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Complex Analysis
Complex Analysis
Introduction
Birth
- Cubic equations $ x^3 = px + q $ always must be a solution
- Del Ferro (1465-1526) and Tartaglia (1499-1577), followed by Cardano (1501-1576), showed that the solution given by
- About 30 years after the discovery of this formula, Bombelli (1526-1572) consider $ \displaystyle\frac{q^2}{4}-\frac{p^3}{27} < 0 $
- It showed that perfectly real problems require complex arithmetic for their solution.
Properties and Definitions
- Complex numbers: expressions of the form $ z = x + iy $, where
- x is called the real part of $z; x = \mathrm{Re} z $
- y is called the imaginary part of $ z; y = \mathrm{Im} z $
- Set of complex numbers: $ \mathbb{C} $ (the complex plane)
- Real numbers: subset of the complex numbers (those whose imaginary part is zero)
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The complex plane can be identified with $ \mathbb{R}^{2} $
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- $ \mathrm{Re}(z + w) = \mathrm{Re}\ z + \mathrm{Re}\ w \ \mathrm{and} \ \mathrm{Im}(z + w) = \mathrm{Im}\ z + \mathrm{Im}\ w $
- Def: The modulus of the complex number z = x + iy is the length of the vector z:
- $ \vert z\vert = \sqrt{x^2 + y^2} $
- $ (x + iy) \cdot (u + iv) = (xu - yv) + i(xv + yu) \in \mathbb{C} $
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$ \displaystyle \frac{x + iy}{u + iv} = \frac{xu + yv}{u^2 + v^2} + i\frac{yu - xv}{u^2 + v^2} \ (\mathrm{for} \ u + iv \neq 0) $
- If $ z = x + iy $ then $ \overline{z} = x − iy $ is the complex conjugate of z
- $ \overline{\overline{z}} = z $
- $ \overline{z + w} = \overline{z} + \overline{w} $
- $ \vert z\vert = \vert \overline{z}\vert $
- $ z\overline{z} = (x + iy)(x - iy) = x^2 + y^2 = \vert z\vert ^2 $
- $ \displaystyle \frac{1}{z} = \frac{\overline{z}}{z\overline{z}} = \frac{\overline{z}}{\vert z\vert ^2} $
- $ \vert z \cdot w \vert = \vert z\vert \cdot \vert w\vert $
- $ \displaystyle \overline{(\frac{z}{w})} = \frac{\overline{z}}{\overline{w}} $
- $ \vert z\vert = 0 \ \mathrm{iff} \ z = 0 $
- $ −\vert z\vert ≤ \mathrm{Re}\ z ≤ \vert z\vert $
- $ −\vert z\vert ≤ \mathrm{Im}\ z ≤ \vert z\vert $
- $ \vert z + w\vert ≤ \vert z\vert + \vert w\vert \ \mathrm{(triangle \ inequality)} $
- $ \vert z − w\vert ≥ \vert z\vert − \vert w\vert \ \mathrm{(reverse\ triangle\ inequality)} $
- The Fundamental Theorem of Algebra
If $ a_{0}, a_{1}, \dots , a_{n} $ are complex numbers with $ a_{n} \neq 0 $ , then the polynomial
$ \displaystyle \displaystyle p(z) = a_{n}z^{n} + a_{n-1}z^{n-1} + \dots + a_{1}z + a_{0} $
has n roots $ z_{1}, z_{2}, \dots , z_{n} \ \mathrm{in} \ \mathbb{C} $
It can be factored as
$ p(z) = a_{n}(z − z_{1})(z − z_{2})\dots(z − z_{n}) $
Polar Representation
- $ r = \vert z\vert $: the distance from the origin
- $ θ $: the angle between the positive x-axis and the line segment from 0 to z
- $ (r, θ) $ are the polar coordinates of z
- Relation between Cartesian and polar coordinates:
- $ x = r cos θ $
- $ y = r sin θ $
- $ z = r(cos θ + i sin θ) $
- Def: The principal argument of z, called Arg z, is the value of θ for which $ −π < θ ≤ π $.
- $ arg z = \lbrace Arg z + 2πk : k = 0, ±1, ±2, . . .\rbrace,\ z \neq 0 $
- Convenient notation: $ e^{i\theta} = cos θ + i sin θ $
- the polar form of z: $ z = r e^{i\theta} $
- $ \vert e^{i\theta}\vert = 1 $
- $ \overline{e^{i\theta}\ =\ e^{-i\theta}} $
- $ \displaystyle \frac{1}{e^{i\theta}}\ =\ e^{-i\theta} $
- $ e^{i(\theta\ + \ \varphi)}\ = e^{i\theta}\ \cdot\ e^{i\varphi} $
- $ arg(\overline{z})\ = \ -arg\ z $
- $ arg(\frac{1}{z})\ = \ - arg\ z $
- $ arg(z_1 z_2)\ = \ arg(z_1) \ + \ arg(z_2) $
- $ (e^{i\theta})^n\ = \ e^{i\cdot n\theta} $
- $ (cos θ + i sin θ)^n \ = \ cos(nθ)\ + \ i sin(nθ) $
Roots of Complex Numbers
- Def: Let $ w $ be a complex number. An nth root of $ w $ is a complex number $ z $ such that $ z^n = w $
- Use the polar form for $ w $ and $ \ z:\ w\ = \ \rho e^{i\varphi}\ \mathrm{and}\ z\ = \ r e^{i\theta} $
- $ z^{n}\ = \ w \ : \ r^{n}e^{in\theta}\ = \ \rho e^{i\varphi}, \mathrm{so} r^n\ = \ \rho \ \mathrm{and} \ e^{in\theta} \ = \ e^{i\varphi} $
- $ r = \sqrt[n]{\rho} \ \mathrm{and} \ n\theta \ = \ \varphi \ + \ 2k\pi, \ k \in \mathbb{Z} $
- $ \displaystyle w^{\frac{1}{n}}\ = \ \sqrt[n]{\rho}\ e^{i(\frac{\varphi}{n} \ + \ \frac{2k\pi}{n})},\ k\ = \ 0, 1, \dots , \ n - 1 $
- Def: The n_th roots of 1 are called the _nth roots of unity
- Since $ 1\ =\ 1\ e^{i\dot 0} $, we find that
- Since $ 1\ =\ 1\ e^{i\dot 0} $, we find that
Topology in the Plane
- Sets in the Complex Plane
- Circles and disks: center $ z_0\ =\ x_0\ +\ i y_0 $, radius $ r $.
- $ B_r(z_0) = {z \in \mathbb{C} : z $ has distance less than $ r $ from $ z_0} $ disk of radius $ r $ , centered at $ z_0 $.
- $ K_r(z_0) = {z \in \mathbb{C} : z $ has distance $ r $ from $ z_0} $ circle of radius $ r $ , centered at $ z_0 $
- Measure distance
- so $ B_r(z_0)\ =\ {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0\vert \ <\ r}\ \mathrm{and}\ K_r(z_0)\ =\ {z ∈ \mathbb{C}\ :\ \vert z\ −\ z_0\vert\ =\ r} $
- Circles and disks: center $ z_0\ =\ x_0\ +\ i y_0 $, radius $ r $.
- Interior Points and Boundary Points
- Def: Let $ E ⊂ \mathbb{C} $. A point $ z_0 $ is an interior point of $ E $ if there is some $ r > 0 $ such that $ B_r(z_0) ⊂ E $.
- Def: Let $ E ⊂ \mathbb{C} $. A point $ b $ is a boundary point of $ E $ if every disk around $ b $ contains a point in $ E $ and a point not in $ E $.
The boundary of the set $ E ⊂ \mathbb{C}, ∂E, $ is the set of all boundary points of $ E $.
- Open and Closed Sets
- A set $ U ⊂ \mathbb{C} $ is open if every one of its points is an interior point.
- A set $ A ⊂ \mathbb{C} $ is closed if it contains all of its boundary points
- Examples:
- $ {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert \ <\ r}\ \mathrm{and} {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert\ >\ r} $ are open
- $ \mathbb{C} $ and $ ∅ $ are open
- $ {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert \ ≤\ r}\ \mathrm{and} {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert\ =\ r} $ are closed
- $ C $ and $ ∅ $ are closed
- $ {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert \ <\ r}\ ∪ {z\ ∈\ \mathbb{C}\ :\ \vert z\ −\ z_0 \vert\ =\ r\ \mathrm{and Im} (z\ −\ z_0)\ >\ 0} $ is neither open nor closed
- Closure and Interior of a Set
- Def:: Let $ E $ be a set in $ \mathbb{C} $.
The closure of $ E $ is the set $ E $ together with all of its boundary points: $ \overline{E}\ =\ E\ ∪\ ∂E $.
The interior of $ E $ , $ \circ{E} $ is the set of all interior points of $ E $. - Examples:
- $ \overline{B_r(z_0)}\ = \ B_r(z_0)\ \cup \ K_r(z_0)\ = \ {z\ \in\ \mathbb{C}\ :\ \vert z\ -\ z_0 \vert \ \leq \ r} $
- $ \overline{K_r(z_0)}\ = \ K_r(z_0) $
- $ \overline{B_r(z_0)\ \setminus \ {z_0}}\ = \ {z\ \in\ \mathbb{C}\ :\ \vert z\ -\ z_0 \vert \ \leq \ r} $
- With $ E\ =\ {z\ \in\ \mathbb{C}\ :\ \vert z\ -\ z_0 \vert \ \leq \ r}, \circ{E}\ = \ B_r(z_0) $
- With $ E\ =\ K_r(z_0),\ \circ{E}\ = \ \emptyset $
- Def:: Let $ E $ be a set in $ \mathbb{C} $.
- Connectedness
- Def: Two sets $ X, Y $ in $ \mathbb{C} $ are separated if there are disjoint open set $ U, V $ so that $ X\ ⊂\ U $ and $ Y\ ⊂\ V $ . A set $ W $ in $ \mathbb{C} $ is connected if it is impossible to find two separated non-empty sets whose union equals $ W $.
- Examples:
- $ X\ =\ [0, 1) $ and $ Y\ =\ (1, 2] $ are separated
- Choose $ U\ =\ B_1(0) $ and $ V\ =\ B_1(2) $ . Thus $ X\ ∪\ Y\ =\ [0, 2]\ \setminus \ {1} $ is not connected
- Theorem: Let $ G $ be an open set in $ \mathbb{C} $ . Then $ G $ is connected if and only if any two points in $ G $ can be joined in $ G $ by successive line segments.
- Def: A set $ A $ in $ \mathbb{C} $ is bounded if there exists a number $ R\ >\ 0 $ such that $ A\ ⊂\ B_R(0) $ . If no such $ R $ exists then $ A $ is called unbounded.
- The Point at Infinity
- In $ \mathbb{R} $ , there are two directions that give rise to $ ±∞ $
- In $ \mathbb{C} $ , there is only one $ ∞ $ which can be attained in many directions.
Function
Complex Functions
- A function $ f:\ A\ \to\ B $ is a rule that assigns to each element $ A $ of exactly one element of $ B $
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$ f^n(z) $ (read: “Eff n”) is called the nth iterate of f
- Sequences and Limits of Complex Numbers
- Def: A sequence {sn} of complex numbers converges to $ s ∈ \mathbb{C} $ if for every $ ε > 0 $ there exists an index $ N ≥ 1 $ such that
We write
- Def: A sequence {sn} of complex numbers converges to $ s ∈ \mathbb{C} $ if for every $ ε > 0 $ there exists an index $ N ≥ 1 $ such that
- Rules for Limits
- Convergent sequences are bounded.
- If $ {s_n} $ converges to $ s $ and $ {t_n} $ converges to $ t $ , then
- $ s_n\ +\ t_n\ \to \ s\ +\ t $
- $ s_n\ \dot \ t_n\ \to \ s\ \dot \ t $ (in particular: $ a\ \dot \ s_n \ \to \ a\ \dot \ s $ for any $ a \in \mathbb{C} $ )
- $ \displaystyle \frac{s_n}{t_n}\ \to \ \frac{s}{t} $ , provided $ t\ \neq \ 0 $
- Facts
- A sequence of complex numbers, $ {s_n} $, converges to $ 0 $ iff the sequence $ {\vert s_n \vert} $ of absolute values converges to $ 0 $
- A sequence of complex numbers, $ {s_n} $, with $ s_n\ =\ x_n\ +\ i y_n $ , converges to $ s\ =\ x\ +\ i y $ iff $ x_n\ \to \ x $ and $ y_n\ \to \ y $ as $ n\ \to \ \infty $
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Squeeze Theorem: Suppose that $ {r_n} $ , $ {s_n} $ and $ {t_n} $ are sequences of real numbers such that $ r_n\ ≤\ s_n\ ≤\ t_n $ for all $ n $ . If both sequences $ {r_n} $ and $ {t_n} $ converge to the same limit, $ L $ , then the sequence $ {s_n} $ converges to the limit $ L $ as well.
-
Theorem: A bounded, monotone sequence of real numbers converges
- Limits of Complex Functions
- Def: The complex-valued function $ f(z) $ has limit $ L $ as $ z\ \to \ z_0 $ if the values of $ f(z) $ are near $ L $ as $ z\ \to \ z_0 $ .
- Also: $ \displaystyle \lim_{z \to z_0}\ f(z)\ = \ L $ if for all $ ε\ >\ 0 $ there exists $ δ\ >\ 0 $ such that $ \vert f(z)\ −\ L\vert \ <\ ε $ whenever $ 0\ <\ \vert z\ −\ z_0\vert \ <\ δ $
- Facts
- If $ f $ has a limit at $ z_0 $ then $ f $ is bounded near $ z_0 $ .
- If $ f(z)\ \to \ L $ and $ g(z)\ \to \ M $ as $ z\ \to \ z_0 $ then
- $ f(z)\ + \ g(z) \ \to \ L \ + \ M $ as $ z\ \to \ z_0 $
- $ f(z)\ \dot \ g(z) \ \to \ L \ \dot \ M $ as $ z\ \to \ z_0 $
- $ \displaystyle \frac{f(z)}{g(z)} \ \to \ \frac{L}{M} $ as $ z\ \to \ z_0 $, provided that $ M\ \neq \ 0 $
- Continuity
- Def: The function $ f $ is continuous at $ z_0 $ if $ f(z)\ \to \ f(z_0) $ as $ z\ \to \ z_0 $ .
- This definition implicitly says that:
- $ f $ is defined at $ z_0 $
- $ f $ has a limit as $ z\ \to \ z_0 $
- The limit equals $ f(z_0) $
