F\in{\cal F}</math> and a routine computation shows that
:::<math>F'(a)=H'(a)/(1-|H(a)|^2)=f'(a)\left(\sqrt{|c|}+\sqrt{|c|^{-1}}\right)/2>f'(a)=M.</math>
::This contradicts the maximality of <math>M</math>, so that <math>f</math> must take all values in <math>D</math>.<ref>{{harvnb|Duren|1983}}</ref><ref>{{harvnb|Jänich|1993}}</ref><ref>{{harvnb|Ahlfors|1978}}</ref>

'''Remark.''' As a consequence of the Riemann mapping theorem, every simply connected domain in the plane is homeomorphic to the unit disk. If points are omitted, this follows from the theorem. For the whole plane, the homeomorphism <math>\phi(z)=z/(1+|z|)</math> gives a homeomorphism of <math>\mathbb{C}</math> onto <math>D</math>.