.. _appendix:

Appendix
========

.. index:: ! Cauchy-Riemann equations
.. index:: ! pair: holomorphic function; analytic function
.. index:: ! entire function

Basic Definitions
-----------------

The two partial differential equations

.. math:: \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

are called the **Cauchy-Riemann equations** for the pair of functions :math:`u, v`.

Let the complex-valued function :math:`f = u + iv` be defined in the open subset :math:`G \subseteq \mathbb C`, and assume that :math:`u` and :math:`v` have first partial derivatives in :math:`G`. Then :math:`f` is differentiable at each point where those partial derivatives are continuous and satisfy the Cauchy-Riemann equations.

A complex-valued function that is defined in an open subset :math:`G \subseteq \mathbb C` and differentiable at every point of :math:`G` is said to be **holomorphic** (or **analytic**) in :math:`G`.

A function that is holomorphic at every point in the entire complex plane is called an **entire function.**

.. index:: ! Cauchy's theorem, Cauchy's formula

.. _cauchy:

Cauchy's Formula
----------------

A continuous function :math:`\gamma:[a,b]\to \mathbb{C}`, where :math:`[a,b]\subset \mathbb R`, is called a *path* in :math:`\mathbb{C}`, and such a path is called *rectifiable* if it is of bounded variation, i.e.,if there is a constant :math:`M>0` such that, for any partition :math:`a=t_1< t_2 < \cdots < t_n=b` of :math:`[a,b]`, :math:`\sum_i |\gamma(t_i) - \gamma(t_{i-1})| \leq M`. In particular, if :math:`\gamma` is a piecewise smooth path, it is rectifiable.

If :math:`\gamma:[a,b]\to \mathbb{C}` is a path in :math:`\mathbb{C}`, the set of points :math:`\{\gamma(t): a\leq t\leq b\}` is called the *trace* of :math:`\gamma`. Some authors denote this set by :math:`\gamma^*`, and others by :math:`\{\gamma\}`. We will write :math:`\gamma^*` if clarity demands it. Otherwise, if we simply write :math:`z\in \gamma`, it should be obvious that we mean :math:`\gamma(t) = z` for some :math:`a\leq t\leq b`. Finally, if :math:`\gamma` is a closed rectifiable path in :math:`\mathbb{C}`, we call the region which has :math:`\gamma` as its boundary the *interior of* :math:`\gamma`.

A *curve* is an equivalence class of paths that are equal modulo a change of parameter. If a path :math:`\gamma` has some (non-parametric) property that interests us (e.g.,it is closed or smooth or rectifiable), then invariably that property is shared by every path in the equivalence class of parametrization of  :math:`\gamma`. Therefore, when parametrization has no relevance to the discussion, we often speak of the “curve” :math:`\gamma`, by which we mean any one of the paths that represent the curve.

If :math:`\gamma` is a closed rectifiable curve in :math:`\mathbb{C}` then, for :math:`w\notin \gamma^*`, the number

.. math:: n(\gamma;w) = \frac{1}{2\pi i} \int_\gamma \frac{dz}{z-w}

is called the **index** of :math:`\gamma` with respect to the point :math:`w`. It is also sometimes called the **winding number** of :math:`\gamma` around :math:`w`.

.. _cauchy-formula:

**Cauchy's Formula.** Let :math:`G\subseteq \mathbb{C}` be an open subset of the plane and suppose :math:`f\in H(G)`. If :math:`\gamma` is a closed rectifiable curve in :math:`G` such that :math:`n(\gamma;w)=0` for all :math:`w \in \mathbb{C}\setminus G`, then for all :math:`z\in G\setminus \gamma^*`,

.. math:: f(z) n(\gamma;z)  = \frac{1}{2\pi i} \int_{\gamma}\frac{f(\zeta)}{\zeta-z}\, d\zeta.

.. index:: homologous to zero

A number of important theorems include the hypothesis that :math:`\gamma` is a closed rectifiable curve with :math:`n(\gamma;w)=0` for all :math:`w \in \mathbb{C}\setminus G` (where :math:`G` is some open subset of the plane). This simply means that :math:`\gamma` is a closed rectifiable curve that is contained *along with its interior* inside :math:`G`. In other words, :math:`\gamma` does not wind around any points in the complement of :math:`G` (e.g.,“holes” in :math:`G`, or points exterior to :math:`G`).

Such a curve :math:`\gamma` is called **homologous to zero** in :math:`G`, denoted :math:`\gamma\approx 0`, and a version of a theorem with this as one of its hypotheses may be called the “homology version” of the theorem.

The following is a slightly more compact statement of Cauchy's formula that uses this terminology.

.. _cauchy-h1:

**Cauchy's Formula.**  Let :math:`G\subseteq \mathbb{C}` be an open set and suppose :math:`f\in H(G)`. If :math:`\gamma` is a closed rectifiable curve that is homologous to zero in :math:`G`, then :math:`\int_\gamma f(z) dz = 0`.

More generally, if :math:`G\subseteq\mathbb{C}` is open and :math:`\gamma_1,\ldots, \gamma_m` are closed rectifiable curves in :math:`G`, then the curve :math:`\gamma = \gamma_1 + \cdots + \gamma_m` is *homologous to zero* in :math:`G` provided :math:`n(\gamma_1,w)+ \cdots +  n(\gamma_m,w) = 0` for all :math:`w\in \mathbb{C}\setminus G`.

We call the next theorem "the full homology version of Cauchy's formula," or more simply, "Cauchy's theorem."

.. _cauchy-theorem:

**Cauchy's Theorem.**  Let :math:`G\subseteq \mathbb{C}` be an open subset of the plane and suppose :math:`f\in H(G)`. If :math:`\gamma_1,\ldots, \gamma_m` are closed rectifiable curves in :math:`G` with :math:`\gamma = \gamma_1 + \cdots + \gamma_m \approx 0`, then for all :math:`z\in G\setminus \gamma^*`,

  .. math:: f(z) \sum_{j=1}^m n(\gamma_j, z) = \frac{1}{2\pi i} \sum_{j=1}^m \int_{\gamma_j}\frac{f(\zeta)}{\zeta-z}\,d\zeta.

A partial converse of Cauchy's theorem is the following:

.. index:: ! Cauchy's theorem (partial converse of), Morera's theorem

.. _conv-cauchy:

**Partial Converse of Cauchy's Theorem.** Let :math:`G` be an open set in the plane and :math:`f\in C(G,\mathbb{C})`. Suppose, for any triangular contour :math:`T\subset G` with :math:`T\approx 0` in :math:`G`, that :math:`\int_T f(z)dz = 0`. Then :math:`f\in H(G)`.

This theorem is still valid (and occasionally easier to apply) if we replace “any triangular contour” with “any rectangular contour with sides parallel to the real and imaginary axes.” This is sometimes called **Morera's theorem**. (The exercise on page 81 of Sarason :cite:`Sarason:2007` asks you to prove it as a corollary of the :ref:`Partial Converse of Cauchy's Theorem <conv-cauchy>`.)

.. index:: ! mean-value property

.. _mean-value:

**Theorem.** (Mean-value property) Let :math:`G\subseteq \mathbb C` be an open set containing a closed disk :math:`\overline{D}(a,r) = \{z : |z - a| \leq r\} \subseteq G`.  Assume :math:`f\in H(G)`.  Then

  .. math:: f(a) = \frac{1}{2\pi}\int_0^{2\pi} f(a+re^{i\theta})\, d\theta

---------------------------------------------

.. _max-mod:

Maximum Modulus Principles
--------------------------

.. index:: ! Rouche's theorem

.. _rouche:

**Rouche's Theorem, version 1.** (:cite:`Sarason:2007`) Let :math:`f, g` be holomorphic functions in a region :math:`G \subseteq \mathbb C`.  Let :math:`K\subseteq G` be compact.  If :math:`|f(z) - g(z)| < |f(z)|` for all :math:`z\in \partial K` (the boundary of :math:`K`), then :math:`f` and :math:`g` have the same number of zeros in the interior of :math:`K`.

**Rouche's Theorem, version 2.** (:cite:`Conway:1978`) Let :math:`f, g` be meromorphic in a neighborhood of the closed disk :math:`\overline{D}(a, R) = \{z : |z -a|\leq R\}`.  Let :math:`Z_f, P_f`, :math:`Z_g, P_g` denote the number of zeros and poles of :math:`f` and :math:`g` (respectively) in :math:`\overline{D}(a, R)`. Suppose that none of the poles of :math:`f` or :math:`g` occur on the circle :math:`|z - a| = R` and suppose

.. math:: |f(z) + g(z)| < |f(z)| + |g(z)| \text{ for all } z \in \partial D(a, R) = \{z : |z - a| = R\}

Then :math:`Z_f - P_f = Z_g - P_g`.

.. index:: ! local mapping theorem

.. _lmt:

**Local Mapping Theorem.** Let :math:`f` be a nonconstant holomorphic function in a region :math:`G`, let :math:`z_0\in G`, and :math:`w_0 = f(z_0)`. Let :math:`m` be the order of the zero of the function :math:`f(z) - w_0` at :math:`z_0`. Then for every sufficiently small :math:`\delta > 0`, there exists :math:`\epsilon > 0` such that every point in the punctured disk :math:`0 < |w - w_0| < \epsilon` is assume by :math:`f` at exactly :math:`m` distinct points in the punctured disk :math:`0 < |z - z_0| < \delta`, with multiplicity 1 at each of those points.

In other words, near the point :math:`z_0`, :math:`f` behaves qualitatively like the polynomial :math:`(z - z_0)^m + w_0`

.. index:: ! open mapping theorem

.. _omt:

**Open Mapping Theorem.** If :math:`f` is any nonconstant holomorphic function on a region :math:`G` and :math:`U\subseteq G` an open set, then :math:`f(U) = \{f(z) : z\in U\}` is open.

.. index:: ! maximum modulus theorems

.. _max-mod-1:

**Maximum Modulus Theorem, version 1.** Suppose :math:`G\subset \mathbb{C}` is open and :math:`f\in H(G)` attains its maximum modulus at some point :math:`a\in G`. Then :math:`f` is constant.

That is, if there is a point :math:`a \in G` with :math:`|f(z)|\leq |f(a)|` for all :math:`z\in G`, then :math:`f` is constant. [1]_

.. _max-mod-2:

**Maximum Modulus Theorem, version 2.** If :math:`G\in \mathbb C` is open and bounded, and if :math:`f \in C(\bar{G})\cap H(G)`, then :math:`\max \{|f(z)| : z\in \bar{G}\} = \max \{|f(z)|: z\in \partial G\}`.

That is, in an open *and bounded* region, if a holomorphic function is continuous on the boundary, then it attains its maximum modulus there.

.. _max-mod-3:

**Maximum Modulus Theorem, version 3.** Let :math:`G\subset \hat{\mathbb{C}} := \mathbb{C}\cup \{\infty\}` be open, let :math:`f\in H(G)`, and suppose there is an :math:`M>0` such that

  .. math:: \lim_{z\to a} |f(z)| \leq M, \; \text{for every } a \in \partial_{\infty}G.

  Then :math:`|f(z)| \leq M` for all :math:`z\in G`.

.. index:: ! Schwarz's lemma

.. _schwarz:

**Schwarz's Lemma.** Let :math:`f \in H(D)`, :math:`|f(z)|\leq 1` for all :math:`z\in D`, and :math:`f(0)=0`. Then

  (a) :math:`|f(z)| \leq |z|`, for all :math:`z\in D`,

  (b) :math:`|f'(0)| \leq 1`, with equality in (a) for some :math:`z\in D\setminus \{0\}` or equality in (b) iff :math:`f(z) = e^{i\theta}z` for some constant :math:`\theta \in \mathbb R`.

.. index:: ! Liouville's theorem

.. _liouville:

**Liouville's Theorem.** The only bounded entire functions are the constant functions.


--------------------------------------------

.. index:: canonical factor, factorization theorems

Factorization Theorems
----------------------

For each integer :math:`k \geq 0` we define **canonical factors** by

.. math:: E_0(z) := 1 - z \quad \text{ and } \quad E_k(z) := (1 - z)e^{z+z^2/2 + \cdots + z^k/k},\; \text{ for } k \geq 1.

The integer :math:`k` is called the **degree** of the canonical factor.

.. index:: ! Weierstrass' factorization theorem

**Weierstrass' Factorization Theorem.**

.. todo:: fill in Weierstrass' Theorem.

.. index:: ! Hadamard's theorem

.. _hadamard:

**Hadamard's Factorization Theorem.** (:cite:`Stein:2003`, 5.1) Suppose :math:`f` is entire and has growth order :math:`\rho_0`. Let :math:`k` be the integer so that :math:`k \leq \rho_0 < k + 1`. If :math:`a_1, a_2, \dots` denote the (non-zero) zeros of :math:`f`, then

.. math:: f(z) = e^{P(z)}z^m \prod_{n=1}^\infty E_k(z/a_n),

where :math:`P` is a polynomial of degree at most :math:`k`, and :math:`m` is the order of the zero of :math:`f` at :math:`z = 0`.

-----------------------------------------

.. _conformal-mapping:

.. index:: conformal mapping, Riemann mapping theorem

Conformal Mapping
-----------------

.. index:: ! Riemann Mapping theorem

.. _riemann:

**Riemann Mapping Theorem.**

  .. todo:: insert statement of rmt

-----------------------------------------

.. _normal-families:

.. index:: pointwise bounded, locally bounded, locally uniformly bounded, equicontinuous

Normal Families
---------------

Let :math:`G \subseteq \mathbb C` be a region, :math:`(X, d)` a metric space, and :math:`\mathcal F` a family of functions mapping :math:`G` into :math:`(X, d)`.

The family :math:`\mathcal{F}` is called **pointwise bounded** if for each :math:`z_0 \in G` the set :math:`\{f(z_0) : f \in \mathcal F\}` is a bounded subset of :math:`(X, d)`.

The family :math:`\mathcal{F}` is called **locally uniformly bounded** (or simply **locally bounded**) if for each compact subset :math:`K \subseteq G` there is a constant :math:`M_K` such that :math:`|f(z)|< M_K` for all :math:`f\in \mathcal{F}` and all :math:`z\in K`.

The family :math:`\mathcal{F}` is called **equicontinuous** at the point :math:`z_0` in :math:`G` if for every :math:`\epsilon > 0` there is a :math:`\delta > 0` such that for :math:`d(f(z),f(z_0)) < \epsilon` for all :math:`|z-z_0|<\delta` and for every :math:`f \in \mathcal F`.

The family :math:`\mathcal{F}` is **equicontinuous** over a set :math:`E \subseteq G` if for every :math:`\epsilon > 0` there is a :math:`\delta > 0` such that for all :math:`z` and :math:`z'` in :math:`E` such that :math:`|z-z'|<\delta`, :math:`d(f(z),f(z_0)) < \epsilon` for all :math:`f \in \mathcal F`.

The important thing about equicontinuity is that the same :math:`\delta` will work for all the functions in :math:`\mathcal F`.

If :math:`\mathcal F` consists of a single function :math:`f`, then the statement that :math:`\mathcal F` is equicontinuous at :math:`z_0` merely asserts that :math:`f` is continuous at :math:`z_0`, and the statement that :math:`\{f\}` is equicontinuous over :math:`E` means that :math:`f` is uniformly continuous on :math:`E`.

**Equicontinuity Theorem.** Suppose :math:`\mathcal F` is equicontinuous at each point of :math:`G`. Then :math:`\mathcal F` is equicontinuous over each compact subset of :math:`G`.

.. index:: ! Arzela-Ascoli theorem

.. _arzela-ascoli:

**Arzela-Ascoli Theorem.** Let :math:`\mathcal{F}\subset C(G, X)` be a family of continuous functions from an open set :math:`G\subseteq \mathbb{C}` into a metric space :math:`(X,d)`. Then :math:`\mathcal{F}` is a normal family if and only if

  (i) :math:`\mathcal{F}` is equicontinuous on each compact subset of :math:`G`, and

  (ii) :math:`\mathcal{F}` is pointwise bounded.

**Remark.** Item (ii) is sometimes presented as the following condition, which of course is equivalent to pointwise boundedness: for each :math:`z\in G`, the set :math:`\{f(z):f\in \mathcal{F}\}` is contained in a compact subset of :math:`X`.

.. index:: ! Montel's theorem

.. _montel:

**Montel's Theorem.** Assume the set-up of the :ref:`Arzela-Ascoli theorem <arzela-ascoli>`, and suppose :math:`X=\mathbb{C}` and :math:`\mathcal{F}\subset H(G)`. Then :math:`\mathcal{F}` is a normal family if and only if it is locally bounded.

.. index:: ! Hurwitz's theorem

.. _hurwitz:

**Hurwitz's Theorem.** Let :math:`G \subset \mathbb C` be a region in the plane and suppose the sequence :math:`\{f_n\} \subseteq H(G)` of holomorphic functions converges to :math:`f`. Let :math:`a \in G` and :math:`R > 0` be such that :math:`\bar{D} := \bar{D}(a; R) \subseteq G` and suppose :math:`f(z) \neq 0` on the boundary :math:`|z-a| = R` of :math:`\bar{D}`. Then there is a number :math:`N>0` such that for all :math:`n\geq N`, :math:`f` and :math:`f_n` have the same number of zeros in :math:`\bar{D}`.

------------------------

.. rubric:: Footnotes

.. [1]
   This version of the maximum modulus principle is an easy consequence of the open mapping theorem, which itself can be proved via the local mapping theorem, which in turn can be proved using Rouché's theorem. Of course, you should know the statements of all of these theorems and, since proving them in this sequence is not hard, you might as well know the proofs too! Two excellent references giving clear and concise proofs are Conway :cite:`Conway:1978` and Sarason :cite:`Sarason:2007`.

----------------------------

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