The definitions and results collected in this document serve as a refresher of some concepts from Analysis I that we will explore in the complex setting throughout the Complex Analysis II module. The definitions and results collected here are taken from the Analysis I lecture notes from 2021–2022 by Prof. D. Schütz and Prof. J. Funke.
Definition 1 (Real sequence). A real sequence is a function from \(\mathbb{N}\) to \(\mathbb{R}\). That is, it assigns to every natural number \(n \in \mathbb{N}\) a real number \(x_n \in \mathbb{R}\). Such a sequence is denoted \(\{x_n\}\).
Definition 2 (Convergence of sequence). A real sequence \(\{x_n\}\) is said to be convergent to the limit \(x \in \mathbb{R}\) if \[\lim_{n \to \infty} | x_n - x | = 0.\] That is, for every \(\epsilon > 0\), there exists and index \(N \in \mathbb{N}\) such that \[| x_n - x | < \epsilon \quad \text{for all } n > N.\] We write \(\lim_{n \to \infty} x_n = x\), or we say “\(x_n \to x\) as \(n \to \infty\).” A sequence that has a limit is called a convergent sequence. If a sequence is not convergent, then it is called divergent.
Definition 3 (Bounded sequence). Let \(\{x_n\}\) be a real sequence and denote the set \(X = \{ x_n \in \mathbb{R} : n \in \mathbb{N}\}\). The sequence \(\{x_n\}\) is called bounded above, respectively below, if \(X\) is bounded above, respectively below. The sequence \(\{x_n\}\) is called bounded if \(X\) is bounded.
Theorem 4 (COLT). Let \(\{x_n\}\) and \(\{y_n\}\) be real sequences that are convergent with limits \(x = \lim_{n \to \infty} x_n\) and \(y = \lim_{n \to \infty} y_n\). Let \(a, b \in \mathbb{R}\). Then we have
\(ax_n + by_n \to ax + by\) as \(n \to \infty\).
\(x_n \cdot y_n \to x \cdot y\) as \(n \to \infty\).
\(\frac{x_n}{y_n} \to \frac{x}{y}\) as \(n \to \infty\).
Definition 5 (Subsequence). Let \(\{x_n\}\) be a sequence. A subsequence of \(\{x_n\}\) is a sequence \(\{x_{n_j}\}\) with \(n_1 < n_2 < n_3 < \cdots\).
Theorem 6 (Bolzano–Weierstrass). Let \(\{x_n\}\) be a bounded real sequence. Then \(\{x_n\}\) has a subsequence which is convergent.
Definition 7 (Lim sup and Lim inf). Let \(\{x_n\}\) be a bounded sequence. The limit superior of \(\{x_n\}\) is defined as \[\limsup_{n \to \infty} x_n = \inf_{n \geq 1} \left\{ \sup_{m \geq n} \{x_m\}\right\},\] and the limit inferior of \(\{x_n\}\) is defined as \[\liminf_{n \to \infty} x_n = \sup_{n \geq 1} \left\{ \inf_{m \geq n} \{x_m\}\right\}.\]
Definition 8 (Series, convergence of series). Let \(\{a_n\}\) be a real sequence. Then the sequence of partial sums \(\{s_k\}\), defined as \[s_k = \sum_{n=0}^k a_n = a_1 + a_2 + \cdots + a_k,\] is called an (infinite) series. If the sequence of partial sums \(\{s_k\}\) is convergent, then we say that the series \(\sum_{n=0}^\infty a_n\) is convergent and we write \[\sum_{n=0}^\infty a_n = \lim_{k \to \infty} s_k.\] Otherwise, we say that the series \(\sum_{n=0}^\infty a_n\) is divergent.
Theorem 9 (COLT for series). Assume the series \(\sum_{n=0}^\infty a_n\) and \(\sum_{n=0}^\infty b_n\) both converge with limits \(a\) and \(b\) respectively. Let \(c \in \mathbb{R}\). Then
\(\sum_{n=0}^\infty (a_n + b_n)\) is convergent with limit \(a+b\).
\(\sum_{n=0}^\infty ca_n\) is convergent with limit \(ca\).
Theorem 10 (Comparison Test). Let \(N \in \mathbb{N}\), \(\{a_n\}_{n\geq N}\) and \(\{b_n\}_{n \geq N}\) be sequences with \(0 \leq a_n \leq b_n\) for all \(n \geq N\).
If \(\sum_{n=0}^\infty b_n\) is convergent with limit \(b\), then \(\sum_{n=0}^\infty a_n\) is also convergent with limit \(a \leq b\).
If \(\sum_{n=0}^\infty a_n\) is divergent, then so is \(\sum_{n=0}^\infty b_n\).
Definition 11 (Absolute convergence). We say that the series \(\sum_{n=0}^\infty a_n\) is absolutely convergent if the series \(\sum_{n=0}^\infty |a_n|\) is convergent.
Theorem 12 (Ratio Test). Let \(\{a_n\}\) be a sequence with \(a_n \neq 0\) for all but possibly finitely many \(n\).
If \(\lim_{n \to \infty} \frac{|a_{n+1}|}{|a_n|} < 1\), then \(\sum_{n=0}^\infty a_n\) converges absolutely.
If \(\lim_{n \to \infty} \frac{|a_{n+1}|}{|a_n|} > 1\), then \(\sum_{n=0}^\infty a_n\) is divergent.
Theorem 13 (Root Test). For a sequence \(\{a_n\}\) set \[a = \limsup |a_n|^{1/n}.\]
If \(a<1\), then \(\sum_{n=0}^\infty a_n\) converges absolutely.
If \(a>1\), then \(\sum_{n=0}^\infty a_n\) is divergent.
Proposition 14 (Properties of image and preimage). Let \(f: X \to Y\) be a function, and assume that \(A, B \subset X\). Then
\(f(A \cap B) \subset f(A) \cap f(B)\).
\(f(A \cup B) = f(A) \cup f(B)\).
\(f(X \setminus A) \supset f(X) \setminus f(A)\).
Assume that \(C, D \subset Y\). Then
\(f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)\).
\(f^{-1}(C \cup D) = f^{-1}(C) \cup f^{-1}(D)\).
\(f^{-1}(Y \setminus C) = X \setminus f^{-1}(C)\).
Definition 15 (Exponential function and Logarithm). The exponential function \(\exp : \mathbb{R} \to (0,\infty)\) is defined as \[\exp(x) := e^x.\] The logarithm function \(\log : (0,\infty) \to \mathbb{R}\) is defined for \(x > 0\) by \[\log(x) = y,\] where \(y\) is the unique real number such that \(\exp{y} = x\).
Let \(a,b \in \mathbb{R}\), \(a \leq b\). We call \((a,b) := \{t \in \mathbb{R}: a<t<b\}\) an open interval inside \(\mathbb{R}\) (where we also allow \(a=-\infty\) or \(b=\infty\)), and \([a,b] := \{t \in \mathbb{R}: a \leq t \leq b\}\) a closed interval or compact interval inside \(\mathbb{R}\) (where we do not allow \(a,b = \pm \infty\)).
We call a subset \(X \subseteq \mathbb{R}\) open if for each \(c \in X\) there exists \(\epsilon > 0\) such that the open interval \((c-\epsilon, c+\epsilon) \subseteq X\).
We call \(c \in X\) and interior point if there exists an open subset \(U\) or an open interval \((a,b)\) containing \(c\) which lies completely in \(U\).
Throughout we let \(f : X \to \mathbb{R}\) be a function on a subset \(X \subseteq \mathbb{R}\).
Definition 16 (Continuous function). Let \(f :X \to \mathbb{R}\) be a function and let \(c\) be an interior point of \(X\). Then \(f\) is called continuous at \(c \in X\) if \[\lim_{x \to c} f(x) = f(c).\] That is, for all \(\epsilon > 0\), there exists a \(\delta > 0\) such that \[|f(x) - f(c)|<\epsilon \quad \text{for all } x \in X \text{ with } |x-c| < \delta.\] The function \(f:X \to \mathbb{R}\) is called continuous if it is continuous for all \(c \in X\).
Proposition 17 (Continuity via sequences). Let \(X \subset \mathbb{R}\), \(c \in X\), and \(f:X \to \mathbb{R}\) be a function. Then \(f\) is continuous at \(c\) if and only if for all sequences \(\{x_n\}\) in \(X\) with \(x_n \to c\) as \(n \to \infty\) we have \(f(x_n) \to f(c)\) as \(n \to \infty\). That is, \[\lim_{n \to \infty} f(x_n) = f\big(\lim_{n\to \infty} x_n\big).\]
Theorem 18 (COLT for continuous functions). Let \(X \subset \mathbb{R}\), \(c \in X\) and \(f, g : X \to \mathbb{R}\) be continuous functions at \(c\). Then
\(a\cdot f(x) + b\cdot g(x)\) is continuous at \(c\) for any \(a, b \in \mathbb{R}\).
\(f(x)\cdot g(x)\) is continuous at \(c\).
\(f(x)/g(x)\) is continuous at \(c\) provided that \(g(c) \neq 0\).
Theorem 19 (Composition of continuous functions is continuous). Let \(X, Y \subset \mathbb{R}\), \(c \in X\), \(f:X \to \mathbb{R}\), \(g:Y \to \mathbb{R}\) with \(f(X) \subset Y\). If \(f\) is continuous at \(c \in X\) and \(g\) is continuous at \(f(c) \in Y\), then \(g \circ f\) is continuous at \(c \in X\).
Theorem 20 (Extreme Value Theorem). Let \(a, b \in \mathbb{R}\) with \(a<b\) and \(f:[a,b] \to \mathbb{R}\) be continuous. Then \(f\) attains its maximum and minimum values on \([a,b]\). That is, every continuous function on a compact interval attains its maximum and minimum.
Definition 21 (Differentiable function). Let \(X \subseteq \mathbb{R}\) be open and \(f:X \to \mathbb{R}\) be a function. We say that \(f\) is differentiable at \(c \in X\) if \[\lim_{x \to c} \frac{f(x) - f(c)}{x-c}\] exists. We denote this limit \(f'(c)\) and call \(f'(c)\) the derivative of \(f\) at \(c\). We call \(f\) a differentiable function if \(f\) is differentiable at all points \(c \in X\). Another formulation of this is \[f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}.\]
Theorem 22 (Sums, products, compositions, quotients of differentiable functions).
Let \(f, g :X \to \mathbb{R}\) be two functions which are differentiable at \(c \in X\) and let \(a \in \mathbb{R}\) be a constant. Then \(f+g\) and \(af\)are also differentiable at \(c\) and \[\begin{aligned} (f+g)'(c) & = f'(c) + g'(c). \\ (af)'(c) & = af'(c). \end{aligned}\] Their product \(fg\) is also differentiable at \(c\) with \[(fg)'(c) = f(c)g'(c) + f'(c)g(c).\]
Let \(f, g :X \to \mathbb{R}\) be two functions such that \(g\) is differentiable at \(c\) and \(f\) is differentiable at \(g(c)\). Then the composition \(f \circ g\) is also differentiable at \(c\) and \[(f \circ g)'(c) = f'(g(c))g'(c).\]
Let \(f: X \to \mathbb{R}\) be a function which is differentiable at \(c\) and such that \(f(c)\neq 0\). Then \(1/f\) is also differentiable at \(c\) and \[\left(\frac{1}{f}\right)'(c) = -\frac{f'(c)}{f^2(c)}.\]
Let \(c \in \mathbb{R}\).
Definition 23 (Power series). A real power series is an infinite series of the form \(\sum_{n=0}^\infty a_n (x-c)^n\) with real \(a_n\) and \(x \in \mathbb{R}\).
Theorem 24 (Cauchy–Hadamard). Let \(\sum_{n=0}^\infty a_n(x-c)^n\) be a power series. Then there exists a constant \(R \in [0,\infty]\) such that
If \(R=0\), then \(\sum_{n=0}^\infty a_n(x-c)^n\) converges only for \(x=c\).
If \(R>0\), then \[\begin{aligned} & \sum_{n=0}^\infty a_n(x-c)^n \text{ converges absolutely for } x \in (c-R,c+R); \\ & \sum_{n=0}^\infty a_n(x-c)^n \text{ diverges for } |x-c|>R. \end{aligned}\]
If we set \[\limsup_n |a_n|^{1/n} = k \in [0,\infty],\] then \(R\) is explicitly given by \[R = \frac{1}{k} \in [0,\infty].\] We call \(R\) the radius of convergence of the power series.
Lemma 25 (Term by term differentiation or integration preserves the radius of convergence). Let \(\sum_{n=0}^\infty a_n(x-c)^n\) be a power series with radius of convergence \(R\). Then the \[\begin{aligned} \text{formal derivative } \sum_{n=0}^\infty n a_n (x-c)^{n-1} & = \frac{1}{x-c} \sum_{n=0}^\infty n a_n (x-c)^n; \\ \text{formal antiderivative } \sum_{n=0}^\infty \frac{a_n}{n+1} (x-c)^{n+1} & = (x-c) \sum_{n=0}^\infty \frac{a_n}{n+1} (x-c)^n \end{aligned}\] also have radius of convergence \(R\).
Theorem 26 (Power series can be differentiated term-by-term in their disc of convergence). Let \(f(x) = \sum_{n=0}^\infty a_n(x-c)^n\) be a power series and \(R \in (0, \infty]\) be its radius of convergence. Then \(f\) is differentiable infinitely many times at all points \(x \in (c-R,c+R)\), and we can differentiate term-by-term: \[f'(x) = \sum_{n=1}^\infty n a_n (x-c)^{n-1}.\]
Definition 27 (Pointwise convergence). Let \(\{f_n\}\) be a sequence of functions on an interval \(I\). We say that \(\{f_n\}\) has a pointwise limit if for all \(x \in I\), the limit \(\lim_{n \to \infty} f_n(x)\) exists (as a sequence of real numbers). In that case, the limit function \(f:I \to \mathbb{R}\) is defined as \[f(x) = \lim_{n \to \infty} f_n(x).\] In other words, we have \[\forall x \in I \quad \forall \epsilon >0 \quad \exists N \in \mathbb{N} \quad \forall n > N : \quad |f_n(x) - f(x)| < \epsilon.\]
Definition 28 (Uniform convergence). Let \(\{f_n\}\) be a sequence of functions on an interval \(I\). We say that \(\{f_n\}\) converges uniformly to \(f\) if for every \(\epsilon > 0\), there exists \(N \in \mathbb{N}\) such that for all \(n \geq N\) and all \(x \in I\), we have \[|f_n(x) - f(x)| < \epsilon.\] If \(f_n\) converges uniformly to \(f\), we write “\(f_n \to f\) uniformly”. In other words, we have \[\forall \epsilon >0 \quad \exists N \in \mathbb{N} \quad \forall n > N \quad \forall x \in I: \quad |f_n(x) - f(x)| < \epsilon.\] Here, \(N\) does not depend on the individual point \(x\), the same \(N\) works for all \(x \in I\).
Theorem 29 (Uniform limits of continuous functions are continuous). Let \(f_n\) be a sequence of continuous functions on an interval \(I\) such that \(f_n \to f\) uniformly. Then the limit function \(f\) is also continuous.
Theorem 30 (Weierstrass \(M\)-test). Let \(I \subset \mathbb{R}\) be an interval and \(f_n: I\to\mathbb{R}\) be a sequence of functions such that \[|f_{n}(x)|\leq M_{n} \text{ for all }x\in I \quad \text{ and }\quad\sum_{n=1}^{\infty}M_{n}<\infty.\] Then \[\sum_{n=1}^{\infty}f_{n}(x)\text{ converges uniformly on }X \text{ to some limit function }f:X\to\mathbb{C}.\]
Theorem 31 (Fundamental Theorem of Calculus). Let \(f\) be a continuous function on \([a,b]\). Then \[F(x) := \int_a^x f(t) \, dt\] is a differentiable function on \([a,b]\) (one-sided at \(a\) and \(b\)), and we have \(F'(x) = f(x)\) for all \(x \in [a,b]\).
Theorem 32 (Integral of uniform limit of continuous functions is limit of integrals). Let \(I=[a,b]\) and \(\{f_n\}\) be a sequence of continuous functions on \(I\) such that \(f_n \to f\) uniformly. Then \[\lim_{n\to\infty}\int_{a}^{c}f_{n}(x)\,dx=\int_{a}^{c}f(x)\,dx,\quad\quad\text{for all }c\in[a,b].\]
The following facts are recalled from the lecture notes from Analysis I, 2021–2022.
Addition of complex numbers is associative and commutative. Multiplication of complex numbers is associative and commutative.
For \(z = x + iy\), we call \(\overline{z} := x - iy\) the complex conjugate of \(z\).
For \(z = x + iy\), we call \(|z| := \sqrt{z \overline{z}} = \sqrt{x^2+y^2}\) the modulus or absolute value of \(z\).
\(| z | = 0\) if and only if \(z=0\).
\(| z \cdot w | = |z| \cdot |w|\).
(Triangle Inequality) For \(z_1, z_2 \in \mathbb{C}\), \(|z_1 + z_2 | \leq |z_1| + |z_2|\).
This document was written by Katie Gittins and Stephen Herrap in the 2022-23 Academic Year↩︎