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Corollary 5.19Abel’s theorem in the contrapositive

If a power series

\begin{align*} \sum _{n=0}^\infty a_n \, (z-z_0)^n \end{align*}

does not converge at $z = w \in \mathbb {C}$, then it does not converge at any $z \in \mathbb {C}$ such that $|z-z_0| {\gt} |w-z_0|$.

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Corollary 5.32Principle of analytic continuation [Palka1991, Cor VIII.1.6]

Let $f, g \colon \mathcal{D}\to \mathbb {C}$ be two analytic functions on a connected open set $\mathcal{D}\subset \mathbb {C}$. If $f(z) = g(z)$ for all $z$ in some subset of $\mathcal{D}$ which has an accumulation point in $\mathcal{D}$, then we have $f(z) = g(z)$ for all $z \in \mathcal{D}$.

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Corollary 5.29Local representation of analytic functions [Palka1991, Cor VIII.1.3]

Suppose that $f \colon \mathcal{D}\to \mathbb {C}$ is a non-constant analytic function on a connected open set $\mathcal{D}\subset \mathbb {C}$. Then for any $z_0 \in \mathcal{D}$, we can write $f$ uniquely in the form

\begin{align*} f(z) = f(z_0) + (z - z_0)^m \, g(z) \qquad \text{ for } z \in \mathcal{D}, \end{align*}

where $m \in \mathbb {N}$ and $g \colon \mathcal{D}\to \mathbb {C}$ is an analytic function such that $g(z_0) \ne 0$.

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Corollary 4.11Cauchy’s integral formula for circles

Suppose that a function $f \colon U \to \mathbb {C}$ is analytic on an open set $U \subset \mathbb {C}$. Let $\overline{\mathcal{B}}(z_0; r) \subset U$ be a closed disk contained in $U$. Then for any point $z \in \mathcal{B}(z_0; r)$ we have

\begin{align*} f(z) = \frac{1}{2\pi \mathfrak {i}} \oint _{\partial \mathcal{B}(z_0; r)} \frac{f(\zeta )}{\zeta - z} \, \mathrm{d}\zeta \end{align*}

where the circle $\partial \mathcal{B}(z_0; r)$ is parametrized in the counterclockwise orientation.

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Corollary 4.26Polynomial factorization [Palka1991, Thm V.3.9]

A complex-coefficient polynomial $p(z) = a_0 + a_1 z + a_2 z^2 + \cdots + a_n z^n$ of degree $n \in \mathbb {N}$ can be factored as

\begin{align*} p(z) = c \, (z - z_1) \, (z - z_2) \, \cdots \, (z - z_n) \end{align*}

where $c = a_n \ne 0$, and $z_1 , \ldots , z_n \in \mathbb {C}$ are the roots of $p$ (with repetition according to the multiplicities of the roots).

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Corollary A.18Continuity of coordinate projections

The coordinate projections

\begin{align*} \Re \mathfrak {e}\colon \; & \mathbb {C}\to \mathbb {R}\qquad \qquad \text{ and } & \quad \Im \mathfrak {m}\colon \; & \mathbb {C}\to \mathbb {R}\\ & z \mapsto \Re \mathfrak {e}(z) & & z \mapsto \Im \mathfrak {m}(z) \end{align*}

are continuous functions.

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Corollary 4.9Local Cauchy’s integral theorem [Palka1991, Thm V.5.1]

Suppose that a function $f \colon U \to \mathbb {C}$ is analytic on a open set $U \subset \mathbb {C}$. Then for any disk $B \subset U$ contained in the domain $U$ and any closed contour $\gamma $ in $B$ we have

\begin{align*} \oint _\gamma f(z) \, \mathrm{d}z = 0 . \end{align*}

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Corollary 4.28Maximum principle for analytic functions continuous up to the boundary [Palka1991, Cor V.3.11]

Let $\mathcal{D}\subset \mathbb {C}$ be a bounded connected open set. Let $f \colon \overline{{\mathcal{D}}} \to \mathbb {C}$ be a continuous function on its closure which is analytic in $\mathcal{D}$. Then $z \mapsto |f(z)|$ attains its maximum in $\overline{{\mathcal{D}}}$ at some point of the boundary $\partial \mathcal{D}$.

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Corollary A.22Real multivariate polynomials are continuous

Let $N \in \mathbb {N}$ be a natural number, and let $c_{n,m} \in \mathbb {R}$ be real numbers for $n,m \in \{ 0,1,\ldots ,N\} $. Then the function $p \colon \mathbb {C}\to \mathbb {R}$ defined by

\begin{align*} p(x + \mathfrak {i}y) = \sum _{m=0}^N \sum _{n=0}^N c_{m,n} \, x^m \, y^n \end{align*}

is continuous.

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Corollary 5.5Contour integration commutes with uniform limits [Palka1991, Thm VII.1.1]

If a sequence $(f_n)_{n \in \mathbb {N}}$ of continuous functions $f_n \colon A \to \mathbb {C}$ on a subset $A \subset \mathbb {C}$ of the complex plane converges uniformly to a function $f \colon A \to \mathbb {C}$, then for any piecewise smooth path $\gamma $ in $A$ we have

\begin{align*} \lim _{n \to \infty } \int _\gamma f_n(z) \, \mathrm{d}z = \int _\gamma f(z) \, \mathrm{d}z . \end{align*}

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Definition 5.11Absolute convergence of a complex series

A complex series $\sum _{n=1}^\infty z_n$ is said to converge absolutely if the series of absolute values $\sum _{n=1}^\infty |z_n|$ converges.

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Definition 1.5Absolute value (modulus) [Palka1991, Sec. I.1.2]

The absolute value (or modulus) of a complex number $z = x + \mathfrak {i}y$ (where $x,y \in \mathbb {R}$) is the nonnegative real number $|z| = \sqrt{x^2 + y^2} \ge 0$.

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Definition 2.12Analytic function [Palka1991, Sec. III.1.3]

A function $f \colon U \to \mathbb {C}$ defined on an open set $U \subset \mathbb {C}$ is said to be analytic (or holomorphic) if it is complex differentiable at every point $z_0 \in U$.

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Definition 3.14Arc-length integral

Let $f : A \to \mathbb {C}$ be a continuous function defined on a subset $A \subset \mathbb {C}$. Let $\gamma \colon [a,b] \to A$ be a piecewise smooth path (i.e., a contour) in $A$, which is a concatenation of smooth paths $\eta _1, \ldots , \eta _n$. We define the integral of $f$ along $\gamma $ as

\begin{align*} \int _\gamma f(z) \, dz \; = \; \sum _{j=1}^n \int _{\eta _j} f(z) \, dz . \end{align*}

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Definition 1.7Argument [Palka1991, Sec. I.1.2]

A real number $\theta \in \mathbb {R}$ is an argument of a complex number $z \in \mathbb {C}$ if

\begin{align*} z = |z| \, \Big( \cos (\theta ) + \mathfrak {i}\, \sin (\theta ) \Big) . \end{align*}

(Note/warning: For a nonzero complex number $z$, it is convenient to denote $\theta = \arg (z)$, but this is an abuse of notation, the argumentis defined only modulo addition of integer multiples of $2 \pi $.)

The principal argument of a nonzero complex number $z \in \mathbb {C}$ is its unique argument on the interval $(-\pi , \pi ]$, and it is denoted by $\mathrm{Arg}(z)$.

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Definition A.3Ball (disk)

Let $X$ be a metric space with metric $\mathsf{d}\colon X \times X \to [0,\infty )$. Let $p_0 \in X$ be a point and let $r{\gt}0$.

The set

\begin{align*} \mathcal{B}(p_0; r) = \Big\{ p \in X \; \Big| \; \mathsf{d}(p,p_0) {\lt} r \Big\} \end{align*}

is called an open ball in $X$, centered at $p_0$, and with radius $r$.

The set

\begin{align*} \overline{\mathcal{B}}(p_0; r) = \Big\{ p \in X \; \Big| \; \mathsf{d}(p,p_0) \le r \Big\} \end{align*}

is called a closed ball in $X$, centered at $p_0$, and with radius $r$.

(In the case of the complex plane $\mathbb {C}$, the term disk is often used instead of the general metric space theory term ball.)

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Definition A.6Boundary

Let $X$ be a metric space, and $A \subset X$ a subset. A point $p \in X$ is said to be a boundary point of $A$ if for all $r {\gt} 0$ we have that $\mathcal{B}(p; r)$ contains points of $A$ and $X \setminus A$ (i.e. $\mathcal{B}(p; r) \cap A \ne \emptyset $ and $\mathcal{B}(p; r) \setminus A \ne \emptyset $).

The set of all boundary points of $A$ is denoted $\partial A$ and called the boundary of $A$.

(It is easy to see that the boundary $\partial A \subset X$ is exactly the set of points of $X$ which are neither interior nor exterior points of $A$.)

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Definition A.9Boundedness

Let $X$ be a metric space. with metric $\mathsf{d}\colon X \times X \to [0,\infty )$.

A subset $A \subset X$ is bounded if there exists a number $M{\gt}0$ such that $\mathsf{d}(p,q) \le M$ for all $p,q \in A$. (If $X$ is nonempty, an equivalent definition would be that $A$ is bounded if it is a subset of some ball in $X$.)

A function $f \colon Z \to X$ with values in a metric space $X$ is bounded if the set $f[Z] \subset X$ of its values is a bounded subset of $X$.

(In the case $X = \mathbb {C}$ we have the following further characterizations: A subset $A \subset \mathbb {C}$ is bounded if and only if there exists an $R{\gt}0$ such that $|z| \le R$ for all $z \in A$. A function $f \colon Z \to \mathbb {C}$ is bounded if and only if there exists an $R{\gt}0$ such that $|f(z)| \le R$ for all $z \in Z$.)

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Definition 1.18Branches of complex logarithm

A branch of the logarithm is a continuous function $\ell \colon U \to \mathbb {C}$ on an open set $U \subset \mathbb {C}$ such that

\begin{align*} e^{\ell (z)} = z \qquad \text{ for all } z \in U . \end{align*}

For example, the principal logarithm $\mathrm{Log}$ restricted to the open set $\mathbb {C}\setminus (-\infty ,0]$ is called the principal branch of the logarithm. Note that this principal branch cannot be extended continuously to the negative real axis.

(Note that since $e^w \ne 0$ for all $w \in \mathbb {C}$, any branch of the logarithm must exclude the origin from its domain of definition, $0 \notin U$.)

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Definition A.13Cauchy sequence

…

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Definition A.8Closed set

Let $X$ be a metric space. A subset $F \subset X$ is said to be a closed set if the complement $X \setminus F \subset X$ is an open set.

(Equivalently, each point $p \in X \setminus F$ in the complement of $F$ is an exterior point of $F$.)

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Definition A.29Compactness

Let $X$ be a metric space. A subset $K \subset X$ is compact if every sequence $(x_n)_{n \in \mathbb {N}}$ of points $x_n \in K$ has a subsequence $(x_{n_k})_{k \in \mathbb {N}}$ which converges to a limit $\lim _{k \to \infty } x_{n_k} \in K$ in the set $K$.

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Definition 1.3Complex conjugate [Palka1991, Sec. I.1.2]

The complex conjugate of a complex number $z = x + \mathfrak {i}y$ (where $x,y \in \mathbb {R}$) is the complex number $\overline{z} = x - \mathfrak {i}y$.

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Definition 2.3Complex derivative [Palka1991, Sec. III.1.1]

Let $f \colon A \to \mathbb {C}$ be a complex-valued function defined on a subset $A \subset \mathbb {C}$ of the complex plane, and let $z_0 \in A$ be an interior point of the subset.

The $f$ is said to have a complex derivative

\begin{align*} f’(z_0) := \lim _{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0} \end{align*}

at $z_0$, if the limit on the right hand side above exists.

(In complex analysis we often drop the epithet “complex” above, and simply call $f'(z_0)$ the derivative of $f$ at $z_0$.)

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Definition 1.9Complex exponential function

We define the complex exponential function $\exp : \mathbb {C}\to \mathbb {C}$ by

\begin{align*} \exp (x + \mathfrak {i}y) = e^x \, \big( \cos (y) + \mathfrak {i}\, \sin (y) \big) \qquad \text{ for } x,y \in \mathbb {R}, \end{align*}

where $e^x$ is the usual real exponential. We also use the notation $e^z = \exp (z)$ for complex exponentials.

The exponential with purely imaginary argument takes the form of Euler’s formula

\begin{align*} e^{\mathfrak {i}\theta } = \cos (\theta ) + \mathfrak {i}\, \sin (\theta ) \qquad \text{ for } \theta \in \mathbb {R}. \end{align*}

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Definition 1.1Complex numbers and their arithmetic operations [Palka1991, Sec. I.1.1]

The set of complex numbers is $\mathbb {C}= \mathbb {R}\times \mathbb {R}$, i.e., the set of pairs $(x,y)$ of real numbers $x,y \in \mathbb {R}$.

The operations of addition and multiplication on $\mathbb {C}$ are defined by the formulas

\begin{align*} (x_1,y_2) + (x_2,y_2) = \; & (x_1+x_2, y_1+y_2) \\ (x_1,y_1) \cdot (x_2,y_2) = \; & (x_1 x_2 - y_1 y_2 , \, x_1 y_2 + y_1 x_2) . \end{align*}

Denote $0 = (0,0) \in \mathbb {C}$ and $1 = (1,0) \in \mathbb {C}$.

For $z = (x,y) \in \mathbb {C}$, denote $-z = (-x,-y) \in \mathbb {C}$ and if $z \ne 0$ then denote $z^{-1} = \Big(\frac{x}{x^2 + y^2} , \frac{-y}{x^2 + y^2} \Big) \in \mathbb {C}$.

We write a complex number $(x,y)$ as $x + \mathfrak {i}\, y$. The compex number $\mathfrak {i}= (0,1) \in \mathbb {C}$ is called the imaginary unit.

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Definition 5.8Complex series [Palka1991, Sec. VII.2.1]

Let $z_1, z_2, z_3, \ldots \in \mathbb {C}$ be complex numbers. For $N \in \mathbb {N}$, define the $N$th partial sum of these as

\begin{align*} S_N \; = \; \sum _{n=1}^N z_n \; = \; z_1 + z_2 + \cdots + z_N \, . \end{align*}

We say that the series $\sum _{n=1}^\infty z_n$ converges if the sequence $(S_N)_{N \in \mathbb {N}}$ of partial sums has a limit, and we then denote

\begin{align*} \sum _{n=1}^\infty z_n = \lim _{N \to \infty } \; \sum _{n=1}^N z_n \, . \end{align*}

(Obvious modifications to the above definition are made if the terms’ indexing starts from $n=0$ or some other index, and the notation is correspondingly changed to, e.g., $\sum _{n=0}^\infty $.)

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Definition 3.1Integral of a complex-valued function

Let $f \colon [a,b] \to \mathbb {C}$ be a complex-valued continuous function defined on a closed interval $[a,b] \subset \mathbb {R}$. We define the integral of $f$ as

\begin{align*} \int _a^b f(t) \, \mathrm{d}t \; = \; \int _a^b \Re \mathfrak {e}\big( f(t) \big) \, \mathrm{d}t \; + \; \mathfrak {i}\int _a^b \Im \mathfrak {m}\big( f(t) \big) \, \mathrm{d}t . \end{align*}

(Note that on the right hand side we have just Riemann integrals of the continuous real-valued functions $t \mapsto \Re \mathfrak {e}\big( f(t) \big)$ and $t \mapsto \Im \mathfrak {m}\big( f(t) \big)$.)

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Definition A.25Connectedness

A set $A \subset X$ in a metric space $X$ is disconnected if there exists a continuous surjective function $f \colon A \to \left\{ 0,1 \right\} $ onto the two-element discrete set $\left\{ 0,1 \right\} $. Otherwise $A$ is connected; then every continuous function $A \to \left\{ 0,1 \right\} $ must be either constant $0$ or constant $1$.

(The usual definition in topology textbooks reads slightly differently, but it is equivalent to the one we chose here by Lemma A.20.)

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Definition A.16Continuity

Let $X$ and $Y$ be metric spaces. A function $f \colon X \to Y$ is said to be continuous at a point $p_0 \in X$ if $\lim _{p \to p_0} f(p) = f(p_0)$.

(Equivalently, for every $\varepsilon {\gt} 0$ there exists a $\delta {\gt} 0$ such that for any $p \in \mathcal{B}(p_0; \delta )$ we have $f(p) \in \mathcal{B}(f(p_0); \varepsilon )$.)

A function $f \colon X \to Y$ is said to be continuous if it is continuous at every point $p_0 \in X$.

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Definition 3.6Contour / piecewise smooth path [Palka1991, Sec. IV.1.2]

A contour (also called a piecewise smooth path) is a continuous function $\gamma \colon [a,b] \to \mathbb {C}$ such that for some finite subdivision $a = t_0 {\lt} t_1 {\lt} \ldots {\lt} t_n = b$, the restrictions $\gamma |_{[t_{j-1},t_j]}$ to the subintervals $[t_{j-1},t_j] \subset [a,b]$ are smooth paths for each $j = 1, \ldots , n$.

If the starting point and the end point of the contour $\gamma $ are the same, $\gamma (a) = \gamma (b)$, then we say that $\gamma $ is a closed contour.

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Definition 3.13Contour integral [Palka1991, Sec. IV.2.1]

Let $f : A \to \mathbb {C}$ be a continuous function defined on a subset $A \subset \mathbb {C}$. Let $\gamma \colon [a,b] \to A$ be a piecewise smooth path in $A$, which is a concatenation of smooth paths $\eta _1, \ldots , \eta _n$. We define the integral of $f$ along $\gamma $ as

\begin{align*} \int _\gamma f(z) \, dz \; = \; \sum _{j=1}^n \int _{\eta _j} f(z) \, dz . \end{align*}

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Definition A.36Contractible path

Let $X$ be a metric space. A closed path $\gamma \colon [a,b] \to X$ is called contractible if it is homotopic to a constant path.

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Definition 4.2Convex set

A subset $A \subset \mathbb {C}$ of the complex plane is called convex if for any two points $z_1, z_2 \in A$, the line segment between them is contained in the subset,

\begin{align*} [z_1, z_2] \, \subset \, A . \end{align*}

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Definition B.1Real differentiability

Let $m,n \in \mathbb {N}$, and let $f \colon U \to \mathbb {R}^m$ be a function defined on a subset $U \subset \mathbb {R}^n$. A linear map $L \colon \mathbb {R}^n \to \mathbb {R}^m$ is said to be a differential of $f$ at $p_0 \in U$ if

\begin{align*} f(p) = f(p_0) + L (p-p_0) + E(p-p_0) \end{align*}

where the error term $E$ is small near $p_0$ in the sense that

\begin{align*} \lim _{p \to p_0} \frac{\| E(p-p_0) \| }{\| p-p_0\| } = 0 . \end{align*}

We say that $f$ is differentiable at $p_0$ if such a linear map $L$ exists.

It is easy to check that the differential $L$ of $f$ at $p_0$ is unique if $p_0$ is an interior point of $U$; we then denote it by $L = \mathrm{d}f (p_0)$.

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Definition 5.33Doubly infinite series [Palka1991, Sec. VII.2.1]

A doubly infinite series of complex numbers is a series of the form

\begin{align*} \sum _{n=-\infty }^\infty z_n = \cdots + z_{-2} + z_{-1} + z_0 + z_1 + z_2 + \cdots , \end{align*}

where $\ldots , z_{-2}, z_{-1}, z_0, z_1, z_2, \ldots \in \mathbb {C}$. We say that such a series converges to $s \in \mathbb {C}$ if for all $\varepsilon {\gt} 0$ there exists an $N \in \mathbb {N}$ such that for all $m_+ \ge N$ and $m_- \le -N$ we have

\begin{align*} \Big| \sum _{n=m_-}^{m_+} z_n - s \Big| {\lt} \varepsilon . \end{align*}

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Definition A.5Exterior point

Let $X$ be a metric space, and $A \subset X$ a subset. A point $p \in X \setminus A$ is said to be an exterior point of $A$ if for some $r {\gt} 0$ we have $\mathcal{B}(p; r) \subset X \setminus A$.

(It is easy to see that the exterior points of $A$ are exactly the interior points)

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Definition 5.14Series of functions [Palka1991, Sec. VII.2.2]

Let $f_1, f_2, f_3, \ldots $ be complex-valued functions on a set $A$. For $N \in \mathbb {N}$, define their $N$th partial sum function $F_N \colon A \to \mathbb {C}$ by

\begin{align*} F_N(z) \; = \; \sum _{n=1}^N f_n(z) \; = \; f_1(z) + \cdots + f_N(z) \, . \end{align*}

We say that the function series $\sum _{n=1}^\infty f_n$ converges pointwise if the sequence $\big(F_N(z)\big)_{N \in \mathbb {N}}$ of partial sums has a limit at every $z \in A$. We say that the function series $\sum _{n=1}^\infty f_n$ converges uniformly on $A$ if the sequence $(F_N)_{N \in \mathbb {N}}$ of partial sum functions converges uniformly on $A$. We say that the function series $\sum _{n=1}^\infty f_n$ converges uniformly on compacts if the sequence $(F_N)_{N \in \mathbb {N}}$ of partial sum functions converges uniformly on compacts.

The limit function is then denoted by $\sum _{n=1}^\infty f_n$.

(Obvious modifications to the above are made if the terms’ indexing starts from $n=0$ or some other index, and the notation is correspondingly changed to, e.g., $\sum _{n=0}^\infty $.)

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Definition 2.22Harmonic conjugate

Suppose that $u \colon U \to \mathbb {R}$ is harmonic function on an open subset $U \subset \mathbb {R}^2$, i.e., a twice continuously differentiable function satisfying $\frac{\partial ^2}{\partial {x}^2} u + \frac{\partial ^2}{\partial {y}^2} u = 0$ on $U$. A function $v \colon U \to \mathbb {R}$ is called a harmonic conjugate of $u$ if the function

\begin{align*} x + \mathfrak {i}y \; \mapsto \; u(x,y) + \mathfrak {i}\, v(x,y) \end{align*}

is analytic on $U$.

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Definition B.9Infimum

The infimum, or the greatest lower bound, of a set $A \subset \mathbb {R}$ is the greatest real number $i$ such that $a \ge i$ for all $a \in A$, and is denoted by $i = \inf A$.

By the completeness axiom of real numbers, every nonempty set ($A \ne \emptyset $) of real numbers which is bounded from below (for some $\ell \in \mathbb {R}$ we have $a \ge \ell $ for all $a \in A$) has an infimum $\inf A \in \mathbb {R}$. We adopt the notational conventions that $\inf \emptyset = +\infty $, and that $\inf A = -\infty $ if $A$ is not bounded from below.

For convenience, we also adopt some flexibility in the notation for infimums of function values or sequence values, similarly as with supremums.

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Definition A.4Interior point

Let $X$ be a metric space, and $A \subset X$ a subset. A point $p \in A$ is said to be an interior point of $A$ if for some $r {\gt} 0$ we have $\mathcal{B}(p; r) \subset A$.

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Definition 6.2Isolated singularity [Palka1991, Sec. VIII.2.1]

Let $f : U \to \mathbb {C}$ be an analytic function on an open set $U \subset \mathbb {C}$. We say that $f$ has an isolated singularity at $z_0 \in \mathbb {C}$ if $\mathcal{B}(z_0; r) \setminus \left\{ z_0 \right\} \subset U$ for some $r{\gt}0$ but $z_0 \notin U$.

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Definition 5.35Laurent series [Palka1991, Sec. VII.3.4]

A Laurent series centered at $z_0 \in \mathbb {C}$ is a doubly infinite series of functions of the form

\begin{align*} z \mapsto & \; \sum _{n=-\infty }^\infty a_n (z - z_0)^n \\ = & \; \cdots + \frac{a_{-2}}{(z-z_0)^2} + \frac{a_{-1}}{z-z_0} + a_0 + a_1 (z-z_0) + a_2 (z-z_0)^2 + \cdots . \end{align*}

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Definition 3.15Length of a path or a contour

Let $\gamma \colon [a,b] \to \mathbb {C}$ be a piecewise smooth path (i.e., a contour) in $\mathbb {C}$. The length $\ell (\gamma )$ of $\gamma $ is defined as

\begin{align*} \ell (\gamma ) = \int _\gamma |\mathrm{d}z| . \end{align*}

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Definition B.11Limit inferior

Let $(x_n)_{n \in \mathbb {N}}$ be a sequence of real numbers. Then the limit inferior of the sequence is defined as

\begin{align*} \liminf _{n \to \infty } x_n \; := \; \lim _{m \to \infty } \Big( \inf _{n \ge m} x_n \Big) . \end{align*}

With the following conventions, the limit inferior of a sequence always exists as either a real number or one of the symbols $\pm \infty $. If the sequence is not bounded from below, then by conventions regarding the infimum, we have $\inf _{n \ge m} x_n = -\infty $ for every $m$, so we correspondingly set $\liminf _{n \to \infty } x_n = -\infty $. Otherwise the sequence $(\inf _{n \ge m} x_n)_{m \in \mathbb {N}}$ is an increasing sequence of real numbers, so either it is bounded from above and converges to $\lim _{m \to \infty } \big( \inf _{n \ge m} x_n \big) \, = \, \sup _{m \in \mathbb {N}} \big( \inf _{n \ge m} x_n \big) \in \mathbb {R}$, or it is not bounded from above and we set $\liminf _{n \to \infty } x_n \, = \, \sup _{m \in \mathbb {N}} \big( \inf _{n \ge m} x_n \big) = +\infty $.

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Definition A.10Limit

Let $X$ be a metric space and let $(x_n)_{n \in \mathbb {N}}$ be a sequence of points in $X$. We say that the sequence $(x_n)_{n \in \mathbb {N}}$ converges to a limit $x \in X$ if for any $\varepsilon {\gt} 0$ there exists an $N \in \mathbb {N}$ such that for all $n \ge N$ we have $x_n \in \mathcal{B}(x; \varepsilon )$ (i.e., $\mathsf{d}(x_n,x) {\lt} \varepsilon $). We then denote

\begin{align*} \lim _{n \to \infty } x_n = x . \end{align*}

(It is straightforward to check that the limit is unique if it exists.)

Let then $X$ and $Y$ be metric spaces, with respective metrics $\mathsf{d}_X$ and $\mathsf{d}_Y$, and let $f \colon X \to Y$ be a function. We say that the function $f$ has a limit $y \in Y$ at a point $p_0 \in X$ if for any $\varepsilon {\gt} 0$ there exists a $\delta {\gt} 0$ such that for all $p \in \mathcal{B}(p_0; \delta ) \setminus \left\{ p_0 \right\} $ we have $f(p) \in \mathcal{B}(y; \varepsilon )$. We then denote

\begin{align*} \lim _{p \to p_0} f(p) = y . \end{align*}

(It is straightforward to check that the limit is unique if it exists.)

(Equivalently, written in terms of distances, $\lim _{p \to p_0} f(p) = y$ means that for any $\varepsilon {\gt} 0$ there exists a $\delta {\gt} 0$ such that we have $\mathsf{d}_Y \big( f(p), y \big) {\lt} \varepsilon $ whenever $0 {\lt} \mathsf{d}_X (p,p_0) {\lt} \delta $.)

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Definition B.10Limit superior

Let $(x_n)_{n \in \mathbb {N}}$ be a sequence of real numbers. Then the limit superior of the sequence is defined as

\begin{align*} \limsup _{n \to \infty } x_n \; := \; \lim _{m \to \infty } \Big( \sup _{n \ge m} x_n \Big) . \end{align*}

With the following conventions, the limit superior of a sequence always exists as either a real number or one of the symbols $\pm \infty $. If the sequence is not bounded from above, then by conventions regarding the supremum, we have $\sup _{n \ge m} x_n = +\infty $ for every $m$, so we correspondingly set $\limsup _{n \to \infty } x_n = +\infty $. Otherwise the sequence $(\sup _{n \ge m} x_n)_{m \in \mathbb {N}}$ is a decreasing sequence of real numbers, so either it is bounded from below and converges to $\lim _{m \to \infty } \big( \sup _{n \ge m} x_n \big) \, = \, \inf _{m \in \mathbb {N}} \big( \sup _{n \ge m} x_n \big) \in \mathbb {R}$, or it is not bounded from below and we set $\limsup _{n \to \infty } x_n \, = \, \inf _{m \in \mathbb {N}} \big( \sup _{n \ge m} x_n \big) = -\infty $.

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Definition 4.1Line segment

Given two points $z_1, z_2 \in \mathbb {C}$ in the complex plane, the line segment between $z_1$ and $z_2$ is the path

\begin{align*} \gamma \colon [0,1] & \; \to \mathbb {C}\\ \gamma (t) & \; = z_1 + t \, (z_2 - z_1) . \end{align*}

We also often view the line segment as a subset of $\mathbb {C}$ rather than a parametrized path, and we then denote it by

\begin{align*} [z_1, z_2] \, = \, \Big\{ z_1 + t \, (z_2 - z_1) \; \Big| \; t \in [0,1] \Big\} \, \subset \, \mathbb {C}. \end{align*}

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Definition 2.1Linear map

Let $\mathbb {K}$ be a field (for example $\mathbb {K}= \mathbb {R}$ or $\mathbb {K}= \mathbb {C}$), and let $V$ and $W$ be vector spaces over $\mathbb {K}$. A function $L : V \to W$ is said to be $\mathbb {K}$-linear if

\begin{align*} L(v_1 + v_2) = \; & L(v_1) + L(v_2) & & \text{ for all } v_1, v_2 \in V, \\ L(c v) = \; & c \, L(v) & & \text{ for all } v \in V, c \in \mathbb {K}. \end{align*}

Such a function $L$ is also called a $\mathbb {K}$-linear map (or a $\mathbb {K}$-linear transformation) between the spaces $V$ and $W$.

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Definition A.1Metric

A metric on a set $X$ is a function $\mathsf{d}\colon X \times X \to [0,\infty )$ such that for all $p_1, p_2, p_3 \in X$ we have

\begin{align*} \mathsf{d}(p_1, p_3) \le \; & \mathsf{d}(p_1,p_2) + \mathsf{d}(p_2,p_3) & \text{(triangle inequality)} \\ \mathsf{d}(p_1, p_2) = \; & \mathsf{d}(p_2,p_1) & \text{(symmetricity)} \\ \mathsf{d}(p_1, p_2) = \; & 0 \text{ if and only if } p_1 = p_2 . & \text{(separation of points)} \end{align*}

The set $X$ equipped with the metric $\mathsf{d}$ on it is called a metric space.

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Definition A.7Open set

Let $X$ be a metric space. A subset $U \subset X$ is said to be an open set if each point $p \in U$ is an interior point of $U$.

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Definition 3.4Path [Palka1991, Sec. IV.1.1]

A path in the complex plane is a continuous function $\gamma \colon [a,b] \to \mathbb {C}$ from a closed interval $[a,b] \subset \mathbb {R}$ to $\mathbb {C}$.

When $A \subset \mathbb {C}$ is a subset of the complex plane, we say that $\gamma $ is a path in $A$ if $\gamma (t) \in A$ for all $t \in [a,b]$.

If the starting point and the end point of the path $\gamma $ are the same, $\gamma (a) = \gamma (b)$, then we say that $\gamma $ is a closed path.

We sometimes want to disregard the parametrization, and view a path $\gamma \colon [a,b] \to \mathbb {C}$ as a subset of the complex plane. This subset is the range $\big\{ \gamma (t) \, \big| \, t \in [a,b] \big\} \subset \mathbb {C}$ of the parametrizing function, but with a slight abuse of notation we often just write $\gamma \subset \mathbb {C}$ also for this subset.

(Note that the subset $\gamma \subset \mathbb {C}$ is compact, by continuity of $\gamma \colon [a,b] \to \mathbb {C}$ and compactness of $[a,b]$.)

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Definition 3.8Concatenation of paths [Palka1991, Sec. IV.1.4]

Given path $\gamma \colon [a,b] \to \mathbb {C}$ and $\eta \colon [c,d] \to \mathbb {C}$ with $\gamma (b) = \eta (c)$ (the starting point of $\eta $ coincides with the end point of $\gamma $), the concatenation of $\gamma $ and $\eta $ is the path $\gamma \oplus \eta \colon [a,b+d-c] \to \mathbb {C}$ defined by

\begin{align*} (\gamma \oplus \eta )(t) = \begin{cases} \gamma (t) & \text{ for } t \in [a,b] , \\ \eta (c+t-b)) & \text{ for } t \in [b,b+d-c] . \end{cases}\end{align*}

(The slightly cumbersome formula in the second case is due to the fact that we need to attach the two parameter intervals of lengths $b-a$ and $d-c$ to each other, and we have, somewhat arbitrarily, chosen to glue them to form the interval $[a,b+d-c]$.)

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Definition A.26Path-connectedness

A set $A \subset X$ in a metric space $X$ is path connected if for any two points $p, q \in X$ there exists a continuous function $\gamma \colon [0,1] \to X$ such that $\gamma (0) = p$ and $\gamma (1) = q$ (a parametrized path in $X$ starting from $p$ and ending at $q$).

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Definition A.35Path homotopy for closed paths

Let $X$ be a metric space and $\gamma _0 \colon [a,b] \to X$ and $\gamma _0 \colon [a,b] \to X$ two closed paths in $X$. If there exists a continuous function (called a homotopy)

\begin{align*} \Gamma \colon [0,1] \times [a,b] \to X \end{align*}

such that

\begin{align*} & \Gamma (0,t) = \gamma _0(t) \quad \text{ and } \quad \Gamma (1,t) = \gamma _1(t) \qquad \text{for all $t \in [a,b]$} \end{align*}

and

\begin{align*} \Gamma (s,a) = \Gamma (s,b) \qquad \text{for all $s \in [0,1]$,} \end{align*}

then we say that the closed paths $\gamma _0$ and $\gamma _1$ are homotopic.

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Definition 3.9Reparametrization of paths [Palka1991, Sec. IV.1.5]

Given a path $\gamma \colon [a,b] \to \mathbb {C}$ and a continuous increasing bijection $\phi \colon [c,d] \to [a,b]$, we define the reparametrization of $\gamma $ by $\phi $ as the path

\begin{align*} \gamma \circ \phi \, \colon \, [c,d] \, \to \; & \mathbb {C}\\ t \, \mapsto \; & \gamma (\phi (t)) . \end{align*}

Note that

$\phi ^{-1} \colon [a,b] \to [c,d]$ is also a continuous increasing bijection (a continuous bijection from the compact $[a,b]$ is automatically a homeomorphism; see Lemma A.33) and reparametrization can be undone by rereparametrizing by $\phi ^{-1}$;
If both $\gamma $ and the reparametrization function $\phi $ are smooth (continuously differentiable), then the reparametrized path $\gamma \circ \phi $ is also smooth;
If both $\gamma $ and the reparametrization function $\phi $ are piecewise smooth, then the reparametrized path $\gamma \circ \phi $ is also piecewise smooth, i.e., a contour.

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Definition 1.15Polynomial

Polynomial functions are functions $p : \mathbb {C}\to \mathbb {C}$ of the form

\begin{align*} p(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0 \end{align*}

where $a_0,a_1,\ldots ,a_{n-1},a_n \in \mathbb {C}$ are coefficients.

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Definition 5.17Power series [Palka1991, Sec. VII.3.3]

Let $z_0 \in \mathbb {C}$ be a point in the complex plane and let $a_0,a_1,a_2\ldots \in \mathbb {C}$ be coefficients. A function series of the form

\begin{align*} \sum _{n=0}^\infty a_n \, (z-z_0)^n = a_0 + a_1 \, (z - z_0) + a_2 \, (z - z_0)^2 + \cdots \end{align*}

is called a power series centered at $z_0$.

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Definition 3.22Primitive of a function [Palka1991, Sec. IV.2.3]

Let $f : U \to \mathbb {C}$ be a function defined on an open subset $U \subset \mathbb {C}$. A primitive of $f$ is a function $F : U \to \mathbb {C}$ such that $F$ is analytic (i.e., complex differentiable) on $U$,

\begin{align*} F’(z) = f(z) \qquad \text{ for all } z \in U . \end{align*}

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Definition 1.17Principal complex logarithm

The principal logarithm is the function

\begin{align*} \mathrm{Log}\colon \; & \mathbb {C}\setminus \left\{ 0 \right\} \to \mathbb {C}\\ \mathrm{Log}(z) \, = \; & \log |z| + \mathfrak {i}\, \mathrm{Arg}(z) , \end{align*}

where $\log |z|$ is the usual natural logarithm of the positive real number $|z|{\gt}0$ and $\mathrm{Arg}(z) \in (-\pi ,\pi ]$ is the principal argument of the nonzero complex number $z \ne 0$.

(Directly from this definition one sees that for $z \in \mathbb {C}\setminus \left\{ 0 \right\} $ we have $e^{\mathrm{Log}(z)} = z$. All complex solutions $w$ to $e^w = z$ are of the form $w = \mathrm{Log}(z) + 2 \pi \mathfrak {i}n$ where $n \in \mathbb {Z}$.)

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Definition 1.20Principal $n$th root function

Let $n \in \mathbb {N}$. The principal (complex) $n$th root of $z \in \mathbb {C}\setminus \left\{ 0 \right\} $ is

\begin{align*} \sqrt[n]{z} \; := \, z^{1/n} \, = \, e^{\frac{1}{n} \mathrm{Log}(z)} . \end{align*}

(It follows directly from the definition and the properties of complex exponential that $(\sqrt[n]{z})^n = z$. All complex solutions $w$ to $w^n = z$ are of the form $w = \zeta \sqrt[n]{z}$ where $\zeta = e^{2 \pi \mathfrak {i}j / n}$ with $j = 0, 1, \ldots , n-1$, i.e., $\zeta $ is one of the $n$ complex $n$th roots of unity.)

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Definition 1.19Principal complex power function

Let $\alpha \in \mathbb {C}$. The principal (complex) $\alpha $th power function is the function $\mathbb {C}\setminus \left\{ 0 \right\} \to \mathbb {C}$ given by

\begin{align*} z \, \mapsto \, z^\alpha \, := \, e^{\alpha \, \mathrm{Log}(z)} . \end{align*}

(Note: Integer powers have more direct natural definitions. For $n \in \mathbb {N}$ we simply define $z^n$ by recursive multiplication and the function $z \mapsto z^n$ is continuous and defined in all of $\mathbb {C}$ and coincides with the principal power function with $\alpha = n$ on $\mathbb {C}\setminus \left\{ 0 \right\} $. We also define $z^{-n}$ by recursive multiplication of the inverse $z^{-1}$ of $z$, and the function $z \mapsto z^{-n}$ is continuous on $\mathbb {C}\setminus \left\{ 0 \right\} $ and coincides with the principal power function with $\alpha = -n$. For $n = 0$ we define $z^0 = 1$ for any $z \in \mathbb {C}$, and this coincides with the principal power function with $\alpha = 0$ on $\mathbb {C}\setminus \left\{ 0 \right\} $.)

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Definition 5.20Radius of convergence

The radius of convergence of a power series

\begin{align*} \sum _{n=0}^\infty a_n \, (z-z_0)^n \end{align*}

is defined as

\begin{align*} R \; := \; \sup \bigg\{ |z-z_0| \; \bigg| \; \sum _{n=0}^\infty a_n \, (z-z_0)^n \, \text{ converges} \bigg\} . \end{align*}

From Lemma 5.18 and Corollary 5.19 it follows that the power series $\sum _{n=0}^\infty a_n \, (z-z_0)^n$ converges for all $z \in \mathbb {C}$ such that $|z-z_0| {\lt} R$ and diverges for all $z \in \mathbb {C}$ such that $|z-z_0| {\gt} R$. The disk $\mathcal{B}(z_0; R)$ is called the disk of convergence of the power series $\sum _{n=0}^\infty a_n \, (z-z_0)^n$.

(If $R=+\infty $, we interpret $\mathcal{B}(z_0; R) = \mathbb {C}$.)

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Definition 1.16Rational function

Rational functions are functions $f : D \to \mathbb {C}$ which can be written as ratios $f(z) = \frac{p(z)}{q(z)}$ of two polynomials $p, q \colon \mathbb {C}\to \mathbb {C}$ on a domain $D \subset \mathbb {C}$ where the denominator polynomial $q$ has no zeroes.

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Definition 6.4Residue at an isolated singularity [Palka1991, Sec. VIII.2.1]

Let $z_0 \in \mathbb {C}$ be an isolated singularity of an analytic function $f : U \to \mathbb {C}$. Let $r {\gt} 0$ be such that $\mathcal{B}(z_0; r) \setminus \left\{ z_0 \right\} \subset U$, so that by Theorem 5.37 $f$ can be represented in $\mathcal{B}(z_0; r) \setminus \left\{ z_0 \right\} $ uniquely as a Laurent series

\begin{align*} f(z) = \sum _{n=-\infty }^\infty a_n (z - z_0)^n . \end{align*}

The coefficient $a_{-1}$ is called the residue of $f$ at $z_0$, and is denoted $\mathrm{Res}_{{z_0}}({f}) = a_{-1} \in \mathbb {C}$.

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Definition 3.7Reverse path [Palka1991, Sec. IV.1.4]

Given a path $\gamma \colon [a,b] \to \mathbb {C}$, the reverse path $\overleftarrow {\gamma } \colon [a,b] \to \mathbb {C}$ is the path defined by

\begin{align*} \overleftarrow {\gamma }(t) = \gamma (a+b-t) \qquad \text{ for } t \in [a,b] . \end{align*}

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Definition B.5Riemann integral

…

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Definition 6.1The Riemann sphere

The extended complex plane is the set

\begin{align*} \hat{\mathbb {C}} = \mathbb {C}\cup \left\{ \infty \right\} \, , \end{align*}

where $\infty $ is a symbol added to $\mathbb {C}$ to represent a single point at infinity. The set $\hat{\mathbb {C}}$ is given a topology in such a way that open sets in $\mathbb {C}$ remain open in $\hat{\mathbb {C}}$, and sets of the form $\big\{ z \in \mathbb {C}\; \big| \; |z|{\gt}M \big\} $ for $M{\gt}0$ form a neighborhood basis at $\infty $.

(This topology makes $\hat{\mathbb {C}}$ homeomorphic to the 2-dimensional sphere in three-dimensional space, and $\hat{\mathbb {C}}$ is also called the Riemann sphere.)

For example a function $f \colon U \to \mathbb {C}$ has limit $\lim _{z \to z_0} f(z) = \infty $ at $z_0$ if for any $M{\gt}0$ there exists a $\delta {\gt} 0$ such that $|f(z)|{\gt}M$ whenever $0 {\lt} |z-z_0| {\lt} \delta $.

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Definition A.37Simple connectedness

A metric space is said to be simply connected if every closed path $\gamma \colon [a,b] \to X$ in $X$ is contractible.

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Definition 6.3Classification of isolated singularities [Palka1991, Sec. VIII.2.1]

\begin{align*} f(z) = \sum _{n=-\infty }^\infty a_n (z - z_0)^n . \end{align*}

Depending on the coefficients $a_n$ of negative indices $n {\lt} 0$, we distinguish three types of singularities:

$f$ has a removable singularity at $z_0$ if $a_n = 0$ for all $n {\lt} 0$;
$f$ has a pole of order $m \in \mathbb {N}$ at $z_0$ if $a_{-m} \ne 0$ and $a_n = 0$ for all $n {\lt} -m$;
$f$ has an essential singularity at $z_0$ if $a_n \ne 0$ for infinitely many $n {\lt} 0$.

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Definition 3.11Arc length integral along a smooth path [Palka1991, Sec. IV.2.1]

Let $f : A \to \mathbb {C}$ be a continuous function defined on a subset $A \subset \mathbb {C}$. Let $\gamma \colon [a,b] \to A$ be a smooth path in $A$. We define the integral of $f$ with respect to the arc length of $\gamma $ as

\begin{align*} \int _\gamma f(z) \, |dz| \; = \; \int _a^b f \big( \gamma (t) \big) \, |\dot{\gamma }(t)| \; \mathrm{d}t . \end{align*}

(Here $\dot{\gamma }(t) = \frac{\mathrm{d}}{\mathrm{d}t} \gamma (t) \in \mathbb {C}$ denotes the derivative of the smooth path $\gamma $ with respect to its parameter $t$, and $|\dot{\gamma }(t)| \ge 0$ denotes the absolute value of this derivative.)

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Definition 3.10Contour integral along a smooth path [Palka1991, Sec. IV.2.1]

\begin{align*} \int _\gamma f(z) \, dz \; = \; \int _a^b f \big( \gamma (t) \big) \, \dot{\gamma }(t) \; \mathrm{d}t . \end{align*}

(Here $\dot{\gamma }(t) = \frac{\mathrm{d}}{\mathrm{d}t} \gamma (t)$ denotes the derivative of the smooth path $\gamma $ with respect to its parameter $t$.)

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Definition 3.5Smooth path [Palka1991, Sec. IV.1.2]

A path $\gamma \colon [a,b] \to \mathbb {C}$ is smooth if it is continuously differentiable, i.e., the derivative

\begin{align*} \dot{\gamma }(t) = \frac{\mathrm{d}}{\mathrm{d}t} \gamma (t) \end{align*}

with respect to the parameter $t$ exists for all $t \in [a,b]$ (one-sided derivatives at the interval end points $a$ and $b$), and defines a continuous complex-valued function $t \mapsto \dot{\gamma }(t)$ on $[a,b]$.

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Definition 4.3Star-shaped set

A subset $A \subset \mathbb {C}$ of the complex plane is called star-shaped if there exists a point $z_* \in A$ such that for any $z \in A$, the line segment between $z_*$ and $z$ is contained in the subset,

\begin{align*} [z_*, z] \, \subset \, A . \end{align*}

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Definition B.8Supremum

The supremum, or the least upper bound, of a set $A \subset \mathbb {R}$ is the smallest real number $s$ such that $a \le s$ for all $a \in A$, and is denoted by $s = \sup A$.

By the completeness axiom of real numbers, every nonempty set ($A \ne \emptyset $) of real numbers which is bounded from above (for some $u \in \mathbb {R}$ we have $a \le u$ for all $a \in A$) has a supremum $\sup A \in \mathbb {R}$. We adopt the notational conventions that $\sup \emptyset = -\infty $, and that $\sup A = +\infty $ if $A$ is not bounded from above.

For convenience, we also adopt some flexibility in the notation: for example the supremum of values of a real-valued function on a set $D$ is denoted by

\begin{align*} \sup _{x \in D} f(x) \; := \; \sup \big\{ f(x) \; \big| \; x \in D \big\} \end{align*}

and the supremum of values in the tail of a real-number sequence $(x_n)$ starting from index $m$ is denoted by

\begin{align*} \sup _{n \ge m} x_n \; := \; \sup \big\{ x_n \; \big| \; m \ge n \big\} \, . \end{align*}

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Definition 5.1Uniform convergence [Palka1991, Sec. VII.1.1]

Let $(f_n)_{n \in \mathbb {N}}$ be a sequence of functions $f_n \colon X \to \mathbb {C}$, and let $f \colon X \to \mathbb {C}$ also be a such function. We say that the sequence $(f_n)_{n \in \mathbb {N}}$ converges uniformly to $f$ (on $X$) if for every $\varepsilon {\gt} 0$ there exists an $N \in \mathbb {N}$ such that for all $n \ge N$ we have

\begin{align*} \big| f_n(x) - f(x) \big| {\lt} \varepsilon \qquad \text{ for all } x \in X . \end{align*}

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Definition A.23Uniform continuity

Let $X$ and $Y$ be metric spaces. A function $f \colon X \to Y$ is uniformly continuous if for every $\varepsilon {\gt} 0$ there exists a $\delta {\gt} 0$ such that for any $p_0 \in X$ and $p \in \mathcal{B}(p_0; \delta )$ we have $f(p) \in \mathcal{B}(f(p_0); \varepsilon )$.

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Definition 5.6Convergence uniformly on compacts [Palka1991, Sec. VII.1.2]

Let $(f_n)_{n \in \mathbb {N}}$ be a sequence of functions $f_n \colon A \to \mathbb {C}$ on $A \subset \mathbb {C}$, and let $f \colon A \to \mathbb {C}$ also be a such function. We say that the sequence $(f_n)_{n \in \mathbb {N}}$ converges uniformly on compacts (UOC) to $f$ if for every compact subset $K \subset A$ the restrictions $f_n|_K \colon K \to \mathbb {C}$ converge uniformly on $K$ to $f|_K \colon K \to \mathbb {C}$. We then write

\begin{align*} f_n \xrightarrow {\mathrm{UOC}} f \qquad \text{ as } n \to \infty . \end{align*}

(This notion is also called by the alternative names locally uniform convergence and normal convergence.)

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Definition 4.14Winding number

Let $z \in \mathbb {C}$, and let $\gamma $ be a closed contour in $\mathbb {C}\setminus \left\{ z \right\} $. The winding number of $\gamma $ around $z$ is defined as

\begin{align*} \mathfrak {n}_{{\gamma }}(z) = \frac{1}{2\pi \mathfrak {i}} \oint _\gamma \frac{\mathrm{d}\zeta }{\zeta - z} . \end{align*}

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Lemma 5.18Abel’s theorem

If a power series

\begin{align*} \sum _{n=0}^\infty a_n \, (z-z_0)^n \end{align*}

converges at $z = w \in \mathbb {C}$, then it converges absolutely for all $z \in \mathbb {C}$ such that $|z-z_0| {\lt} |w-z_0|$.

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Lemma 5.12Absolute convergence implies convergence

If a complex series converges absolutely, then it converges.

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Lemma 1.6Properties of absolute value [Palka1991, Sec. I.1.2 (1.2)]

For any $z, w \in \mathbb {C}$, we have

\begin{align*} |z|^2 = z \, \overline{z} , \qquad |z w| = |z| \, |w| , \end{align*}

\begin{align*} \Re \mathfrak {e}(z) \le |z| , \qquad \Im \mathfrak {m}(z) \le |z| , \end{align*}

\begin{align*} |z + w| \le |z| + |w| , \qquad |z + w| \ge \big| |z| - |w| \big| . \end{align*}

Also, if $z \ne 0$, then

\begin{align*} z^{-1} = \frac{\overline{z}}{|z|^2} , \qquad \qquad \Big| \frac{w}{z} \Big| = \frac{|w|}{|z|} . \end{align*}

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Lemma 2.14Analytic functions are continuous

Every function $f \colon U \to \mathbb {C}$ which is analytic on an open set $U \subset \mathbb {C}$ is also continuous on $U$.

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Lemma 4.23Cauchy’s estimate for derivatives [Palka1991, Thm V.3.6]

Suppose that a function $f \colon U \to \mathbb {C}$ is analytic on an open set $U \subset \mathbb {C}$ containing the disk $\mathcal{B}(z_0; r) \subset U$, and suppose that there exists a constant $M{\gt}0$ such that $|f(z)| \le M$ for all $z \in \mathcal{B}(z_0; r)$. Then for any $n \in \mathbb {N}$ and any $z \in \mathcal{B}(z_0; r)$ we have the following bound for the $n$th derivative $f^{(n)}$ of $f$:

\begin{align*} \big| f^{(n)}(z) \big| \le \frac{n! \, M \, r}{( r - |z-z_0| )^{n+1}} . \end{align*}

In particular, for the center point $z_0$ of the disk, we have

\begin{align*} \big| f^{(n)}(z_0) \big| \le n! \, M \, r^{-n} . \end{align*}

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Lemma 5.2Cauchy criterion for uniform convergence [Palka1991, Thm VII.1.2]

Let $f_n \colon A \to \mathbb {C}$, $n \in \mathbb {N}$, be complex-valued functions defined on the same set $A$. Then the sequence $(f_n)_{n \in \mathbb {N}}$ converges uniformly on $A$ if and only if for every $\varepsilon {\gt} 0$ there exists an $N \in \mathbb {N}$ such that for all $m,n \ge N$ and all $z \in A$ we have $|f_n(z) - f_m(z)| {\lt} \varepsilon $.

(When $(f_n)_{n \in \mathbb {N}}$ satisfies the condition above, it could be called a uniform Cauchy sequence on $A$.)

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Lemma 2.10Chain rule [Palka1991, Thm III.1.1]

If $f \colon A \to B \subset \mathbb {C}$ is differentiable at $z_0 \in A$ and $g \colon B \to \mathbb {C}$ is differentiable at $f(z_0) \in B$, then the composition $g \circ f \colon A \to \mathbb {C}$ is differentiable at $z_0$, with derivative

\begin{align*} (g \circ f)’(z_0) = f’(z_0) \; g’\big( f(z_0) \big) . \end{align*}

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Lemma A.15Every complex Cauchy sequence converges

If a complex number sequence $(z_n)_{n \in \mathbb {N}}$ is Cauchy, then it converges to a limit $\lim _{n \to \infty } z_n \in \mathbb {C}$.

(This property is known as completeness of the metric space $\mathbb {C}$.)

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Lemma 1.4Properties of complex conjugate [Palka1991, Sec. I.1.2 (1.1)]

For any $z , w \in \mathbb {C}$, we have

\begin{align*} \overline{\overline{z}} = z , \qquad \overline{z + w} = \overline{z} + \overline{w} , \qquad \overline{z w} = \overline{z} \, \overline{w} , \end{align*}

\begin{align*} \Re \mathfrak {e}(z) = \frac{z + \overline{z}}{2} , \qquad \Im \mathfrak {m}(z) = \frac{z - \overline{z}}{2 \mathfrak {i}} . \end{align*}

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Lemma A.19Operations with continuous complex-valued functions

Let $X$ be a metric space, let $p_0 \in X$ be a point, and let $f,g \colon X \to \mathbb {C}$ be two complex-valued functions on $X$ which are continuous at $p_0$. Then also the functions

\begin{align*} p \mapsto f(p) + g(p) \quad \text{and} \quad p \mapsto f(p) \, g(p) \end{align*}

are continuous at $p_0$.

If moreover $g(p_0) \ne 0$, then also the function $p \mapsto \frac{f(p)}{g(p)}$ is continuous at $p_0$.

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Lemma 2.5Complex differentiability implies continuity [Palka1991, Sec. III.1.1]

If a function $f \colon A \to \mathbb {C}$ has a complex derivative $f'(z_0)$ at a point $z_0 \in A$, then it is continuous at $z_0$.

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Lemma 2.6Complex derivative implies differentiability

Let $f \colon A \to \mathbb {C}$ be a function defined on a set $A \subset \mathbb {C}$, and let $u \colon A \to \mathbb {R}$ and $v \colon A \to \mathbb {R}$ be its real and imaginary parts, viewed as real-valued functions of two real variables, $u(x,y) = \Re \mathfrak {e}\big( f (x + \mathfrak {i}y) \big)$ and $v(x,y) = \Im \mathfrak {m}\big( f (x + \mathfrak {i}y) \big)$, so that $f = u + \mathfrak {i}\, v$. If $f$ has a complex derivative $f'(z_0)$ at an interior point $z_0 = x_0 + \mathfrak {i}y_0 \in A$, then $u$ and $v$ are differentiable at $(x_0,y_0)$ and their partial derivatives satisfy the Cauchy-Riemann equations

\begin{align*} \frac{\partial {u}}{\partial {x}}(x_0,y_0) = \frac{\partial {v}}{\partial {y}}(x_0,y_0) \quad \text{and} \quad \frac{\partial {u}}{\partial {y}}(x_0,y_0) = - \frac{\partial {v}}{\partial {x}}(x_0,y_0). \end{align*}

(These equations are equivalent to the differential $\mathrm{d}f (x_0, y_0) \colon \mathbb {R}^2 \to \mathbb {R}^2$ being $\mathbb {C}$-linear when we identify $\mathbb {R}^2 = \mathbb {C}$.)

We can then write the derivative at $z_0$ in any of the following ways:

\begin{align*} f’(z_0) \, = \; & \frac{\partial {u}}{\partial {x}}(x_0,y_0) + \mathfrak {i}\, \frac{\partial {v}}{\partial {x}}(x_0,y_0) = \frac{\partial {v}}{\partial {y}}(x_0,y_0) - \mathfrak {i}\, \frac{\partial {u}}{\partial {y}}(x_0,y_0) \\ = \; & \frac{\partial {u}}{\partial {x}}(x_0,y_0) - \mathfrak {i}\, \frac{\partial {u}}{\partial {y}}(x_0,y_0) = \frac{\partial {v}}{\partial {y}}(x_0,y_0) + \mathfrak {i}\, \frac{\partial {v}}{\partial {x}}(x_0,y_0) . \end{align*}

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Lemma 1.10Properties of the complex exponential

For any $z,w \in \mathbb {C}$ we have

\begin{align*} e^{z + w} = e^z \, e^w. \end{align*}

For any $z \in \mathbb {C}$ we have

\begin{align*} e^{\overline{z}} = \overline{e^z}, \quad |e^z| = e^{\Re \mathfrak {e}(z)}, \quad \arg (e^z) = \Im \mathfrak {m}(z) \; \; (\mathrm{mod}~ 2\pi ). \end{align*}

For $z,w \in \mathbb {C}$ we have $e^{z} = e^{w}$ if and only if $z = w + 2 \pi \mathfrak {i}n$ for some $n \in \mathbb {Z}$.

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Lemma A.11Limits in the complex plane

For a sequence $(z_n)_{n \in \mathbb {N}}$ of complex numbers we have

\begin{align*} \lim _{n \to \infty } z_n = z \end{align*}

if and only if

\begin{align*} \lim _{n \to \infty } \Re \mathfrak {e}(z_n) = \Re \mathfrak {e}(z) \quad \text{and} \quad \lim _{n \to \infty } \Im \mathfrak {m}(z_n) = \Im \mathfrak {m}(z) . \end{align*}

Let $X$ be a metric space, let $f \colon X \to \mathbb {C}$ a complex-valued function on $X$, and let $p_0 \in X$ be a point. Then we have

\begin{align*} \lim _{p \to p_0} f(p) = z \end{align*}

if and only if

\begin{align*} \lim _{p \to p_0} \Re \mathfrak {e}\big( f(p) \big) = \Re \mathfrak {e}(z) \quad \text{and} \quad \lim _{p \to p_0} \Im \mathfrak {m}\big( f(p) \big) = \Im \mathfrak {m}(z) . \end{align*}

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Lemma A.12Operations with complex limits

Let $(z_n)_{n \in \mathbb {N}}$ and $(w_n)_{n \in \mathbb {N}}$ be complex number sequences converging to limits

\begin{align*} \lim _{n \to \infty } z_n = z \quad \text{and} \quad \lim _{n \to \infty } w_n = w . \end{align*}

Then we have

\begin{align*} \lim _{n \to \infty } (z_n + w_n) \, = \, z + w , \quad \lim _{n \to \infty } (z_n w_n) \, = \, z w , \quad \lim _{n \to \infty } \frac{z_n}{w_n} \, = \, \frac{z}{w} \; \text{ if $w \ne 0$}. \end{align*}

Let $X$ be a metric space, let $p_0 \in X$ be a point, and let $f,g \colon X \to \mathbb {C}$ be two complex-valued functions on $X$ such that

\begin{align*} \lim _{p \to p_0} f(p) = z \quad \text{and} \quad \lim _{p \to p_0} g(p) = w . \end{align*}

Then we have

\begin{align*} \lim _{p \to p_0} \big( f(p) + g(p) \big) \, = \, z + w , \quad \lim _{p \to p_0} \big( f(p) \, g(p) \big) \, = \, z w , \quad \lim _{p \to p_0} \frac{f(p)}{g(p)} \, = \, \frac{z}{w} \; \text{ if $w \ne 0$}. \end{align*}

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Lemma 2.2Complex linear versus real linear maps of $\mathbb {C}$

Let $L : \mathbb {C}\to \mathbb {C}$ be a $\mathbb {R}$-linear map represented in the basis $1,\mathfrak {i}$ by the matrix $M = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \in \mathbb {R}^{2 \times 2}$. Then the following are equivalent:

$L$ is $\mathbb {C}$-linear;
$b = - c$ and $a = d$.

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Lemma A.21Composition of continuous functions

Let $X$, $Y$, and $Z$ be metric spaces, and let $f \colon X \to Y$ and $g \colon Y \to Z$ be functions. If $f$ is continuous at $x_0 \in X$ and $g$ is continuous at $f(x_0) \in Y$, then the composition $g \circ f \colon X \to Z$ is continuous at $x_0$.

(The composition $g \circ f$ is defined by the formula $(g \circ f)(x) = g \big( f(x) \big)$.)

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Lemma A.20Continuity characterization

Let $X$ and $Y$ be metric spaces, and let $f \colon X \to Y$ be a function. Then the following are equivalent:

$f$ is a continuous function;
for every open set $V \subset Y$, the preimage $f^{-1}[V] = \big\{ x \in X \, \big| \, f(x) \in V \big\} $ is an open set in $X$;
for every closed set $A \subset Y$, the preimage $f^{-1}[A] = \big\{ x \in X \, \big| \, f(x) \in A \big\} $ is a closed set in $X$.

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Lemma A.33Continuous bijection from a compact domain is a homeomorphism

Let $X$ and $Y$ be metric spaces and assume that $X$ is compact. Then for any continuous bijection $f \colon X \to Y$, also the inverse $f^{-1} \colon Y \to X$ is continuous.

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Lemma A.17Continuity of complex-valued functions

Let $X$ be a metric space, and let $f \colon X \to \mathbb {C}$ be a complex-valued function on $X$. Then $f$ is continuous at $p_0 \in X$ if and only if its real and imaginary parts $p \mapsto \Re \mathfrak {e}\big( f(p) \big)$ and $p \mapsto \Im \mathfrak {m}\big( f(p) \big)$ are continuous at $p_0$.

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Lemma B.6Riemann integrability of continuous functions

Any continuous function $f \colon [a,b] \to \mathbb {R}$ is Riemann integrable on $[a,b]$.

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Corollary 3.20An a priori bound for contour integrals

Let $f : A \to \mathbb {C}$ be a continuous function defined on $A \subset \mathbb {C}$, and let $\gamma $ be a contour in $A$. Assume that $|f(z)| \le M$ for all points $z$ on the contour $\gamma $. Then we have

\begin{align*} \left| \int _\gamma f(z) \, \mathrm{d}z \right| \; \le \; M \, \ell (\gamma ) , \end{align*}

where $\ell (\gamma ) = \int _\gamma |\mathrm{d}z|$ denotes the length of the contour $\gamma $.

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Lemma 5.34Convergence of doubly infinite series [Palka1991, Lem VII.2.1]

A doubly infinite series

\begin{align*} \sum _{n=-\infty }^\infty z_n = \cdots + z_{-2} + z_{-1} + z_0 + z_1 + z_2 + \cdots , \end{align*}

of complex numbers converges if and only if both the series $\sum _{n=0}^\infty z_n$ and $\sum _{n=1}^\infty z_{-n}$ converge.

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Lemma 4.4Convex sets are star-shaped

Any nonempty convex set is star-shaped.

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Theorem 1.13De Moivre’s formula [Palka1991, Sec. I.1.2 (1.7)]

For any $\theta \in \mathbb {R}$ and $n \in \mathbb {Z}$, we have

\begin{align*} \big( \cos (\theta ) + \mathfrak {i}\, \sin (\theta ) \big)^n = \cos (n \theta ) + \mathfrak {i}\, \sin (n \theta ) . \end{align*}

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Lemma 4.19Analyticity of derivatives [Palka1991, Thm V.3.1]

If a function $f \colon U \to \mathbb {C}$ is analytic on an open set $U \subset \mathbb {C}$, then its derivative $f'$ is also analytic on $U$. In particular, then $f$ is continuously differentiable, $f \in \mathcal{C}^{{1}}(U)$.

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Lemma 3.21[Palka1991, Lemma V.1.6]

Let $\gamma $ be a contour in $\mathbb {C}$, and let $h \colon \gamma \to \mathbb {C}$ be a continuous function on the contour (we slightly abuse the notation here to identify the contour as a subset $\gamma \subset \mathbb {C}$). Let $k \in \mathbb {N}$ be a positive integer. Define $H \colon \mathbb {C}\setminus \gamma \to \mathbb {C}$ by

\begin{align*} H(z) = \int _\gamma \frac{h(\zeta )}{(\zeta - z)^{k}} \, \mathrm{d}\zeta . \end{align*}

Then $H$ is analytic on $\mathbb {C}\setminus \gamma $, and its derivative at $z \in \mathbb {C}\setminus \gamma $ is given by

\begin{align*} H’(z) = k \, \int _\gamma \frac{h(\zeta )}{(\zeta - z)^{k+1}} \, \mathrm{d}\zeta . \end{align*}

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Lemma 2.7Linearity of the derivative [Palka1991, Sec. III.1.2 (3.4)]

If two functions $f, g \colon A \to \mathbb {C}$ have complex derivatives $f'(z_0), g'(z_0)$ at a point $z_0 \in A$, then the sum function $f + g$ has a complex derivative at $z_0$ given by

\begin{align*} (f + g)’(z_0) = f’(z_0) + g’(z_0) . \end{align*}

If a function $f \colon A \to \mathbb {C}$ is has a complex derivative $f'(z_0)$ at a point $z_0 \in A$ and $c \in \mathbb {C}$ is a complex number, then the function $c f$ has complex derivative

\begin{align*} (c f)’(z_0) = c \, f’(z_0) \end{align*}

at $z_0$.

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Lemma B.2Differentiability implies continuity

If a function $f \colon U \to \mathbb {R}^m$ defined on a subset $U \subset \mathbb {R}^n$ is differentiable at $p_0 \in U$, then it is continuous at $p_0$.

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Lemma 1.8Discontinuity of the principal argument

The principal argument $\mathrm{Arg}\colon \mathbb {C}\setminus \left\{ 0 \right\} \to (-\pi ,\pi ]$ is continuous on the subset $\mathbb {C}\setminus (-\infty ,0]$, but it is discontinuous on the negative real axis $(-\infty ,0]$.

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Lemma 2.17The complex exponential is analytic

The complex exponential function $\exp \colon \mathbb {C}\to \mathbb {C}$ is analytic. Its (complex) derivative at $z \in \mathbb {C}$ is $\exp '(z) = \exp (z)$.

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Lemma 3.3Fundamental theorem of calculus for complex-valued integrals

Suppose that $f \colon [a,b] \to \mathbb {C}$ is a continuously differentiable complex-valued function on a closed interval $[a,b] \subset \mathbb {R}$. Denote its derivative by $\dot{f}(t) = \frac{\mathrm{d}}{\mathrm{d}t} f(t)$. Then for the integral of the derivative of $f$ we have

\begin{align*} \int _a^b \dot{f}(t) \, \mathrm{d}t \; = \; f(b) - f(a) . \end{align*}

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Lemma 5.10Geometric series

The geometric series

\begin{align*} \sum _{n=0}^\infty z^n \; = \; 1 + z + z^2 + z^3 + \cdots , \end{align*}

with ratio $z \in \mathbb {C}$ converges if and only if $|z| {\lt} 1$. In that case its sum is

\begin{align*} \sum _{n=0}^\infty z^n \; = \; \frac{1}{1-z} . \end{align*}

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Lemma 4.6Goursat’s lemma [Palka1991, Lem V.1.1]

Suppose that a function $f \colon U \to \mathbb {C}$ is analytic on an open set $U \subset \mathbb {C}$. Then for any closed triangle $\triangle \subset U$, we have

\begin{align*} \oint _{\partial \triangle } f(z) \, \mathrm{d}z = 0 . \end{align*}

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Lemma 2.23Local existence of harmonic conjugates

Let $B = \mathcal{B}(z_0; r) \subset \mathbb {C}$ be a disk in the complex plane. Suppose that $u \colon B \to \mathbb {R}$ is harmonic function on $B$. Then a harmonic conjugate $v \colon B \to \mathbb {R}$ of $u$ in the disk $B$ exists and is unique up to an additive constant.

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Corollary 4.20Analyticity of higher derivatives [Palka1991, Cor V.3.2]

If a function $f \colon U \to \mathbb {C}$ is analytic on an open set $U \subset \mathbb {C}$, then its derivatives $f', f'', \ldots , f^{(k)}, \ldots $ of all orders are also analytic on $U$. In particular, then $f$ is infinitely differentiable, $f \in \mathcal{C}^{{\infty }}(U)$.

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Lemma 4.12Homotopy invariance of contour integrals

Let $f \colon U \to \mathbb {C}$ be an analytic function on an open set $U \subset \mathbb {C}$, and let $\gamma _0$ and $\gamma _1$ be two closed contours in $U$ which are homotopic to each other in $U$. Then we have

\begin{align*} \oint _{\gamma _0} f(z) \, \mathrm{d}z = \oint _{\gamma _1} f(z) \, \mathrm{d}z . \end{align*}

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Lemma 3.12Contour integrals and smooth path concatenation [Palka1991, Lem IV.2.1(iv)]

If a smooth path $\gamma $ in $A$ is a concatenation of smooth paths $\eta _1, \ldots , \eta _n$, and $f : A \to \mathbb {C}$ is a continuous function defined on $A \subset \mathbb {C}$, then we have

\begin{align*} \int _\gamma f(z) \, \mathrm{d}z \; = \; \sum _{j=1}^n \int _{\eta _j} f(z) \, \mathrm{d}z \end{align*}

and

\begin{align*} \int _\gamma f(z) \, |\mathrm{d}z| \; = \; \sum _{j=1}^n \int _{\eta _j} f(z) \, |\mathrm{d}z| . \end{align*}

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Lemma 3.18Linearity of integrals [Palka1991, Lem IV.2.1(i-ii)]

Let $A \subset \mathbb {C}$ be a subset of the complex plane let and $\gamma \colon [a,b] \to A$ be a contour in $A$.

If $f, g \colon A \to \mathbb {C}$ are continuous functions defined on $A$, then the contour integral and the arc length integral of their sum are

\begin{align*} \int _\gamma \big( f(z) + g(z) \big) \, \mathrm{d}z \; = \; & \int _\gamma f(z) \, \mathrm{d}z + \int _\gamma g(z) \, \mathrm{d}z \\ \int _\gamma \big( f(z) + g(z) \big) \, |\mathrm{d}z| \; = \; & \int _\gamma f(z) \, |\mathrm{d}z| + \int _\gamma g(z) \, |\mathrm{d}z| . \end{align*}

If $f \colon A \to \mathbb {C}$ is a complex-valued continuous function defined on $A$, and $\lambda \in \mathbb {C}$ is a complex number, then the contour integral and the arc length integral of the scalar multiple of $f$ are

\begin{align*} \int _\gamma \lambda \, f(z) \, \mathrm{d}z \; = \; & \lambda \int _\gamma f(z) \, \mathrm{d}z \\ \int _\gamma \lambda \, f(z) \, |\mathrm{d}z| \; = \; & \lambda \int _\gamma f(z) \, |\mathrm{d}z| . \end{align*}

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Lemma 3.17Contour integrals and path reversal [Palka1991, Lem IV.2.1(iii)]

If $f : A \to \mathbb {C}$ is a continuous function defined on $A \subset \mathbb {C}$, and $\gamma $ is a piecewise path in $A$, then for the contour integral and the arc length integral behave as follows under path reversal: then we have

\begin{align*} \int _{\overleftarrow {\gamma }} f(z) \, \mathrm{d}z \; = \; - \int _{\gamma } f(z) \, \mathrm{d}z \end{align*}

and

\begin{align*} \int _{\overleftarrow {\gamma }} f(z) \, |\mathrm{d}z| \; = \; \int _{\gamma } f(z) \, |\mathrm{d}z| \end{align*}

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Lemma 3.19Triangle inequality for contour integrals [Palka1991, Lem IV.2.1(vi)]

Let $f : A \to \mathbb {C}$ be a continuous function defined on $A \subset \mathbb {C}$, and let $\gamma $ be a contour in $A$. Then we have

\begin{align*} \left| \int _\gamma f(z) \, \mathrm{d}z \right| \; \le \; \int _\gamma |f(z)| \, |dz| . \end{align*}

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Lemma 2.11Derivative of inverse [Palka1991, Thm III.4.1]

Suppose that $f$ is a complex-valued function defined on a subset of the complex plane, which has a nonzero complex derivative $f'(z_0) \ne 0$ at a point $z_0$ and which has a local inverse function near $z_0$ in the sense that there are open sets $U, V \subset \mathbb {C}$ with $z_0 \in U$ and $f(z_0) \in V$, and the restriction of $f$ to $U$ is continuous $U \to V$ with a continuous inverse. Then the local inverse function $f^{-1} \colon V \to U$ has complex derivative at $w_0 := f(z_0)$ given by

\begin{align*} (f^{-1})’(w_0) = \frac{1}{f'(z_0)} . \end{align*}

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Lemma B.3Jacobian matrix of the differential

If a function $f \colon U \to \mathbb {R}^m$ defined on a subset $U \subset \mathbb {R}^n$ is differentiable at an interior point $p_0$ of $U$, then it has all first order partial derivatives at $p_0$, and the matrix representation of the differential $\mathrm{d}f (p_0)$ in the standard bases of $\mathbb {R}^m$ and $\mathbb {R}^n$ is

\begin{align*} \mathrm{d}f (p_0) = \left[ \begin{array}{ccc} \frac{\partial f_1}{\partial x_1}(p_0) & \cdots & \frac{\partial f_1}{\partial x_n}(p_0) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(p_0) & \cdots & \frac{\partial f_m}{\partial x_n}(p_0) \end{array} \right] \in \mathbb {R}^{m \times n} , \end{align*}

where $f_1, \ldots , f_m \colon U \to \mathbb {R}$ denote component functions of $f$.

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Lemma 5.36Annulus of convergence of Laurent power series [Palka1991, Thm VII.3.5]

Consider a Laurent series

\begin{align*} f(z) = & \; \sum _{n=-\infty }^\infty a_n (z - z_0)^n . \end{align*}

Denote

\begin{align*} \rho _- = \limsup _{n \to \infty } \sqrt[n]{|a_{-n}|}, \qquad \rho _+ = \Big( \limsup _{n \to \infty } \sqrt[n]{|a_{n}|} \Big)^{-1} . \end{align*}

Then the series $\sum _{n=-\infty }^\infty a_n (z - z_0)^n$ converges for all $z$ is the annulus

\begin{align*} \mathcal{A}_{\rho _-,\rho _+}(z_0) := \Big\{ z \in \mathbb {C}\; \Big| \; \rho _- {\lt} |z-z_0| {\lt} \rho _+ \Big\} . \end{align*}

Moreover, the convergence is uniform on compact subsets of $\mathcal{A}_{\rho _-,\rho _+}(z_0)$, and the series defines an analytic function $f(z)$ on the annulus $\mathcal{A}_{\rho _-,\rho _+}(z_0)$.

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Lemma 2.8Leibniz rule [Palka1991, Sec. III.1.2 (3.4)]

If two functions $f, g \colon A \to \mathbb {C}$ have complex derivatives $f'(z_0), g'(z_0)$ at a point $z_0 \in A$, then the product function $f g$ has complex derivative

\begin{align*} (f g)’(z_0) = f’(z_0) \, g(z_0) + f(z_0) \, g’(z_0) \end{align*}

at $z_0$.

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Lemma B.12Limit with limsup and liminf

Let $(x_n)_{n \in \mathbb {N}}$ be a sequence of real numbers, and let $x \in \mathbb {R}$. Then the following are equivalent:

The limit $\lim _{n \to \infty } x_n$ exists and equals $x$.
We have both $\limsup _{n \to \infty } x_n = x$ and $\liminf _{n \to \infty } x_n = x$.

(With the usual conventions of $\pm \infty $ as possible limits of real-number sequences, the above equivalence of conditions also extends to the cases $x = \pm \infty $.)

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Lemma 3.2Complex linearity of complex-valued integrals

If $f, g \colon [a,b] \to \mathbb {C}$ are complex-valued continuous functions defined on a closed interval $[a,b] \subset \mathbb {R}$, then the integral of their sum is

\begin{align*} \int _a^b \big( f(t) + g(t) \big) \, \mathrm{d}t \; = \; \int _a^b f(t) \, \mathrm{d}t + \int _a^b g(t) \, \mathrm{d}t . \end{align*}

If $f \colon [a,b] \to \mathbb {C}$ is a complex-valued continuous function defined on a closed interval $[a,b] \subset \mathbb {R}$, and $\lambda \in \mathbb {C}$ is a complex number, then the integral of the scalar multiple of $f$ is

\begin{align*} \int _a^b \lambda \, f(t) \, \mathrm{d}t \; = \; \lambda \int _a^b f(t) \, \mathrm{d}t . \end{align*}

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Lemma 2.4Local linear approximation

If a function $f \colon A \to \mathbb {C}$ has complex derivative $f'(z_0) = \lambda \in \mathbb {C}$ at a point $z_0 \in A$, then we can write a linear approximation

\begin{align*} f(z) = f(z_0) + (z - z_0) \, \lambda + \epsilon (z) , \end{align*}

where the error term $\epsilon $ is small near $z_0$ in the sense that $\lim _{z \to z_0} \frac{\epsilon (z)}{|z-z_0|} = 0$.

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Lemma A.2Metric in the complex plane

The formula

\begin{align*} \mathsf{d}(z,w) = |z - w| \qquad \text{ for } z,w \in \mathbb {C}\end{align*}

defines a metric on the complex plane $\mathbb {C}$.

(Thus $\mathbb {C}$ becomes a metric space. Also any subset of $\mathbb {C}$, in particular $\mathbb {R}\subset \mathbb {C}$, becomes a metric space when equipped with the metric given by the above formula restricted to the subset.)

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Lemma 3.24Existence of primitives for monomials

For $n \in \left\{ 0,1,2,\ldots \right\} $, the monomial function $f(z) = z^n$ has a primitive $F(z) = \frac{1}{n+1} z^{n+1} + c$ (with $c \in \mathbb {C}$ arbitrary) in the whole complex plane $\mathbb {C}$.

For $n \in \left\{ -2,-3,-4,\ldots \right\} $, the monomial function $f(z) = z^n$ has a primitive $F(z) = \frac{1}{n+1} z^{n+1} + c$ (with $c \in \mathbb {C}$ arbitrary) in the punctured complex plane $\mathbb {C}\setminus \left\{ 0 \right\} $.

The monomial function $f(z) = z^{-1} = \frac{1}{z}$ does not have a primitive in the punctured complex plane $\mathbb {C}\setminus \left\{ 0 \right\} $.

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Lemma 1.12Multiplication in polar form [Palka1991, Sec. I.1.2 (1.6)]

For any $z,w \in \mathbb {C}$, written in polar form as $z = r e^{\mathfrak {i}\theta }$ and $w = r' e^{\mathfrak {i}\theta '}$, the product can be written in polar form as

\begin{align*} z w = r r’ \, e^{\mathfrak {i}(\theta + \theta ')} . \end{align*}

In other words,

\begin{align*} |z w| = |z| \, |w| \qquad \text{ and } \qquad \arg (z w) = \arg (z) + \arg (w) \; \; (\mathrm{mod}~ 2\pi ). \end{align*}

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Lemma 5.27No vanishing of all derivatives at a point [Palka1991, Thm VIII.1.1]

Suppose that $f \colon \mathcal{D}\to \mathbb {C}$ is an analytic function on a connected open set $\mathcal{D}\subset \mathbb {C}$. If there exists a point $z_0 \in \mathcal{D}$ such that $f^{(n)}(z_0) = 0$ for all $n \in \mathbb {N}$, then $f$ is a constant function.

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Lemma A.32On a compact domain continuity implies uniform continuity

If $X$ is compact and a function $f \colon X \to Y$ is continuous, then it is uniformly continuous.

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Lemma A.28Open connected sets are path-connected

Suppose that $U \subset \mathbb {C}$ is an open subset of the complex plane. Then $U$ is connected if and only if it is path-connected.

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Lemma A.27Path-connectedness implies connectedness

If a metric space $X$ is path-connected, then it is connected.

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Lemma 1.11Polar form

Every complex number $z \in \mathbb {C}$ can be written in the polar form

\begin{align*} z = r \, e^{\mathfrak {i}\theta } \qquad \text{ where $r \ge 0$ and $\theta \in \mathbb {R}$.} \end{align*}

The modulus of $z$ is the number $r = |z|$ above. If $z \ne 0$, then $\theta $ above is a choice of the argument of $z$, i.e., $\theta = \mathrm{Arg}(z) + 2 \pi m$ for some $m \in \mathbb {Z}$.

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Lemma 2.15Polynomials are analytic

Every polynomial function $p \colon \mathbb {C}\to \mathbb {C}$ is analytic.

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Lemma 5.23Analyticity of power series [Palka1991, Thm VII.3.3]

Let $z_0 \in \mathbb {C}$ be a point in the complex plane and let $a_0,a_1,a_2\ldots \in \mathbb {C}$ be coefficients. Suppose that the power series

\begin{align*} f(z) = \sum _{n=0}^\infty a_n \, (z-z_0)^n \end{align*}

has radius of convergence $R {\gt} 0$. Then it defines an analytic function $f$ on the disk $\mathcal{B}(z_0; R)$. The derivative of $f$ is given by the power series

\begin{align*} f’(z) = \sum _{n=1}^\infty n \, a_n \, (z-z_0)^{n-1} . \end{align*}

Moreover, the coefficients $a_k$ are related to the $k$th derivatives of $f$ at $z_0$ through the formula

\begin{align*} a_k = \frac{f^{(k)}(z_0)}{k!} . \end{align*}

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Lemma 5.24Uniqueness of power series representation

Suppose that two power series $\sum _{n=0}^\infty a_n \, (z-z_0)^n$ and $\sum _{n=0}^\infty b_n \, (z-z_0)^n$ converge in a disk $\mathcal{B}(z_0; r)$ of radius $r {\gt} 0$ and represent the same function

\begin{align*} \sum _{n=0}^\infty a_n \, (z-z_0)^n \; = \; \sum _{n=0}^\infty b_n \, (z-z_0)^n \qquad \text{ for } z \in \mathcal{B}(z_0; r) . \end{align*}

Then their coefficients must be equal: $a_n = b_n$ for all $n$.

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Lemma 4.7Existence of primitives in star-shaped domains

Every analytic function $f \colon U \to \mathbb {C}$ on a star-shaped domain $U \subset \mathbb {C}$ has a primitive in $U$.

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Lemma 2.9Derivative of a quotient [Palka1991, Sec. III.1.2 (3.4)]

If two functions $f, g \colon A \to \mathbb {C}$ have complex derivatives $f'(z_0), g'(z_0)$ at a point $z_0 \in A$ and $g(z_0) \ne 0$, then the quotient function $f / g$ has complex derivative

\begin{align*} \left( \frac{f}{g} \right)’(z_0) = \frac{f'(z_0) \, g(z_0) - f(z_0) \, g'(z_0)}{g(z_0)^2} . \end{align*}

at $z_0$.

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Lemma 5.13D’Alembert’s ratio test

Suppose that $\sum _{n=1}^\infty z_n$ is a complex series such that the limit

\begin{align*} r = \lim _{n \to \infty } \frac{|z_{n+1}|}{|z_n|} \end{align*}

exists. Then:

If $r {\lt} 1$, then the series $\sum _{n=1}^\infty z_n$ converges absolutely.
If $r {\gt} 1$, then the series $\sum _{n=1}^\infty z_n$ does not converge.

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Lemma 5.21D’Alembert’s ratio test for the radius of convergence

Suppose that for the coefficients of a power series

\begin{align*} \sum _{n=0}^\infty a_n \, (z-z_0)^n \end{align*}

the limit

\begin{align*} \rho = \lim _{n \to \infty } \frac{|a_n|}{|a_{n+1}|} \end{align*}

exists. Then the radius of convergence $R$ of the power series is $R = \rho $.

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Lemma 2.16Rational functions are analytic

Every rational function $f \colon U \to \mathbb {C}$ is analytic on its domain of definition $U \subset \mathbb {C}$.

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Lemma 2.21Harmonicity of real and imaginary parts

Suppose that $f \colon U \to \mathbb {C}$ is a analytic function on an open subset $U \subset \mathbb {C}$ of the complex plane. Let $u,v \colon U \to \mathbb {R}$ denote the real and imaginary parts of $f$ defined by $u(x,y) = \Re \mathfrak {e}\big( f(x + \mathfrak {i}y)\big)$ and $v(x,y) = \Im \mathfrak {m}\big( f(x + \mathfrak {i}y)\big)$. Assume moreover that that $u$ and $v$ are twice continuously differentiable (later it will be shown that this assumption holds automatically by the analyticity of $f$). Then $u$ and $v$ are harmonic functions, i.e., they satisfy

\begin{align*} \triangle u = 0 \qquad \text{ and } \qquad \triangle v = 0 , \qquad \text{ where} \qquad \triangle = \frac{\partial ^2}{\partial {x}^2} + \frac{\partial ^2}{\partial {y}^2} . \end{align*}

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Lemma A.14Every real Cauchy sequence converges

If a real number sequence $(x_n)_{n \in \mathbb {N}}$ is Cauchy, then it converges to a limit $\lim _{n \to \infty } x_n \in \mathbb {R}$.

(This property is known as completeness of the metric space $\mathbb {R}$.)

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Lemma 3.16Reparametrization invariance of integrals [Palka1991, Lem IV.2.1(v)]

Let $\gamma $ be a piecewise smooth path in $A$, and let $\widetilde{\gamma }$ be obtained from $\gamma $ by an orientation-preserving reparametrization. Then for any continuous function $f : A \to \mathbb {C}$ we have

\begin{align*} \int _{\widetilde{\gamma }} f(z) \, dz \; = \; \int _{\gamma } f(z) \, dz \end{align*}

and

\begin{align*} \int _{\widetilde{\gamma }} f(z) \, |dz| \; = \; \int _{\gamma } f(z) \, |dz| . \end{align*}

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Lemma 2.18Branches of $n$th root functions are analytic

The principal branch of the $n$th root function $z \mapsto \sqrt[n]{z}$ is analytic on its domain $\mathbb {C}\setminus (-\infty ,0]$.

(Different branch choices can be made to obtain analyticity on other domains, but for $n \ge 2$, no branch of $\sqrt[n]{z}$ can be made analytic on all of $\mathbb {C}$.)

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Lemma 1.14Roots of unity

For any $n \in \mathbb {N}$, the solutions $z \in \mathbb {C}$ to the equation

\begin{align*} z^n = 1 \end{align*}

are the $n$ distinct complex numbers

\begin{align*} z_j \, = \, e^{\mathfrak {i}2 \pi j / n} \, = \, \cos \Big( \frac{2 \pi j}{n} \Big) + \mathfrak {i}\, \sin \Big( \frac{2 \pi j}{n} \Big) \qquad \text{ where } j = 0, 1, \ldots , n-1 . \end{align*}

These solutions are called the (complex) $n$th roots of unity.

In particular, we have the polynomial factorization

\begin{align*} z^n - 1 \, = \, \prod _{j=0}^{n-1} \Big( z - e^{\mathfrak {i}2 \pi j / n} \Big) . \end{align*}

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Lemma 4.29Schwarz’s lemma [Palka1991, Thm V.3.14]

Let $f \colon \mathcal{B}(0; 1) \to \mathbb {C}$ be an analytic function on the open unit disk such that $|f(z)| \le 1$ for all $z \in \mathcal{B}(0; 1)$ and $f(0) = 0$. Then we have

\begin{align*} |f’(0)| \le 1 \qquad \text{ and } \qquad |f(z)| \le |z| \quad \text{ for all } z \in \mathcal{B}(0; 1) . \end{align*}

Furthermore, unless $f$ is of the form $f(z) = \lambda z$ for some $\lambda \in \mathbb {C}$ with $|\lambda | = 1$, then we have

\begin{align*} |f’(0)| {\lt} 1 \qquad \text{ and } \qquad |f(z)| {\lt} |z| \quad \text{ for all } z \in \mathcal{B}(0; 1) \setminus \left\{ 0 \right\} . \end{align*}

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Lemma 4.5Star-shaped sets are path connected and simply connected

Any star-shaped set $U \subset \mathbb {C}$ is path connected and simply connected.

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Lemma 5.9Terms of a convergent series tend to zero

If a complex series $\sum _{n=1}^\infty z_n$ converges, then we have

\begin{align*} \lim _{n \to \infty } z_n = 0 . \end{align*}

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Lemma B.7Trigonometric angle sum identities

Let $\alpha , \beta \in \mathbb {R}$. Then we have

\begin{align*} \cos (\alpha + \beta ) \; = \; \cos (\alpha ) \, \cos (\beta ) - \sin (\alpha ) \, \sin (\beta ) \\ \sin (\alpha + \beta ) \; = \; \cos (\alpha ) \, \sin (\beta ) + \sin (\alpha ) \, \cos (\beta ) . \end{align*}

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Lemma 5.3Continuity is preserved in uniform limits [Palka1991, Thm VII.1.1]

Let $X$ be a metric space (e.g., $\mathbb {R}$, $\mathbb {C}$, or a subset of these). If a sequence $(f_n)_{n \in \mathbb {N}}$ of continuous functions $f_n \colon X \to \mathbb {C}$ converges uniformly to a function $f \colon X \to \mathbb {C}$, then $f$ is continuous.

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Lemma 5.4Integration commutes with uniform limits [Palka1991, Thm VII.1.1]

If a sequence $(f_n)_{n \in \mathbb {N}}$ of continuous functions $f_n \colon [a,b] \to \mathbb {C}$ on a closed interval $[a,b] \subset \mathbb {R}$ converges uniformly to a function $f \colon [a,b] \to \mathbb {C}$, then we have

\begin{align*} \lim _{n \to \infty } \int _a^b f_n(x) \, \mathrm{d}x = \int _a^b f(x) \, \mathrm{d}x . \end{align*}

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Lemma A.24Uniform continuity implies continuity

If a function $f \colon X \to Y$ is uniformly continuous, then it is continuous.

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Lemma 5.16Series of analytic functions [Palka1991, Thm VII.3.2]

Suppose that functions $f_1, f_2, \ldots \colon U \to \mathbb {C}$ are analytic functions on an open set $U \subset \mathbb {C}$ such that the series $\sum _{n=1}^\infty f_n$ converges uniformly on compacts to a function $f \colon U \to \mathbb {C}$. Then $f$ is analytic on $U$. Moreover, for any $k \in \mathbb {N}$, the series $\sum _{n=1}^\infty f^{(k)}_n$ of $k$th derivatives converges uniformly on compacts to $f^{(k)}$.

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Lemma 5.7UOC limit of analytic functions [Palka1991, Thm VII.3.1]

Suppose that functions $f_1, f_2, \ldots \colon U \to \mathbb {C}$ are analytic functions on an open set $U \subset \mathbb {C}$ and the sequence $(f_n)_{n \in \mathbb {N}}$ converges uniformly on compacts to a function $f$. Then $f$ is analytic on $U$. Moreover, for any $k \in \mathbb {N}$, the sequence $(f^{(k)}_n)_{n \in \mathbb {N}}$ of $k$th derivatives converges uniformly on compacts to $f^{(k)}$.

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Lemma B.4Vanishing partial derivatives implies locally constant

Suppose that $f \colon U \to \mathbb {R}^m$ is a function defined on an open and connected subset $U \subset \mathbb {R}^n$ of $\mathbb {R}^n$ whose first order partial derivatives exist and are zero at all points of $U$. Then $f$ is a constant function.

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Lemma 5.15Weierstrass M-test [Palka1991, Thm VII.2.2]

Suppose that $M_1,M_2,\ldots \ge 0$ are nonnegative numbers such that the series $\sum _{n=1}^\infty M_n$ converges. Suppose also that for each $n \in \mathbb {N}$, $f_n \colon X \to \mathbb {C}$ is a function on $X$ such that $|f_n(x)| \le M_n$ for all $x \in X$. Then the series $\sum _{n=1}^\infty f_n$ converges absolutely and uniformly on $X$.

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Lemma 4.15Winding number concatenation and reversal

Let $\gamma $ and $\eta $ be closed contours in $\mathbb {C}$ both starting and ending at the same point $z_0 \in \mathbb {C}$. Then for any point $z \in \mathbb {C}\setminus (\gamma \cup \eta )$ we have

\begin{align*} \mathfrak {n}_{{\gamma \oplus \eta }}(z) \; = \; \mathfrak {n}_{{\gamma }}(z) + \mathfrak {n}_{{\eta }}(z) \end{align*}

and for any point $z \in \mathbb {C}\setminus \gamma $ we have

\begin{align*} \mathfrak {n}_{{\overleftarrow {\gamma }}}(z) \; = \; -\mathfrak {n}_{{\gamma }}(z) . \end{align*}

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Lemma 4.17Homotopy invariance of winding numbers

Let $z \in \mathbb {C}$ be a point and let $\gamma $ and $\eta $ be two closed contours in $\mathbb {C}\setminus \left\{ z \right\} $ which are homotopic to each other in $\mathbb {C}\setminus \left\{ z \right\} $. Then we have

\begin{align*} \mathfrak {n}_{{\gamma }}(z) = \mathfrak {n}_{{\eta }}(z) . \end{align*}

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Lemma 4.16Winding number properties

Let $\gamma $ be a closed contour in $\mathbb {C}$. Then the winding numbers $\mathfrak {n}_{{\gamma }}(z)$ of points $z \in \mathbb {C}\setminus \gamma $ satisfy:

$z \mapsto \mathfrak {n}_{{\gamma }}(z)$ is constant on each connected component of $\mathbb {C}\setminus \gamma $;
$\mathfrak {n}_{{\gamma }}(z) = 0$ for all $z$ in the unbounded connected component of $\mathbb {C}\setminus \gamma $;
If $\gamma $ is a Jordan contour and $V \subset \mathbb {C}\setminus \gamma $ is the bounded connected component of $\mathbb {C}\setminus \gamma $, then either $\mathfrak {n}_{{\gamma }}(z) = 1$ for all $z \in V$ or $\mathfrak {n}_{{\gamma }}(z) = -1$ for all $z \in V$.

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Lemma 2.19Analytic functions of vanishing derivative

Suppose that $f \colon D \to \mathbb {C}$ is a analytic function on a connected open subset $D \subset \mathbb {C}$ of the complex plane such that $f'(z) = 0$ for all $z \in D$. Then $f$ is a constant function.

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Theorem 5.37Laurent series for analytic functions [Palka1991, Thm VII.3.6]

Suppose that $f \colon U \to \mathbb {C}$ is an analytic function on an open set $U \subset \mathbb {C}$ which contains an annulus

\begin{align*} \mathcal{A}_{r_1,r_2}(z_0) = \Big\{ z \in \mathbb {C}\; \Big| \; r_1 {\lt} |z - z_0| {\lt} r_2 \Big\} \end{align*}

for some $z_0 \in \mathbb {C}$ and $0 \le r_1 {\lt} r_2$. Then the function $f$ can be uniquely represented in $\mathcal{A}_{r_1,r_2}(z_0)$ as a series

\begin{align*} f(z) = \sum _{n=-\infty }^\infty a_n \, (z-z_0)^n , \end{align*}

where the coefficients $a_n$, for $n \in \mathbb {Z}$, are given by

\begin{align*} a_n = \frac{1}{2\pi \mathfrak {i}} \oint _{\partial \mathcal{B}(z_0; r)} \frac{f(z)}{(z - z_0)^{n+1}} \, \mathrm{d}z \qquad \text{ for any } r \in (r_1, r_2) . \end{align*}

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Theorem 5.25Taylor series of analytic functions [Palka1991, Thm VII.3.4]

Suppose that $f \colon U \to \mathbb {C}$ is an analytic function on an open set $U \subset \mathbb {C}$ which contains a disk $\mathcal{B}(z_0; r) \subset U$. Then the function $f$ can be represented in $\mathcal{B}(z_0; r)$ as a power series

\begin{align*} f(z) = \sum _{n=0}^\infty \frac{f^{(n)}(z_0)}{n!} \, (z-z_0)^n . \end{align*}

Moreover, this is the unique power series centered at $z_0$ that representats $f$ in a neighborhood of $z_0$.

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Theorem A.30Bolzano-Weierstrass theorem

A subset $B \subset \mathbb {R}$ of the real line is compact if an only if it is closed and bounded.

A subset $A \subset \mathbb {C}$ of the complex plane is compact if an only if it is closed and bounded.

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Theorem 4.18Cauchy’s integral formula [Palka1991, Thm V.2.3]

Suppose that a function $f \colon U \to \mathbb {C}$ is analytic on an open simply connected subset $U \subset \mathbb {C}$ of the complex plane. Then for any closed contour $\gamma $ in $U$ we have

\begin{align*} \oint _\gamma \frac{f(\zeta )}{\zeta - z} \, \mathrm{d}\zeta \; = \; 2\pi \mathfrak {i}\, \mathfrak {n}_{{\gamma }}(z) \, f(z) . \end{align*}

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Theorem 4.22Cauchy’s integral formula for derivatives

\begin{align*} \mathfrak {n}_{{\gamma }}(z) \, f^{(n)}(z) = \frac{n!}{2\pi \mathfrak {i}} \oint _\gamma \frac{f(\zeta )}{(\zeta - z)^{n+1}} \, \mathrm{d}\zeta . \end{align*}

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Theorem 4.10Cauchy’s integral formula for star-shaped subdomains [Palka1991, Thm V.2.3]

Suppose that a function $f \colon U \to \mathbb {C}$ is analytic on an open set $U \subset \mathbb {C}$, and suppose that $\gamma $ is a closed contour in $U$ parametrizing the boundary of a star-shaped Jordan subdomain $V \subset U$ in a counterclockwise orientation. Then for any point $z \in V$ we have

\begin{align*} f(z) = \frac{1}{2\pi \mathfrak {i}} \oint _\gamma \frac{f(\zeta )}{\zeta - z} \, \mathrm{d}\zeta . \end{align*}

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Theorem 2.13Cauchy-Riemann equations [Palka1991, Thm III.2.2]

Let $f \colon U \to \mathbb {C}$ be a function defined on an open set $U \subset \mathbb {C}$, and let $u \colon U \to \mathbb {R}$ and $v \colon U \to \mathbb {R}$ be its real and imaginary parts, viewed as real-valued functions of two real variables,

\begin{align*} u(x,y) = \Re \mathfrak {e}\Big( f (x + \mathfrak {i}y) \Big) \quad \text{and} \quad v(x,y) = \Im \mathfrak {m}\Big( f (x + \mathfrak {i}y) \Big) \end{align*}

so that $f = u + \mathfrak {i}\, v$.

Then the following are equivalent:

The functions $u$ and $v$ are differentiable at every point in $U$ and their partial derivatives satisfy the Cauchy-Riemann equations

\begin{align*} \frac{\partial {u}}{\partial {x}} = \frac{\partial {v}}{\partial {y}} \quad \text{and} \quad \frac{\partial {u}}{\partial {y}} = - \frac{\partial {v}}{\partial {x}} \end{align*}

in $U$.
The function $f$ is analytic.

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Theorem 4.13Cauchy’s integral theorem [Palka1991, Thm V.5.1]

Suppose that a function $f \colon U \to \mathbb {C}$ is analytic on a open set $U \subset \mathbb {C}$. Then for any contractible closed contour $\gamma $ we have

\begin{align*} \oint _\gamma f(z) \, \mathrm{d}z = 0 . \end{align*}

In particular, if $U$ is simply connected, then for any closed contour $\gamma $ in $U$ we have

\begin{align*} \oint _\gamma f(z) \, \mathrm{d}z = 0 , \end{align*}

and the analytic function $f$ has a primitive in $U$.

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Theorem 4.8Cauchy’s integral theorem for star-shaped domains [Palka1991, Thm V.1.5]

Suppose that a function $f \colon U \to \mathbb {C}$ is analytic on a star-shaped open subset $U \subset \mathbb {C}$. Then for any closed contour $\gamma $ in $U$ we have

\begin{align*} \oint _\gamma f(z) \, \mathrm{d}z = 0 . \end{align*}

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Theorem 1.2The field of complex numbers [Palka1991, Sec. I.1.1]

The set $\mathbb {C}$ of compex numbers with its operations of addition and multiplication, is a field, i.e., the following properties hold for all $z, w, z_1, z_2, z_3 \in \mathbb {C}$:

$z + w = w + z$ (commutativity of addition)
$z w = w z$ (commutativity of multiplication)
$z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3$ (associativity of addition)
$z_1 (z_2 z_3) = (z_1 z_2) z_3$ (associativity of multiplication)
$0 = 0 + 0 \, \mathfrak {i}= (0,0) \in \mathbb {C}$ satisfies $z + 0 = z$ (neutral element for addition)
$1 = 1 + 0 \, \mathfrak {i}= (1,0) \in \mathbb {C}$ satisfies $z \cdot 1= z$ (neutral element for multiplication)
$z + (-z) = 0$ for any $z \in \mathbb {C}$ (opposite element / additive inverse)
$z \, z^{-1} = 1$ for any $z \in \mathbb {C}\setminus \left\{ 0 \right\} $ (multiplicative inverse)
$(z_1 + z_2) w = z_1 w + z_2 w$ (distributivity).

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Theorem 2.20Criteria for constantness of a analytic function

Suppose that $f \colon D \to \mathbb {C}$ is a analytic function on a connected open subset $D \subset \mathbb {C}$ of the complex plane. If any of the functions $u = \Re \mathfrak {e}(f) \colon D \to \mathbb {R}$, $v = \Im \mathfrak {m}(f) \colon D \to \mathbb {R}$, $|f| \colon D \to \mathbb {R}$, is constant on $D$, then $f$ is itself a constant function.

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Theorem A.31Boundedness of continuous functions on compacts

Suppose that $X$ is compact. Then every continuous function $f \colon X \to \mathbb {R}$ is bounded.

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Theorem 5.31Discrete mapping theorem [Palka1991, Thm VIII.1.5]

Suppose that $f \colon \mathcal{D}\to \mathbb {C}$ is a non-constant analytic function on a connected open set $\mathcal{D}\subset \mathbb {C}$. Then the set of zeros of $f$ is discrete, i.e., for every $z_0 \in \mathcal{D}$ such that $f(z_0)=0$, there exists a $r{\gt}0$ such that $f(z) \ne 0$ for all $z \in \mathcal{B}(z_0; r) \setminus \left\{ z_0 \right\} $.

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Theorem 6.7Characterization of essential singularities [Palka1991, Thm VIII.2.6 and Thm VIII.2.7]

Let $z_0 \in \mathbb {C}$ be an isolated singularity of an analytic function $f : U \to \mathbb {C}$. Then the following conditions are equivalent:

(E-1): The singularity of $f$ at $z_0$ is essential (i.e., infinitely many Laurent series coefficients of $f$ near $z_0$ are nonzero).
(E-2): For any small $\delta {\gt} 0$, the image $f \big[ \mathcal{B}(z_0; \delta ) \setminus \left\{ z_0 \right\} \big]$ is dense in $\mathbb {C}$.
(E-3): The limit $\lim _{z \to z_0} f(z)$ does not exist in the extended complex plane $\hat{\mathbb {C}} = \mathbb {C}\cup \left\{ \infty \right\} $.

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Theorem 5.28Factor theorem for analytic functions [Palka1991, Thm VIII.1.2]

Suppose that $f \colon \mathcal{D}\to \mathbb {C}$ is a non-constant analytic function on a connected open set $\mathcal{D}\subset \mathbb {C}$, and $z_0 \in \mathcal{D}$ is a point where $f(z_0) = 0$. Then $f$ can be uniquely represented as

\begin{align*} f(z) = (z - z_0)^m \, g(z) \qquad \text{ for } z \in \mathcal{D}, \end{align*}

where $m \in \mathbb {N}$ and $g \colon \mathcal{D}\to \mathbb {C}$ is an analytic function such that $g(z_0) \ne 0$.

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Theorem 3.23Fundamental theorem of calculus for contour integrals [Palka1991, Thm IV.2.2]

Suppose that $f \colon U \to \mathbb {C}$ is a continuous function on an open set $U \subset \mathbb {C}$, and that $f$ has a primitive $F \colon U \to \mathbb {C}$. Then for any contour $\gamma \colon [a,b] \to U$ we have

\begin{align*} \int _\gamma f(z) \, dz \; = \; F \big( \gamma (b) \big) - F \big( \gamma (a) \big) . \end{align*}

In particular for any closed contour $\gamma $ in $U$, we have

\begin{align*} \oint _\gamma f(z) \, dz \; = \; 0 . \end{align*}

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Theorem 4.25Fundamental theorem of algebra [Palka1991, Thm V.3.8]

Every non-constant polynomial function $p \colon \mathbb {C}\to \mathbb {C}$ has a root, i.e., there exists a $z_0 \in \mathbb {C}$ such that $p(z_0) = 0$.

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Theorem 5.22Hadamard’s formula for the radius of convergence [Palka1991, Thm VII.3.3]

Let $z_0 \in \mathbb {C}$ be a point in the complex plane and let $a_0,a_1,a_2\ldots \in \mathbb {C}$ be coefficients. The radius of convergence of a power series

\begin{align*} \sum _{n=0}^\infty a_n \, (z-z_0)^n \end{align*}

is given by the formula

\begin{align*} R = \frac{1}{\limsup _{n \to \infty } \sqrt[n]{|a_n|}} , \end{align*}

with the conventions $\frac{1}{+\infty }=0$ and $\frac{1}{0} = +\infty $.

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Theorem A.34Cantor’s intersection theorem [Palka1991, Thm II.4.5]

Let $X$ be a metric space. Suppose that $K_1, K_2, K_3, \ldots $ are nonempty compact subsets of $X$ nested so that $K_1 \supset K_2 \supset K_3 \supset \cdots $. Then the intersection $\bigcap _{n=1}^\infty K_n$ is nonempty.

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Corollary 6.9Residue theorem for Jordan contours [Palka1991, Cor VIII.3.2]

Let $U \subset \mathbb {C}$ be an open set and $S \subset U$ a discrete subset of it. Let $\mathcal{D}$ be a Jordan domain such that $\overline{{\mathcal{D}}} \subset U$ and $\partial \mathcal{D}\cap S = \emptyset $. Let $\gamma $ be a closed contour traversing the boundary $\partial \mathcal{D}$ of the Jordan domain in the positive orientation. Let $f : U \setminus S \to \mathbb {C}$ be an analytic function with isolated singularities at the points of $S$. Then

\begin{align*} \oint _\gamma f(z) \, dz = 2 \pi \mathfrak {i}\sum _{w \in S \cap \mathcal{D}} \mathrm{Res}_{{w}}({f}) . \end{align*}

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Theorem 5.30L’Hospital’s rule for analytic functions [Palka1991, Thm VIII.1.4]

Let $f$ and $g$ be functions that are analytic in a neighborhood of $z_0$ such that $f(z_0) = 0$ and $g(z_0) = 0$. Then we have

\begin{align*} \lim _{z \to z_0} \frac{f(z)}{g(z)} = \lim _{z \to z_0} \frac{f'(z)}{g'(z)} , \end{align*}

understood in the sense that either both limits exist and are equal to each other, or else neither limit exists.

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Theorem 4.24Liouville’s theorem [Palka1991, Thm V.3.7]

If a function $f \colon \mathbb {C}\to \mathbb {C}$ on the entire complex plane is analytic and bounded, then $f$ is a constant function.

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Theorem 4.27Maximum principle for analytic functions [Palka1991, Thm V.3.10]

Let $f \colon \mathcal{D}\to \mathbb {C}$ be an analytic function on a connected open set $\mathcal{D}\subset \mathbb {C}$. Suppose that there exists a point $z_0 \in \mathcal{D}$ such that

\begin{align*} |f(z)| \le |f(z_0)| \qquad \text{ for all } z \in \mathcal{D}. \end{align*}

Then $f$ is a constant function.

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Theorem 4.30Mean value property for analytic functions

Suppose that a function $f \colon U \to \mathbb {C}$ is analytic on an open set $U \subset \mathbb {C}$ containing the closed disk $\overline{\mathcal{B}}(z; r) \subset U$. Then we have

\begin{align*} f(z) = \frac{1}{2 \pi r} \oint _{\partial \mathcal{B}(z; r)} \frac{f(\zeta )}{\zeta - z} \, |\mathrm{d}\zeta | . \end{align*}

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Theorem 4.21Morera’s theorem [Palka1991, Thm V.3.3]

Let $f \colon U \to \mathbb {C}$ be a continuous function on an open set $U \subset \mathbb {C}$. If $f$ has the property that

\begin{align*} \oint _{\partial \triangle } f(z) \, \mathrm{d}z = 0 \end{align*}

for any closed triangle $\triangle \subset U$, then $f$ is analytic on $U$.

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Theorem 6.6Characterization of poles [Palka1991, Thm VIII.2.3 and Thm VIII.2.4]

Let $z_0 \in \mathbb {C}$ be an isolated singularity of an analytic function $f : U \to \mathbb {C}$. Then the following conditions are equivalent:

(P-1): The singularity of $f$ at $z_0$ is a pole (i.e., finitely many Laurent series coefficients of $f$ near $z_0$ are nonzero).
(P-2): There exists an $m \in \mathbb {N}= \left\{ 1,2,\ldots \right\} $ such that $z \mapsto (z-z_0)^m \, f(z)$ has a removable singularity and a nonzero limit as $z \to z_0$.
(P-3): The function $f$ has the limit $\lim _{z \to z_0} f(z) = \infty $ at $z_0$.

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Theorem 3.25Characterization of the existence of primitives

Let $f \colon U \to \mathbb {C}$ be a continuous function on an open set $U \subset \mathbb {C}$. Then the following conditions are equivalent:

$f$ has a primitive on $U$;
the contour integrals $\int _\gamma f(z) \, \mathrm{d}z$ of $f$ along contours $\gamma $ in $U$ only depend on the starting point and the end point of $\gamma $;
for all closed contours $\gamma $ in $U$ we have $\oint _\gamma f(z) \, \mathrm{d}z = 0$.

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Theorem 6.5Removable singularity characterization [Palka1991, Thm VIII.2.1 and Thm VIII.2.2]

Let $z_0 \in \mathbb {C}$ be an isolated singularity of an analytic function $f : U \to \mathbb {C}$. Then the following conditions are equivalent:

(R-1): The singularity of $f$ at $z_0$ is removable (i.e., all negative index Laurent series coefficients of $f$ expanded near $z_0$ vanish).
(R-2): There exists an analytic function $\tilde{f} \colon U \cup \left\{ z_0 \right\} \to \mathbb {C}$ such that $f(z) = \tilde{f}(z)$ for all $z \in U$.
(R-3): The limit $\lim _{z \to z_0} f(z)$ exists in $\mathbb {C}$.
(R-4): The function $f$ is bounded in some punctured disk $\mathcal{B}(z_0; r) \setminus \left\{ z_0 \right\} $ with $r{\gt}0$.

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Theorem 6.8Residue theorem [Palka1991, Thm VIII.3.1]

Let $U \subset \mathbb {C}$ be an open set and $\gamma $ a contractible closed contour in $U$. Let $f : U \setminus S \to \mathbb {C}$ be an analytic function with isolated singularities at a countable set $S \subset U$ of points. Then

\begin{align*} \oint _\gamma f(z) \, dz = 2 \pi \mathfrak {i}\sum _{w \in S} \mathfrak {n}_{{w}}(\gamma ) \, \mathrm{Res}_{{w}}({f}) . \end{align*}

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Theorem 5.26Equivalent characterizations of analyticity

Let $f \colon U \to \mathbb {C}$ be a continuous function on an open set $U \subset \mathbb {C}$. Then the following are equivalent:

$f$ is analytic on $U$;
for any $z \in U$ there exists a neighborhood of $z$ in which $f$ has a primitive;
for any $z \in U$ there exists a neighborhood of $z$ in which $f$ can be represented as a convergent power series.

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