7.2: B- Results from one Complex Variable
- Page ID
- 74253
We review some results from one complex variable that may be useful for reading this book. The reader should first look through Section 0.1 for basic notation and motivation, although we will review some of the results again here. Let \(U \subset \mathbb{C}\) be open. A function \(f \colon U \to \mathbb{C}\) is holomorphic if it is complex differentiable at every point, that is, \[f'(z) = \lim_{h \in \mathbb{C} \to 0} \frac{f(z+h) - f(z)}{h}\] exists for all \(z \in U\). For example, polynomials and rational functions in \(z\) are holomorphic. Perhaps the most important holomorphic function is the solution to the differential equation \(f'(z) = f(z)\), \(f(0) = 1\), that is the complex exponential, defined in the entire complex plane: \[f(z) = e^z = e^{x+iy} = e^x \bigl( \cos y + i \sin(y) \bigr) .\]
A piecewise-\(C^1\) path (or curve) in \(\mathbb{C}\) is a continuous function \(\gamma \colon [a,b] \to \mathbb{C}\), continuously differentiable except at finitely many points, such that one-sided limits of \(\gamma'\) exist at all points of \([a,b]\) and such that \(\gamma'\) (and its one sided limits) is never zero. By abuse of notation, when \(\gamma\) is used as a set, we mean the image \(\gamma\bigl([a,b]\bigr)\). For a continuous function \(f \colon \gamma \to \mathbb{C}\), define \[\int_\gamma f(z) \, dz \overset{\text{def}}{=} \int_a^b f\bigl(\gamma(t)\bigr) \gamma'(t) \, dt .\] As \(\gamma'\) is defined at all but finitely many points and is otherwise continuous, the integral is well-defined. Similarly, one defines the more general path integral in \(dz = dx + i\,dy\) and \(d\bar{z} = dx - i\, dy\). Let \(z = \gamma(t) = \gamma_1(t) + i \, \gamma_2(t) = x + i\, y\) parametrize the path. Then \[\begin{align}\begin{aligned} \int_\gamma f(z) \, dz + g(z) \, d\bar{z} & = \int_\gamma \Bigl(f\bigl(x+i\, y\bigr) + g\bigl(x+i\, y\bigr) \Bigr) \, dx + i \, \Bigl( f\bigl(x+i\, y\bigr) - g\bigl(x+i\, y\bigr) \Bigr) \, dy \\ & = \int_a^b \Bigl( f\bigl(\gamma(t)\bigr) \gamma'(t) + f\bigl(\gamma(t)\bigr) \overline{\gamma'(t)} \Bigr) \, dt \\ & = \int_a^b \biggl( \Bigl(f\bigl(\gamma(t)\bigr) + g\bigl(\gamma(t)\bigr) \Bigr) \, \gamma_1'(t) + i \, \Bigl( f\bigl(\gamma(t)\bigr) - g\bigl(\gamma(t)\bigr) \Bigr) \gamma_2'(t) \biggr) \, dt . \end{aligned}\end{align}\]
A path is closed if \(\gamma(a) = \gamma(b)\), and a path is simple if \(\gamma|_{(a,b]}\) is one-to-one with the possible exception of \(\gamma(a) = \gamma(b)\).
An open \(U \subset \mathbb{C}\) has piecewise-\(C^{1}\) boundary if for each \(p \in \partial U\) there is an open neighborhood \(W\) of \(p\) such that \(\partial U \cap W = \gamma\bigl((a,b)\bigr)\) where \(\gamma \colon [a,b] \to \mathbb{C}\) is an injective piecewise-\(C^1\) path, and such that each \(p \in \partial U\) is in the closure of \(\mathbb{C} \setminus \overline{U}\). Intuitively the boundary is a piecewise-\(C^1\) curve that locally cuts the plane into two open pieces. If at each point where the parametrization of \(\partial U\) is differentiable the domain is on the left (\(\gamma'(t)\) rotated by \(\frac{\pi}{2}\) points into the domain), then the boundary is positively oriented. As in the introduction, we have the following version of Cauchy integral formula.
Cauchy Integral Formula
Let \(U \subset \mathbb{C}\) be a bounded open set with piecewise-\(C^1\) boundary \(\partial U\) oriented positively, and let \(f \colon \overline{U} \to \mathbb{C}\) be a continuous function holomorphic in \(U\). Then for \(z \in U\), \[f(z) = \frac{1}{2\pi i} \int_{\partial U} \frac{f(\zeta)}{\zeta-z} \, d \zeta .\]
The theorem follows from Green’s theorem, which is the Stokes’ theorem in two dimensions. In the versions we state, one needs to approximate the open set by smaller open sets from the inside to insure the partial derivatives are bounded. See Theorem 4.1.1. Let us state Green’s theorem using the \(dz\) and \(d\bar{z}\) for completeness. See Appendix C for an overview of differential forms.
Green's Theorem
Let \(U \subset \mathbb{C}\) be a bounded open set with piecewise-\(C^1\) boundary \(\partial U\) oriented positively, and let \(f \colon \overline{U} \to \mathbb{C}\) be continuous with bounded continuous partial derivatives in \(U\). Then
\[\begin{align}\begin{aligned} \int_{\partial U} f(z) \, dz + g(z) \, d\bar{z}& = \int_{U} d \Bigl( f(z) \, dz + g(z) \, d\bar{z} \Bigr) = \int_{U} \left( \frac{\partial g}{\partial z} - \frac{\partial f}{\partial \bar{z}} \right) \, dz \wedge d\bar{z} \\ &= (-2i) \int_{U} \left( \frac{\partial g}{\partial z} - \frac{\partial f}{\partial \bar{z}} \right) \, dx \wedge dy = (-2i) \int_{U} \left( \frac{\partial g}{\partial z} - \frac{\partial f}{\partial \bar{z}} \right) \, dA.\end{aligned}\end{align}\]
The Cauchy integral formula is equivalent to what is usually called just Cauchy’s theorem:
Cauchy
Let \(U \subset \mathbb{C}\) be a bounded open set with piecewise-\(C^1\) boundary \(\partial U\) oriented positively, and let \(f \colon \overline{U} \to \mathbb{C}\) be a continuous function holomorphic in \(U\). Then \[\int_{\partial U} f(z) \, dz = 0 .\]
There is a converse to Cauchy as well. A triangle \(T \subset \mathbb{C}\) is the convex hull of the three vertices (we include the inside of the triangle), and \(\partial T\) is the boundary of the triangle oriented counter-clockwise. Let us state the following theorem as an “if and only if,” even though, usually it is only the reverse direction that is called Morera’s theorem.
Morera
Suppose \(U \subset \mathbb{C}\) is an open set, and \(f \colon U \to \mathbb{C}\) is continuous. Then \(f\) is holomorphic if and only if \[\int_{\partial T} f(z) \, dz = 0\] for all triangles \(T \subset U\).
As we saw in the introduction, a holomorphic function has a power series.
If \(U \subset \mathbb{C}\) is open and \(f \colon U \to \mathbb{C}\) is holomorphic, then \(f\) is infinitely differentiable, and if \(\Delta_\rho(p) \subset \mathbb{C}\) is a disc, then \(f\) has a power series that converges absolutely uniformly on compact subsets of \(\Delta_\rho(p)\): \[f(z) = \sum_{k=0}^\infty c_k {(z-p)}^k ,\] where given a simple closed (piecewise-\(C^1\)) path \(\gamma\) going once counter-clockwise around \(p\) inside \(\Delta_\rho(p)\),
\[c_{k}=\frac{f^{(k)}(p)}{k!}=\frac{1}{2\pi i}\int_{\gamma}\frac{f(\zeta )}{(\zeta -z)^{k+1}}d\zeta .\]
Cauchy estimates follow: If \(M\) is the maximum of \(|f|\) on the circle \(\partial \Delta_r(p)\), then \[|c_k| \leq \frac{M}{r^k} .\] Conversely, if a power series satisfies such estimates, it converges on \(\Delta_r(p)\).
A holomorphic \(f \colon \mathbb{C} \to \mathbb{C} \) that is entire. An immediate application of Cauchy estimates is Liouville’s theorem:
Liouville
If \(f\) is entire and bounded, then \(f\) is constant.
And as a holomorphic function has a power series it satisfies the identity theorem:
Identity
Suppose \(U \subset \mathbb{C}\) is a domain and \(f \colon U \to \mathbb{C}\) is holomorphic. If the zero set \(f^{-1}(0)\) has a limit point in \(U\), then \(f \equiv 0\).
Another consequence of the Cauchy integral formula is that there is a differential equation characterizing holomorphic functions.
Cauchy-Riemann Equations
Let \(U \subset \mathbb{C}\) be open. A function \(f \colon U \to \mathbb{C}\) is holomorphic if and only if \(f\) is continuously differentiable and \[\frac{\partial f}{\partial \bar{z}} = \frac{1}{2} \left( \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} \right) = 0 \qquad \text{on $U$.}\]
Yet another consequence of the Cauchy formula (and one can make an argument that everything in this appendix is a consequence of the Cauchy formula) is the open mapping theorem.
Open Mapping Theorem
Suppose \(U \subset \mathbb{C}\) is a domain and \(f \colon U \to \mathbb{C}\) is holomorphic and not constant. Then \(f\) is an open mapping, that is, \(f(V)\) is open whenever \(V\) is open.
The real and imaginary parts \(u\) and \(v\) of a holomorphic function \(f = u+iv\) are harmonic, that is \(\nabla^2 u = \nabla^2 v = 0\), where \(\nabla^2\) is the Laplacian. A domain \(U\) is simply connected if every simple closed path is homotopic in \(U\) to a constant, in other words, if the domain has no holes. For example a disc is simply connected.
If \(U \subset \mathbb{C}\) is a simply connected domain and \(u \colon U \to \mathbb{R}\) is harmonic, then there exists a harmonic function \(v \colon U \to \mathbb{R}\) such that \(f = u+iv\) is holomorphic.
The function \(v\) is called the harmonic conjugate of \(u\). For further review of harmonic functions see Section 2.4 on harmonic functions. We have the following versions of the maximum principle.
Maximum Principles
Suppose \(U \subset \mathbb{C}\) is a domain.
- If \(f \colon U \to \mathbb{C}\) is holomorphic and \(|f|\) achieves a local maximum in \(U\), then \(f\) is constant.
- If \(U\) is bounded and \(f \colon \overline{U} \to \mathbb{C}\) is holomorphic in \(U\) and continuous, then \(|f|\) achieves its maximum on \(\partial U\).
- If \(f \colon U \to \mathbb{R}\) is harmonic achieves a local maximum or a minimum in \(U\), then \(f\) is constant.
- If \(U\) is bounded and \(f \colon \overline{U} \to \mathbb{R}\) is harmonic in \(U\) and continuous, then \(f\) achieves its maximum and minimum on \(\partial U\).
The first two items are sometimes called the maximum modulus principle. The maximum principle immediately implies the following lemma.
Schwarz's Lemma
Suppose \(f \colon \mathbb{D} \to \mathbb{D}\) is holomorphic and \(f(0) = 0\), then
- \(|f(z)| \leq |z|\), and
- \(|f'(0)| \leq 1\).
Furthermore, if \(|f(z_0)| = |z_0|\) for some \(z_0 \in \mathbb{D} \setminus \{ 0 \}\) or \(|f'(0)| = 1\), then for some \(\theta \in \mathbb{R}\) we have \(f(z) = e^{i\theta} z\) for all \(z \in \mathbb{D}\).
The theorem above is actually quite general.
Riemann Mapping Theorem
If \(U \subset \mathbb{C}\) is a nonempty simply connected domain such that \(U \neq \mathbb{C}\), then \(U\) is biholomorphic to \(\mathbb{D}\). Given \(z_0 \in U\) there exists a unique biholomorphic \(f \colon U \to \mathbb{D}\) such that \(f(z_0) = 0\), \(f'(z_0) > 0\), and \(f\) maximizes \(|f'(z_0)|\) among all injective holomorphic maps to \(\mathbb{D}\) such that \(f(z_0) = 0\).
Schwarz’s lemma can also be used to classify the automorphisms of the disc (and hence any simply connected domain). Let \(\text{Aut}(\mathbb{D})\) denote the group of biholomorphic (both \(f\) and \(f^{-1}\) are holomorphic) self maps of the disc to itself.
If \(f \in \text{Aut}(\mathbb{D})\), then there exists an \(a \in \mathbb{D}\) and \(\theta \in \mathbb{R}\) such that \[f(z) = e^{i\theta} \frac{z-a}{1-\bar{a}z} .\]
Speaking of automorphisms. We have the following version of inverse function theorem.
Suppose \(U\) and \(V\) are open subsets of \(\mathbb{C}\).
- If \(f \colon U \to V\) is holomorphic and bijective (one-to-one and onto), then \(f'(z) \not= 0\) for all \(z \in V\), and \(f^{-1} \colon V \to U\) is holomorphic. If \(f(p) = q\), then \[\left(f^{-1}\right)(q) = \frac{1}{f'(p)} .\]
- If \(f \colon U \to V\) is holomorphic, \(f(p) = q\), and \(f'(p) \not= 0\), then there exists a neighborhood \(W\) of \(q\) and a holomorphic function \(g \colon W \to U\) that is one-to-one and \(f\bigl(g(z)\bigr) = z\) for all \(z \in W\).
The Riemann mapping theorem actually follows from the following theorem about existence of branches of the logarithm.
Suppose \(U \subset \mathbb{C}\) is a simply connected domain, and \(f \colon U \to \mathbb{C}\) is a holomorphic function without zeros in \(U\). Then there exists a holomorphic function \(L \colon U \to \mathbb{C}\) such that \[e^L = f .\] In particular, we can take roots: For every \(k \in \mathbb{N}\), there exists a holomorphic function \(g \colon U \to \mathbb{C}\) such that \[g^k = f .\]
In one complex variable, zeros of holomorphic functions can be divided out. Moreover, zeros of holomorphic functions are of finite order unless the function is identically zero.
Suppose \(U \subset \mathbb{C}\) is a domain and \(f \colon U \to \mathbb{C}\) is holomorphic, not identically zero, and \(f(p) = 0\) for some \(p \in U\). There exists a \(k \in \mathbb{N}\) and a holomorphic function \(g \colon U \to \mathbb{C}\), such that \(g(p) \not= 0\) and \[f(z) = {(z-p)}^k g(z) \qquad \text{for all $z \in U$.}\]
The number \(k\) above is called the order or multiplicity of the zero at \(p\). We can use this fact and the existence of roots to show that every holomorphic function is locally like \(z^k\). The function \(\varphi\) below can be thought of as a local change of coordinates.
Suppose \(U \subset \mathbb{C}\) is a domain and \(f \colon U \to \mathbb{C}\) is holomorphic, not identically zero, and \(p \in U\). Then there exists a \(k \in \mathbb{N}\), a neighborhood \(V \subset U\) of \(p\), and a holomorphic function \(\varphi \colon V \to \mathbb{C}\) with \(\varphi'(p) \not= 0\), such that \[{\bigl(\varphi(z)\bigr)}^k = f(z) - f(p) \qquad \text{for all $z \in V$.}\]
Convergence of holomorphic functions is the same as for continuous functions: uniform convergence on compact subsets. Sometimes this is called normal convergence.
Suppose \(U \subset \mathbb{C}\) is open and \(f_k \colon U \to \mathbb{C}\) is a sequence of holomorphic functions which converge uniformly on compact subsets of \(U\) to \(f \colon U \to \mathbb{C}\). Then \(f\) is holomorphic, and every derivative \(f_k^{(\ell)}\) converges uniformly on compact subsets to the derivative \(f^{(\ell)}\).
Holomorphic functions satisfy a Heine–Borel-like property:
Montel
Suppose \(U \subset \mathbb{C}\) is open and \(f_n \subset U \to \mathbb{C}\) is a sequence of holomorphic functions. If \(\{ f_n \}\) is uniformly bounded on compact subsets of \(U\), then there exists a subsequence converging uniformly on compact subsets of \(U\).
A sequence of holomorphic functions cannot create or delete zeros out of thin air:
Hurwitz
Suppose \(U \subset \mathbb{C}\) is a domain and \(f_n \subset U \to \mathbb{C}\) is a sequence of holomorphic functions converging uniformly on compact subsets of \(U\) to \(f \colon U \to \mathbb{C}\). If \(f\) is not identically zero and \(z_0\) is a zero of \(f\), then there exists a disc \(\Delta_r(z_0)\) and an \(N\), such that for all \(n \geq N\), \(f_n\) has the same number of zeros (counting multiplicity) in \(\Delta_r(z_0)\) as \(f\) (counting multiplicity).
A common application, and sometimes the way the theorem is stated, is that if \(f_n\) have no zeros in \(U\), then either the limit \(f\) is identically zero, or it also has no zeros.
If \(U \subset \mathbb{C}\) is open, \(p \in U\), and \(f \colon U \setminus \{ p \} \to \mathbb{C}\) is holomorphic, we say that \(f\) has an isolated singularity at \(p\). An isolated singularity is removable if there exists a holomorphic function \(F \colon U \to \mathbb{C}\) such that \(f(z) = F(z)\) for all \(z \in U \setminus \{ p \}\). An isolated singularity is a pole if \[\lim_{z \to p} f(z) = \infty \qquad \text{(that is $|f(z)| \to \infty$ as $|z-p| \to 0$)}.\] An isolated singularity that is neither removable nor a pole is said to be essential.
At nonessential isolated singularities the function blows up to a finite integral order. The first part of the following proposition is usually called the Riemann extension theorem.
Suppose \(U \subset \mathbb{C}\) is an open set, \(p \in U\), and \(f \colon U \setminus \{p\} \to \mathbb{C}\) holomorphic.
- If \(f\) is bounded (near \(p\) is enough), then \(p\) is a removable singularity.
- If \(p\) is a pole, there exists a \(k \in \mathbb{N}\) such that \[g(z) = {(z-p)}^k f(z)\] is bounded near \(p\) and hence \(g\) has a removable singularity at \(p\).
The number \(k\) above is called the order of the pole. There is a symmetry between zeros and poles: If \(f\) has a zero of order \(k\), then \(\frac{1}{f}\) has a pole of order \(k\). If \(f\) has a pole of order \(k\), then \(\frac{1}{f}\) has a removable singularity, and the extended function has a zero of order \(k\).
Let \(\mathbb{P}^{1} = \mathbb{C} \cup \{\infty\}\) be the Riemann sphere. The topology on \(\mathbb{P}^{1}\) is given by insisting that the function \(\frac{1}{z}\) is a homeomorphism of \(\mathbb{P}^{1}\) to itself, where \(\frac{1}{\infty} = 0\) and \(\frac{1}{0} = \infty\). A function \(f \colon U \to \mathbb{P}^{1}\) is called meromorphic, if it is not identically \(\infty\), is holomorphic on \(U \setminus f^{-1}(\infty)\), and has poles at \(f^{-1}(\infty)\). A holomorphic function with poles is meromorphic by setting the value to be \(\infty\) at the poles. A meromorphic function is one that can locally be written as a quotient of holomorphic functions.
At an isolated singularity we can expand a holomorphic function via the so-called Laurent series by adding all negative powers. The Laurent series also characterizes the type of the singularity.
If \(\Delta \subset \mathbb{C}\) is a disc centered at \(p \in \mathbb{C}\), and \(f \colon \Delta \setminus \{p\} \to \mathbb{C}\) holomorphic, then there exists a double sequence \(\{ c_{k} \}_{k = -\infty}^\infty\) such that \[f(z) = \sum_{k=-\infty}^\infty c_k {(z-p)}^k ,\] converges absolutely uniformly on compact subsets of \(\Delta\). If \(\gamma\) is a simple closed piecewise-\(C^1\) path going once counter-clockwise around \(p\) in \(\Delta\), then
\[c_{k}=\frac{1}{2\pi i}\int_{\gamma}\frac{f(\zeta )}{(\zeta -z)^{k+1}}d\zeta .\] The singularity at \(p\) is
- removable if \(c_k = 0\) for all \(k < 0\).
- pole of order \(\ell \in \mathbb{N}\) if \(c_k = 0\) for all \(k < -\ell\) and \(c_{-\ell} \not= 0\).
- essential if for every exist infinitely negative \(k\) such that \(c_k \not= 0\).
If \(p\) is an isolated singularity of \(f\), then call the corresponding \(c_{-1}\) the residue of \(f\) at \(p\), and write it as \(\operatorname{Res}(f,p)\). The proposition says that for a small \(\gamma\) around \(p\) in the positive direction, \[\operatorname{Res}(f,p) = c_{-1} = \frac{1}{2\pi i} \int_\gamma f(z) \, dz .\] Combining this equation with Cauchy’s theorem tells us that to compute integrals of functions with isolated singularities we simply need to find the residues, which tend to be simpler to compute. For example, if \(p\) is a simple pole (of order 1), then \[\operatorname{Res}(f,p) = \lim_{z \to p} (z-p)f(z) .\]
Residue Theorem
Suppose \(U \subset \mathbb{C}\) is an open set, and \(\gamma\) is a piecewise-\(C^1\) simple closed path in \(U\) such that the interior of \(\gamma\) is in \(U\). Suppose that \(f \colon U \setminus S \to \mathbb{C}\) is a holomorphic function with isolated singularities in a finite set \(S\), and suppose \(S\) lies in the interior of \(\gamma\). Then \[\int_{\gamma} f(z) \, dz = 2\pi i \sum_{p \in S} \operatorname{Res}(f,p) .\]
The identity theorem says that zeros of a nonconstant holomorphic \(f\) have no limit points, and so are isolated points. Since \(\frac{1}{f}\) is a meromorphic function with zeros at the poles of \(f\), poles are also isolated. Zeros and poles of can be counted fairly easily.
Argument Principle
Suppose \(U \subset \mathbb{C}\) is an open set, and \(\gamma\) is a piecewise-\(C^1\) simple closed path in \(U\) such that the interior of \(\gamma\) is in \(U\). Suppose that \(f \colon U \to \mathbb{P}^{1}\) is a meromorphic function with no zeros or poles on \(\gamma\). Then \[\frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)} \, dz = N - P ,\] where \(N\) is the number of zeros of \(f\) inside \(\gamma\) and \(P\) is the number of poles inside \(\gamma\), both counted with multiplicity.
Furthermore, suppose \(h \colon U \to \mathbb{C}\) is holomorphic. Let \(z_1,\ldots,z_N\) be the zeros of \(f\) inside \(\gamma\) and \(w_1,\ldots,w_P\) be the poles of \(f\) inside \(\gamma\). Then \[\frac{1}{2\pi i} \int_\gamma h(z) \frac{f'(z)}{f(z)} \, dz = \sum_{j=1}^N h(z_j) \quad - \quad \sum_{j=1}^P h(w_j) .\]
The proof is an immediate application of the residue theorem. Simply compute the residues at the zeros and poles of \(f\). In particular, if \(f\) has a zero at \(p\) or multiplicity \(k\), then \(h(z) \frac{f'(z)}{f(z)}\) has a simple pole at \(p\) with residue \(k\, h(p)\). Similarly, if \(f\) has a pole at \(p\) of order \(k\), then \(h(z) \frac{f'(z)}{f(z)}\) has a simple pole with residue \(-k\, h(p)\) at \(p\).
In the couple of theorems above, we avoided introducing winding numbers by making \(\gamma\) a simple closed curve, so the statements above may be slightly different from what you have seen in your one-variable course.
Another useful way to count zeros is Rouché’s theorem.
Rouché
Suppose \(U \subset \mathbb{C}\) is an open set, and \(\gamma\) is a piecewise-\(C^1\) simple closed path in \(U\) such that the interior of \(\gamma\) is in \(U\). Suppose that \(f \colon U \to \mathbb{C}\) and \(g \colon U \to \mathbb{C}\) are holomorphic functions such that \[|f(z)-g(z)| < |f(z)|+|g(z)|\] for all \(z \in \gamma\). Then \(f\) and \(g\) have the same number of zeros inside \(\gamma\) (up to multiplicity).
In the classical statement of the theorem the weaker inequality \(|f(z)-g(z)| < |f(z)|\) is used. Notice that either inequality precludes any zeros on \(\gamma\) itself.
A holomorphic function with an essential singularity achieves essentially every value. A weak version of this result (and an easy to prove one) is the Casorati-Weierstrass theorem: If a holomorphic \(f\) has an essential singularity at \(p\), then for every neighborhood \(W\) of \(p\), \(f\bigl(W \setminus \{p\}\bigr)\) is dense in \(\mathbb{C}\). Let us state the much stronger theorem of Picard: A function with an essential singularity is very wild. It achieves every value (except possibly one) infinitely often.
Picard's Big Theorem
Suppose \(U \subset \mathbb{C}\) is open, \(f \colon U \setminus \{ p \} \to \mathbb{C}\) is holomorphic, and \(f\) has an essential singularity at \(p\). Then for every neighborhood \(W\) of \(p\), \(f\bigl(W \setminus \{ p \}\bigr)\) is either \(\mathbb{C}\) or \(\mathbb{C}\) minus a point.
For example, \(e^{1/z}\) has an essential singularity at the origin and the function is never 0. Since we stated the big theorem, let us also state the little theorem.
Picard's Little Theorem
If \(f \colon \mathbb{C} \to \mathbb{C}\) is holomorphic, then \(f(\mathbb{C})\) is either \(\mathbb{C}\) or \(\mathbb{C}\) minus a point.
One theorem from algebra that is important in complex analysis, and becomes perhaps even more important in several variables is the fundamental theorem of algebra. It really is a theorem of complex analysis and its standard proof is via the maximum principle.
Fundamental Theorem of Algebra
If \(P \colon \mathbb{C} \to \mathbb{C}\) is a nonzero polynomial of degree \(k\),then \(P\) has exactly \(k\) zeros (roots) in \(\mathbb{C}\) counted with multiplicity.
The set of rational functions is dense in the space of holomorphic functions, and we even have control over where the poles need to be. Note that a nonconstant polynomial has a “pole at infinity” meaning \(P(z) \to \infty\) as \(z \to \infty\). Letting \(\mathbb{P}^{1}\) again be the Riemann sphere, we have Runge’s approximation theorem.
Runge
Suppose \(U \subset \mathbb{C}\) is an open set and \(A \subset \mathbb{P}^1 \setminus U\) is a set containing at least one point from each component of \(\mathbb{P}^1 \setminus U\). Suppose \(f \colon U \to \mathbb{C}\) is holomorphic. Then for any \(\epsilon > 0\) and any compact \(K \subset \subset U\), there exists a rational function \(R\) with poles in \(A\) such that \[|R(z) - f(z)| < \epsilon \qquad \text{for all $z \in K$}.\]
Perhaps a surprising generalization of the classical Weierstrass approximation theorem, and one of my favorite one-variable theorems, is Mergelyan’s theorem. It may be good to note that Mergelyan does not follow from Runge.
Mergelyan
Suppose \(K \subset \subset \mathbb{C}\) is a compact set such that \(\mathbb{C} \setminus K\) is connected and \(f \colon K \to \mathbb{C}\) is a continuous function that is holomorphic in the interior \(K^\circ\). Then for any \(\epsilon > 0\) and any compact \(K \subset \subset U\), there exists a polynomial \(P\) such that \[|P(z) - f(z)| < \epsilon \qquad \text{for all $z \in K$}.\]
The reason why the theorem is perhaps surprising is that \(K\) may have only a small or no interior. Using a closed interval \(K=[a,b]\) of the real line we recover the Weierstrass approximation theorem.
Given an open set \(U \subset \mathbb{C}\), we say \(U\) is symmetric with repect to the real axis if \(z \in U\) implies \(\bar{z} \in U\). We divide \(U\) into three parts \[U_+ = \{ z \in U : \Im z > 0 \}, \qquad U_0 = \{ z \in U : \Im z = 0 \}, \qquad U_- = \{ z \in U : \Im z < 0 \}.\] We have the following theorem for extending (reflecting) holomorphic functions past boundaries.
Schwarz Reflection Principle
Suppose \(U \subset \mathbb{C}\) is a domain symmetric with respect to the real axis, \(f \colon U_+ \cup U_0 \to \mathbb{C}\) a continuous function holomorphic on \(U_+\) and real valued on \(U_0\). Then the function \(g \colon U \to \mathbb{C}\) defined by \[g(z) = f(z) \quad \text{if $z \in U_+ \cup U_0$}, \qquad g(z) = \overline{f(\bar{z})} \quad \text{if $z \in U_-$},\] is holomorphic on \(U\).
In fact, the reflection is really about harmonic functions.
Schwarz Reflection Principle for Harmonic Functions
Suppose \(U \subset \mathbb{C}\) is a domain symmetric with respect to the real axis, \(f \colon U_+ \cup U_0 \to \mathbb{R}\) a continuous function harmonic on \(U_+\) and zero on \(U_0\). Then the function \(g \colon U \to \mathbb{R}\) defined by \[g(z) = f(z) \quad \text{if $z \in U_+ \cup U_0$}, \qquad g(z) = -f(\bar{z}) \quad \text{if $z \in U_-$},\] is harmonic on \(U\).
Functions may be defined locally, and continued along paths. Suppose \(p\) is a point and \(D\) is a disc centered at \(p \in D\). A holomorphic function \(f \colon D \to \mathbb{C}\) can be analytically continued along a path \(\gamma \colon [0,1] \to \mathbb{C}\), \(\gamma(0) = p\), if for every \(t \in [0,1]\) there exists a disc \(D_t\) centered at \(\gamma(t)\), where \(D_0=D\), and a holomorphic function \(f_t \colon D_t \to \mathbb{C}\), where \(f_0 = f\), and for each \(t_0 \in [0,1]\) there is an \(\epsilon > 0\) such that if \(|t-t_0| < \epsilon\), then \(f_t = f_{t_0}\) in \(D_t \cap D_{t_0}\). The monodromy theorem says that as long as there are no holes, analytic continuation defines a function uniquely.
Monodromy Theorem
If \(U \subset \mathbb{C}\) is a simply connected domain, \(D \subset U\) a disc and \(f \colon D \to \mathbb{C}\) a holomorphic function that can be analytically continued from \(p \in D\) to every \(q \in U\) along any path from \(p\) to \(q\), then there exists a unique holomorphic function \(F \colon U \to \mathbb{C}\) such that \(F|_D = f\).
An interesting and useful theorem getting an inequality in the opposite direction from Schwarz’s lemma, and one which is often not covered in a one-variable course is the Koebe \(\frac{1}{4}\)-theorem. Think of why no such theorem could possibly hold for just smooth functions. At first glance the theorem should seem quite counterintuitive, and at second glance, it should seem outright outrageous.
Koebe Quarter Theorem
Suppose \(f \colon \mathbb{D} \to \mathbb{C}\) is holomorphic and injective. Then \(f(\mathbb{D})\) contains a disc centered at \(f(0)\) and radius \(\frac{|f'(0)|}{4}\).
The \(\frac{1}{4}\) is sharp, that is, it is the best it can be.
Finally, it is useful to factor out all the zeros of a holomorphic function, not just finitely many. Similarly, we can work with poles.
Weierstrass Product Theorem
Suppose \(U \subset \mathbb{C}\) is a domain, \(\{ a_k \}\), \(\{ b_k \}\) are countable sets in \(U\) with no limit points in \(U\), and \(\{ n_k \}\), \(\{ m_k \}\) any countable sets of natural numbers. Then there exists a meromorphic function \(f\) of \(U\) whose zeros are exactly at \(a_k\), with orders given by \(n_k\), and poles are exactly at \(b_k\), with orders given by \(m_k\).
For a more explicit statement, we need infinite products. The product \(\prod_{j=1}^\infty (1+a_j)\) converges if the sequence of partial products \(\prod_{j=1}^n (1+a_j)\) converges. We say that the product converges absolutely if \(\prod_{j=1}^\infty (1+|a_j|)\) converges, which is equivalent to \(\sum_{j=1}^\infty |a_j|\) converging.
Define \[E_0(z) = (1-z), \qquad E_m(z) = (1-z) \exp\left( z +\frac{z^2}{2} + \cdots + \frac{z^m}{m} \right) .\] The function \(E_m\bigl(\frac{z}{a}\bigr)\) has a zero of order 1 at \(a\).
Weierstrass Factorization Theorem
Let \(f\) be an entire function with zeros (repeated according to multiplicity) at points of the sequence \(\{ a_k \}\) except the zero at the origin, whose order is \(m\) (possibly \(m=0\)). Then there exists an entire function \(g\) and a sequence \(\{ p_k \}\) such that \[f(z) = z^m e^{g(z)} \prod_{k=1}^\infty E_{p_k}\left(\frac{z}{a_k}\right) ,\] converges uniformly absolutely on compact subsets.
The \(p_k\) are chosen such that \[\sum\limits_{j=1}^\infty \left|\frac{r}{a_{k}}\right|^{1+p_{k}}\] converges for all \(r > 0\).
\[\ast\:\ast\:\ast\nonumber\]
There are many other useful theorems in one complex variable, and we could spend a lot of time listing them all. However, hopefully the listing above is useful as a refresher for the reader of the most common results, some of which are used in this book, some of which are useful in the exercises, and some of which are just too interesting not to mention.