A family \(\mathcal F\) of holomorphic functions on \(\Omega\) is normal if every sequence in \(\mathcal F\) has a subsequence converging uniformly on compact subsets of \(\Omega\). If holomorphic \(f_n\) converge to \(f\) uniformly on compact subsets of \(\Omega\), then \(f\) is holomorphic by Morera’s theorem, and \(f_n'\to f'\) uniformly on compact subsets by Cauchy’s integral formula. The Arzelà–Ascoli theorem: a family of continuous functions on \(\Omega\) that is uniformly bounded and equicontinuous on each compact subset has the property that every sequence in it has a subsequence converging uniformly on compact subsets. Montel’s theorem: a family of holomorphic functions that is uniformly bounded on each compact subset of \(\Omega\) is normal, because Cauchy’s estimate makes it equicontinuous on compact subsets. Riemann mapping theorem: every simply connected domain \(\Omega\subsetneq\CC\) is biholomorphic to \(\DD\); the map is obtained by maximizing \(\abs{f'(z_0)}\) over the injective holomorphic \(f\colon\Omega\to\DD\) with \(f(z_0)=0\), a maximum that Montel’s theorem and Hurwitz’s theorem show is attained. Montel’s theorem for omitted values: a family of holomorphic functions on \(\Omega\) that all omit the same two complex values is normal in the sense that every sequence has a subsequence converging uniformly on compact subsets either to a holomorphic function or to \(\infty\). It implies Picard’s great theorem: in every punctured neighbourhood of an essential singularity, a function takes every complex value, with at most one exception, infinitely often.
Locally uniform convergence (Tag D-AIQG3)
Let \(\Omega\subseteq\CC\) be open and let \(f_n,f\colon\Omega\to\CC\) for \(n\ge1\). The sequence \((f_n)\) converges locally uniformly to \(f\) on \(\Omega\) if \(f_n\to f\) uniformly on every compact subset \(K\subseteq\Omega\).
Locally uniform limits of holomorphic functions are holomorphic (Tag C-TODSQ)
Let \(\Omega\subseteq\CC\) be open, and let \((f_n)_{n\in\NN}\) be holomorphic functions on \(\Omega\) that converge to \(f\) uniformly on every compact subset of \(\Omega\). Then \(f\) is holomorphic on \(\Omega\), and \(f_n'\to f'\) uniformly on every compact subset of \(\Omega\).
Equicontinuity on a set (Tag D-PPYCK)
Let \(\Omega\subseteq\CC\) be open, let \(K\subseteq\Omega\), and let \(\mathcal F\) be a family of holomorphic functions on \(\Omega\). The family \(\mathcal F\) is equicontinuous on \(K\) if for every \(\varepsilon>0\) there exists \(\delta>0\) such that \[ \abs{f(z)-f(w)}<\varepsilon\qquad\text{for all } f\in\mathcal F \text{ and all } z,w\in K \text{ with } \abs{z-w}<\delta. \]
Families uniformly bounded on compact subsets (Tag D-MBDTR)
Let \(\Omega\subseteq\CC\) be open and let \(\mathcal F\) be a family of holomorphic functions on \(\Omega\). The family \(\mathcal F\) is uniformly bounded on compact subsets of \(\Omega\) if for every compact set \(K\subseteq\Omega\) there exists \(M>0\) such that \[ \abs{f(z)}\le M\qquad\text{for all } z\in K \text{ and all } f\in\mathcal F. \]
Arzelà--Ascoli theorem (Tag T-6F3GO)
Let \(X\) be a compact Hausdorff space, and give \(C(X;\RR)\) the uniform norm \[ \norm{f}_{\infty,X}\coloneqq\sup_{x\in X}\abs{f(x)}, \] under which it is a Banach space. A subset \(A\subseteq C(X;\RR)\) is compact if and only if \(A\) is closed, uniformly bounded, and equicontinuous at every point of \(X\).
In particular, for \(X=[a,b]\subseteq\RR\), every uniformly bounded and uniformly equicontinuous sequence in \(C([a,b];\RR)\) has a uniformly convergent subsequence.
Normal families (Tag D-IJMPJ)
Let \(\Omega\subseteq\CC\) be open. A family \(\mathcal F=\{f_j\}_{j\in J}\) of holomorphic functions on \(\Omega\) is normal if every sequence in \(\mathcal F\) has a subsequence that converges locally uniformly on \(\Omega\).
Montel's theorem (Tag T-4ALS2)
Let \(\Omega\subseteq\CC\) be open and let \(\mcf\) be a family of holomorphic functions on \(\Omega\) that is uniformly bounded on compact subsets of \(\Omega\). Then
\(\mcf\) is equicontinuous on every compact subset of \(\Omega\), and
\(\mcf\) is a normal family.
Let \(K\subseteq\Omega\) be compact and choose \(r>0\) with \(3r<\operatorname{dist}(K,\CC\sm\Omega)\) (any \(r>0\) if \(\Omega=\CC\)). The set \(K'\coloneqq\ts{z\st\operatorname{dist}(z,K)\le2r}\) is a compact subset of \(\Omega\), so there is \(M\) with \(\abs f\le M\) on \(K'\) for all \(f\in\mcf\). For \(z,w\in K\) with \(\abs{z-w}<r\), the circle \(C=\ts{\abs{\zeta-w}=2r}\) lies in \(K'\), and the Cauchy integral formula gives \[ \abs{f(z)-f(w)}=\abs{\frac{1}{2\pi i}\int_Cf(\zeta)\Big(\frac{1}{\zeta-z}-\frac{1}{\zeta-w}\Big)d\zeta}\le\frac{1}{2\pi}\cdot2\pi(2r)\cdot M\cdot\frac{\abs{z-w}}{r\cdot2r}=\frac{M}{r}\abs{z-w}, \] which is independent of \(f\); this is equicontinuity on \(K\). Normality follows from the Arzelà–Ascoli theorem applied on an exhaustion of \(\Omega\) by compact sets, with a diagonal argument.
Montel’s second theorem uses the extended convention for normality in the remark on D-QTJ7T, where a subsequence may instead tend to \(\infty\) uniformly on compact subsets: a family of meromorphic functions on a region \(\Omega\) that omits the same three values of the Riemann sphere is normal in that sense.
Locally uniformly bounded holomorphic families are equicontinuous and normal (Tag T-MCB7V)
Let \(\Omega\subseteq\CC\) be open and let \(\mcf\) be a family of holomorphic functions on \(\Omega\) that is uniformly bounded on compact subsets of $\Omega$. Then \(\mcf\) is equicontinuous on every compact subset of \(\Omega\), and \(\mcf\) is a normal family.
Hurwitz's theorem (Tag T-FZWEC)
Let \(\Omega\subseteq\CC\) be open and let \((f_k)_{k\geq1}\) be a sequence of holomorphic functions on \(\Omega\) converging locally uniformly on \(\Omega\) to \(f\). Suppose \(f\) has a zero of order \(n\) at \(z_0\in\Omega\). Then there is \(\delta>0\) such that for all sufficiently large \(k\), \(f_k\) has exactly \(n\) zeros in the disc \(\{z : \abs{z-z_0}<\delta\}\), counted with multiplicity. Moreover, for every \(\varepsilon\in(0,\delta)\) there is \(K\) such that for all \(k\geq K\) these \(n\) zeros lie in \(\{z : \abs{z-z_0}<\varepsilon\}\).
Locally uniform limits of univalent functions are univalent or constant (Tag T-SULVA)
Let \(\Omega\subseteq\CC\) be a connected open set and let \((f_k)_{k\geq1}\) be univalent functions on \(\Omega\) converging locally uniformly on \(\Omega\) to \(f\). Then \(f\) is either univalent or constant.
Riemann mapping theorem (Tag T-4MDS6)
Let \(\Omega\subsetneq\CC\) be a nonempty simply connected open set. Then for every \(z_0\in\Omega\) there exists a unique biholomorphism \(F\colon\Omega\to\DD\) such that \(F(z_0)=0\) and \(F'(z_0)>0\). In particular, any two nonempty simply connected open sets \(\Omega_1,\Omega_2\subsetneq\CC\) are biholomorphic.
This is [@SS03].
Riemann mapping theorem (Tag T-55MPA)
Let \(\Omega\subsetneq\CC\) be a nonempty simply connected open set. Then for every \(z_0\in\Omega\) there exists a unique biholomorphism \(F\colon\Omega\to\DD\) such that \(F(z_0)=0\) and \(F'(z_0)>0\). In particular, any two nonempty simply connected open sets \(\Omega_1,\Omega_2\subsetneq\CC\) are biholomorphic.
Picard's little theorem (Tag T-HWBWI)
Let \(f\) be a nonconstant entire function. Then \(f\) takes every value in \(\CC\) with at most one exception. Equivalently, an entire function that omits two distinct values of \(\CC\) is constant.
Picard's big theorem (Tag T-DDOWW)
Let \(z_0\in\CC\), let \(\delta>0\), and let \(f\) be meromorphic on the punctured disc \(\{z : 0<\abs{z-z_0}<\delta\}\). If there are three distinct points of \(\CC\cup\{\infty\}\) that \(f\) does not take on this punctured disc, then \(f\) extends to a meromorphic function on the disc \(\{z : \abs{z-z_0}<\delta\}\).