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A First Course in
Complex Analysis
Version 1.
Matthias Beck Gerald Marchesi Department of Mathematics Department of Mathematical Sciences San Francisco State University Binghamton University (SUNY) San Francisco, CA 94132 Binghamton, NY 13902- [email protected] [email protected]
Dennis Pixton Lucas Sabalka Department of Mathematical Sciences Department of Mathematics & Computer Science Binghamton University (SUNY) Saint Louis University Binghamton, NY 13902-6000 St Louis, MO 63112 [email protected] [email protected]
Copyright 2002–2012 by the authors. All rights reserved. The most current version of this book is available at the websites
http://www.math.binghamton.edu/dennis/complex.pdf http://math.sfsu.edu/beck/complex.html.
This book may be freely reproduced and distributed, provided that it is reproduced in its entirety from the most recent version. This book may not be altered in any way, except for changes in format required for printing or other distribution, without the permission of the authors.
These are the lecture notes of a one-semester undergraduate course which we have taught several times at Binghamton University (SUNY) and San Francisco State University. For many of our students, complex analysis is their first rigorous analysis (if not mathematics) class they take, and these notes reflect this very much. We tried to rely on as few concepts from real analysis as possible. In particular, series and sequences are treated “from scratch." This also has the (maybe disadvantageous) consequence that power series are introduced very late in the course. We thank our students who made many suggestions for and found errors in the text. Spe- cial thanks go to Joshua Palmatier, Collin Bleak, Sharma Pallekonda, and Dmytro Savchuk at Binghamton University (SUNY) for comments after teaching from this book.
- 1 Complex Numbers
- 1.0 Introduction
- 1.1 Definitions and Algebraic Properties
- 1.2 From Algebra to Geometry and Back
- 1.3 Geometric Properties
- 1.4 Elementary Topology of the Plane
- 1.5 Theorems from Calculus
- Exercises
- Optional Lab
- 2 Differentiation
- 2.1 First Steps
- 2.2 Differentiability and Holomorphicity
- 2.3 Constant Functions
- 2.4 The Cauchy–Riemann Equations
- Exercises
- 3 Examples of Functions
- 3.1 Möbius Transformations
- 3.2 Infinity and the Cross Ratio
- 3.3 Stereographic Projection
- 3.4 Exponential and Trigonometric Functions
- 3.5 The Logarithm and Complex Exponentials
- Exercises
- 4 Integration
- 4.1 Definition and Basic Properties
- 4.2 Cauchy’s Theorem
- 4.3 Cauchy’s Integral Formula
- Exercises
- CONTENTS
- 5 Consequences of Cauchy’s Theorem
- 5.1 Extensions of Cauchy’s Formula
- 5.2 Taking Cauchy’s Formula to the Limit
- 5.3 Antiderivatives
- Exercises
- 6 Harmonic Functions
- 6.1 Definition and Basic Properties
- 6.2 Mean-Value and Maximum/Minimum Principle
- Exercises
- 7 Power Series
- 7.1 Sequences and Completeness
- 7.2 Series
- 7.3 Sequences and Series of Functions
- 7.4 Region of Convergence
- Exercises
- 8 Taylor and Laurent Series
- 8.1 Power Series and Holomorphic Functions
- 8.2 Classification of Zeros and the Identity Principle
- 8.3 Laurent Series
- Exercises
- 9 Isolated Singularities and the Residue Theorem
- 9.1 Classification of Singularities
- 9.2 Residues
- 9.3 Argument Principle and Rouché’s Theorem
- Exercises
- 10 Discrete Applications of the Residue Theorem
- 10.1 Infinite Sums
- 10.2 Binomial Coefficients
- 10.3 Fibonacci Numbers
- 10.4 The ‘Coin-Exchange Problem’
- 10.5 Dedekind sums
- Solutions to Selected Exercises
Chapter 1
Complex Numbers
Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk. (God created the integers, everything else is made by humans.) Leopold Kronecker (1823–1891)
1.0 Introduction
The real numbers have many nice properties. There are operations such as addition, subtraction, multiplication as well as division by any real number except zero. There are useful laws that govern these operations such as the commutative and distributive laws. You can also take limits and do calculus. But you cannot take the square root of −1. Equivalently, you cannot find a root of the equation x^2 + 1 = 0. (1.1) Most of you have heard that there is a “new” number i that is a root of the Equation (1.1). That is, i^2 + 1 = 0 or i^2 = −1. We will show that when the real numbers are enlarged to a new system called the complex numbers that includes i, not only do we gain a number with interesting properties, but we do not lose any of the nice properties that we had before. Specifically, the complex numbers, like the real numbers, will have the operations of addi- tion, subtraction, multiplication as well as division by any complex number except zero. These operations will follow all the laws that we are used to such as the commutative and distributive laws. We will also be able to take limits and do calculus. And, there will be a root of Equation (1.1). In the next section we show exactly how the complex numbers are set up and in the rest of this chapter we will explore the properties of the complex numbers. These properties will be both algebraic properties (such as the commutative and distributive properties mentioned already) and also geometric properties. You will see, for example, that multiplication can be described geometrically. In the rest of the book, the calculus of complex numbers will be built on the properties that we develop in this chapter.
The definition of our multiplication implies the innocent looking statement
(0, 1) · (0, 1) = (−1, 0). (1.13)
This identity together with the fact that
(a, 0) · (x, y) = (ax, ay)
allows an alternative notation for complex numbers. The latter implies that we can write
(x, y) = (x, 0) + (0, y) = (x, 0) · (1, 0) + (y, 0) · (0, 1).
If we think—in the spirit of our remark on the embedding of R in C —of (x, 0) and (y, 0) as the real numbers x and y, then this means that we can write any complex number (x, y) as a linear combination of (1, 0) and (0, 1), with the real coefficients x and y. (1, 0), in turn, can be thought of as the real number 1. So if we give (0, 1) a special name, say i, then the complex number that we used to call (x, y) can be written as x · 1 + y · i, or in short,
x + iy.
The number x is called the real part and y the imaginary part^1 of the complex number x + iy, often denoted as Re(x + iy) = x and Im(x + iy) = y. The identity (1.13) then reads
i^2 = −.
We invite the reader to check that the definitions of our binary operations and Theorem 1.1 are coherent with the usual real arithmetic rules if we think of complex numbers as given in the form x + iy. This algebraic way of thinking about complex numbers has a name: a complex number written in the form x + iy where x and y are both real numbers is in rectangular form. In fact, much more can now be said with the introduction of the square root of −1. It is not just that the polynomial z^2 + 1 has roots, but every polynomial has roots in C :
Theorem 1.2. (see Theorem 5.7) Every non-constant polynomial of degree d has d roots (counting multi- plicity) in C.
The proof of this theorem requires some important machinery, so we defer its proof and an extended discussion of it to Chapter 5.
1.2 From Algebra to Geometry and Back
Although we just introduced a new way of writing complex numbers, let’s for a moment return to the (x, y)-notation. It suggests that one can think of a complex number as a two-dimensional real vector. When plotting these vectors in the plane R^2 , we will call the x-axis the real axis and the y-axis the imaginary axis. The addition that we defined for complex numbers resembles vector addition. The analogy stops at multiplication: there is no “usual" multiplication of two
D D
k k
W W z 1
z 2
z 1 + z 2
Figure 1.1: Addition of complex numbers.
vectors in R^2 that gives another vector, and certainly not one that agrees with our definition of the product of two complex numbers. Any vector in R^2 is defined by its two coordinates. On the other hand, it is also determined by its length and the angle it encloses with, say, the positive real axis; let’s define these concepts thoroughly. The absolute value (sometimes also called the modulus) r = |z| ∈ R of z = x + iy is
r = |z| :=
x^2 + y^2 ,
and an argument of z = x + iy is a number φ ∈ R such that
x = r cos φ and y = r sin φ.
A given complex number z = x + iy has infinitely many possible arguments. For instance, the number 1 = 1 + 0 i lies on the x-axis, and so has argument 0, but we could just as well say it has argument 2 π , 4 π , − 2 π , or 2 π ∗ k for any integer k. The number 0 = 0 + 0 i has modulus 0, and every number φ is an argument. Aside from the exceptional case of 0, for any complex number z, the arguments of z all differ by a multiple of 2 π , just as we saw for the example z = 1. The absolute value of the difference of two vectors has a nice geometric interpretation:
Proposition 1.3. Let z 1 , z 2 ∈ C be two complex numbers, thought of as vectors in R^2 , and let d(z 1 , z 2 ) denote the distance between (the endpoints of) the two vectors in R^2 (see Figure 1.2). Then
d(z 1 , z 2 ) = |z 1 − z 2 | = |z 2 − z 1 |.
Proof. Let z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2. From geometry we know that d(z 1 , z 2 ) =
(x 1 − x 2 )^2 + (y 1 − y 2 )^2. This is the definition of |z 1 − z 2 |. Since (x 1 − x 2 )^2 = (x 2 − x 1 )^2 and (y 1 − y 2 )^2 = (y 2 − y 1 )^2 , this is also equal to |z 2 − z 1 |.
That |z 1 − z 2 | = |z 2 − z 1 | simply says that the vector from z 1 to z 2 has the same length as its inverse, the vector from z 2 to z 1. It is very useful to keep this geometric interpretation in mind when thinking about the abso- lute value of the difference of two complex numbers. The first hint that the absolute value and argument of a complex number are useful concepts is the fact that they allow us to give a geometric interpretation for the multiplication of two
(^1) The name has historical reasons: people thought of complex numbers as unreal, imagined.
Formal (x, y)
Algebraic:
Geometric:
rectangular exponential
cartesian polar
x + iy rei θ
r θ x
y
z z
Figure 1.4: Five ways of thinking about a complex number z ∈ C.
At this point, this exponential notation is indeed purely a notation. We will later see in Chapter 3 that it has an intimate connection to the complex exponential function. For now, we motivate this maybe strange-seeming definition by collecting some of its properties. The reader is encouraged to prove them.
Lemma 1.4. For any φ , φ 1 , φ 2 ∈ R , (a) ei φ^1 ei φ^2 = ei( φ^1 + φ^2 )
(b) 1/ei φ^ = e−i φ
(c) ei( φ +^2 π )^ = ei φ
(d)
ei φ
(e) (^) dd φ ei φ^ = i ei φ. With this notation, the sentence “The complex number x + iy has absolute value r and argu- ment φ " now becomes the identity x + iy = rei φ. The left-hand side is often called the rectangular form, the right-hand side the polar form of this complex number. We now have five different ways of thinking about a complex number: the formal definition, in rectangular form, in polar form, and geometrically using Cartesian coordinates or polar coor- dinates. Each of these five ways is useful in different situations, and translating between them is an essential ingredient in complex analysis. The five ways and their corresponding notation are listed in Figure 1.4.
1.3 Geometric Properties
From very basic geometric properties of triangles, we get the inequalities
−|z| ≤ Re z ≤ |z| and − |z| ≤ Im z ≤ |z|. (1.14)
The square of the absolute value has the nice property
|x^ +^ iy|^2 =^ x^2 +^ y^2 = (x^ +^ iy)(x^ −^ iy)^.
This is one of many reasons to give the process of passing from x + iy to x − iy a special name: x − iy is called the (complex) conjugate of x + iy. We denote the conjugate by
x + iy = x − iy.
Geometrically, conjugating z means reflecting the vector corresponding to z with respect to the real axis. The following collects some basic properties of the conjugate. Their easy proofs are left for the exercises. Lemma 1.5. For any z, z 1 , z 2 ∈ C , (a) z 1 ± z 2 = z 1 ± z 2
(b) z 1 · z 2 = z 1 · z 2
(c)
z 1 z 2
= z z^12
(d) z = z
(e) |z| = |z|
(f) (^) |z|^2 = zz
(g) Re z = 12 (z + z)
(h) Im z = (^21) i (z − z)
(i) ei φ^ = e−i φ. From part (f) we have a neat formula for the inverse of a non-zero complex number:
z−^1 =
z
z |z|^2
A famous geometric inequality (which holds for vectors in R n) is the triangle inequality
|z 1 +^ z 2 | ≤ |z 1 | +^ |z 2 |.
By drawing a picture in the complex plane, you should be able to come up with a geometric proof of this inequality. To prove it algebraically, we make extensive use of Lemma 1.5:
|z 1 + z 2 |^2 = (z 1 + z 2 ) (z 1 + z 2 ) = (z 1 + z 2 ) (z 1 + z 2 ) = z 1 z 1 + z 1 z 2 + z 2 z 1 + z 2 z 2 = (^) |z 1 |^2 + z 1 z 2 + z 1 z 2 + (^) |z 2 |^2 = (^) |z 1 |^2 + 2 Re (^) (z 1 z 2 ) + (^) |z 2 |^2.
(c) A point c is an accumulation point of E if every open disk centered at c contains a point of E different from c.
(d) A point d is an isolated point of E if it lies in E and some open disk centered at d contains no point of E other than d.
The idea is that if you don’t move too far from an interior point of E then you remain in E; but at a boundary point you can make an arbitrarily small move and get to a point inside E and you can also make an arbitrarily small move and get to a point outside E.
Definition 1.8. A set is open if all its points are interior points. A set is closed if it contains all its boundary points.
Example 1.9. For R > 0 and z 0 ∈ C , {z ∈ C : |z − z 0 | < R} and {z ∈ C : |z − z 0 | > R} are open. {z ∈ C : |z − z 0 | ≤ R} is closed.
Example 1.10. C and the empty set ∅ are open. They are also closed!
Definition 1.11. The boundary of a set E, written ∂ E, is the set of all boundary points of E. The interior of E is the set of all interior points of E. The closure of E, written E, is the set of points in E together with all boundary points of E.
Example 1.12. If G is the open disk {z ∈ C : |z − z 0 | < R} then
G = {z ∈ C : |z − z 0 | ≤ R} and ∂ G = {z ∈ C : |z − z 0 | = R}.
That is, G is a closed disk and ∂ G is a circle.
One notion that is somewhat subtle in the complex domain is the idea of connectedness. Intu- itively, a set is connected if it is “in one piece.” In the reals a set is connected if and only if it is an interval, so there is little reason to discuss the matter. However, in the plane there is a vast variety of connected subsets, so a definition is necessary.
Definition 1.13. Two sets X, Y ⊆ C are separated if there are disjoint open sets A and B so that X ⊆ A and Y ⊆ B. A set W ⊆ C is connected if it is impossible to find two separated non-empty sets whose union is equal to W. A region is a connected open set.
The idea of separation is that the two open sets A and B ensure that X and Y cannot just “stick together.” It is usually easy to check that a set is not connected. For example, the intervals X = [0, 1) and Y = (1, 2] on the real axis are separated: There are infinitely many choices for A and B that work; one choice is A = D 1 ( 0 ) (the open disk with center 0 and radius 1) and B = D 1 ( 2 ) (the open disk with center 2 and radius 1). Hence their union, which is [0, 2] \ { 1 }, is not connected. On the other hand, it is hard to use the definition to show that a set is connected, since we have to rule out any possible separation. One type of connected set that we will use frequently is a curve.
Definition 1.14. A path or curve in C is the image of a continuous function γ : [a, b] → C , where [a, b] is a closed interval in R. The path γ is smooth if γ is differentiable.
We say that the curve is parametrized by γ. It is a customary and practical abuse of notation to use the same letter for the curve and its parametrization. We emphasize that a curve must have a parametrization, and that the parametrization must be defined and continuous on a closed and bounded interval [a, b]. Since we may regard C as identified with R^2 , a path can be specified by giving two continuous real-valued functions of a real variable, x(t) and y(t), and setting γ (t) = x(t) + y(t)i. A curve is closed if γ (a) = γ (b) and is a simple closed curve if γ (s) = γ (t) implies s = a and t = b or s = b and t = a, that is, the curve does not cross itself. The following seems intuitively clear, but its proof requires more preparation in topology:
Proposition 1.15. Any curve is connected.
The next theorem gives an easy way to check whether an open set is connected, and also gives a very useful property of open connected sets.
Theorem 1.16. If W is a subset of C that has the property that any two points in W can be connected by a curve in W then W is connected. On the other hand, if G is a connected open subset of C then any two points of G may be connected by a curve in G; in fact, we can connect any two points of G by a chain of horizontal and vertical segments lying in G.
A chain of segments in G means the following: there are points z 0 , z 1 ,... , zn so that, for each k, zk and zk+ 1 are the endpoints of a horizontal or vertical segment which lies entirely in G. (It is not hard to parametrize such a chain, so it determines a curve.) As an example, let G be the open disk with center 0 and radius 2. Then any two points in G can be connected by a chain of at most 2 segments in G, so G is connected. Now let G 0 = G \ { 0 }; this is the punctured disk obtained by removing the center from G. Then G is open and it is connected, but now you may need more than two segments to connect points. For example, you need three segments to connect −1 to 1 since you cannot go through 0.
Warning : The second part of Theorem 1.16 is not generally true if G is not open. For example, circles are connected but there is no way to connect two distinct points of a circle by a chain of segments which are subsets of the circle. A more extreme example, discussed in topology texts, is the “topologist’s sine curve,” which is a connected set S ⊂ C that contains points that cannot be connected by a curve of any sort inside S.
The reader may skip the following proof. It is included to illustrate some common techniques in dealing with connected sets.
Proof of Theorem 1.16. Suppose, first, that any two points of G may be connected by a path that lies in G. If G is not connected then we can write it as a union of two non-empty separated subsets X and Y. So there are disjoint open sets A and B so that X ⊆ A and Y ⊆ B. Since X and Y are non-empty we can find points a ∈ X and b ∈ Y. Let γ be a path in G that connects a to b. Then X γ := X ∩ γ and Y γ := Y ∩ γ are disjoint, since X and Y are disjoint, and are non-empty since the former contains a and the latter contains b. Since G = X ∪ Y and γ ⊂ G we have γ = X γ ∪ Y γ. Finally, since X γ ⊂ X ⊂ A and Y γ ⊂ Y ⊂ B, X γ and Y γ are separated by A and B. But this means that γ is not connected, and this contradicts Proposition 1.15.
Finally, we can apply differentiation and integration with respect to different variables in either order:
Theorem 1.22 (Leibniz’s^4 Rule). Suppose f is continuous on the rectangle R given by a ≤ x ≤ b and
c ≤ y ≤ d, and suppose the partial derivative ∂ ∂^ xf exists and is continuous on R. Then
d dx
∫ (^) d
c
f (x, y) dy =
∫ (^) d
c
∂ f ∂ x
(x, y) dy.
Exercises
- Let z = 1 + 2 i and w = 2 − i. Compute:
(a) z + 3 w. (b) w − z. (c) z^3. (d) Re(w^2 + w). (e) z^2 + z + i.
- Find the real and imaginary parts of each of the following:
(a) zz−+aa (a ∈ R ). (b) (^37) i++^51 i.
(c)
− 1 +i √ 3 2
(d) in^ for any n ∈ Z.
- Find the absolute value and conjugate of each of the following:
(a) − 2 + i. (b) ( 2 + i)( 4 + 3 i). (c) √^32 −+i 3 i. (d) ( 1 + i)^6.
- Write in polar form:
(a) 2i. (b) 1 + i. (c) − 3 +
3 i. (d) −i. (e) ( 2 − i)^2. (^4) Named after Gottfried Wilhelm Leibniz (1646–1716). For more information about Leibnitz, see http://www-groups.dcs.st-and.ac.uk/∼history/Biographies/Leibnitz.html.
(f) | 3 − 4 i|. (g)
5 − i.
(h)
(^1) √−i 3
- Write in rectangular form:
(a)
2 ei^3 π /4. (b) 34 ei π /2. (c) −ei^250 π. (d) 2e^4 π i.
- Write in both polar and rectangular form:
(a) 2i (b) eln(^5 )i (c) e^1 +i π / (d) (^) dd φ e φ +i φ
- Prove the quadratic formula works for complex numbers, regardless of whether the dis- criminant is negative. That is, prove, the roots of the equation az^2 + bz + c = 0, where a, b, c ∈ C , are −b±
√ b^2 − 4 ac 2 a as long as^ a^6 =^ 0.
- Use the quadratic formula to solve the following equations. Put your answers in standard form.
(a) z^2 + 25 = 0. (b) 2z^2 + 2 z + 5 = 0. (c) 5z^2 + 4 z + 1 = 0. (d) z^2 − z = 1. (e) z^2 = 2 z.
- Fix A ∈ C and B ∈ R. Show that the equation |z^2 | + Re(Az) + B = 0 has a solution if and only if |A^2 | ≥ 4 B. When solutions exist, show the solution set is a circle.
- Find all solutions to the following equations:
(a) z^6 = 1. (b) z^4 = −16. (c) z^6 = −9. (d) z^6 − z^3 − 2 = 0.
- Show that |z| = 1 if and only if (^1) z = z.
(e) |z − 1 | + |z + 1 | < 3.
- What are the boundaries of the sets in the previous exercise?
- The set E is the set of points z in C satisfying either z is real and − 2 < z < −1, or |z| < 1, or z = 1 or z = 2.
(a) Sketch the set E, being careful to indicate exactly the points that are in E. (b) Determine the interior points of E. (c) Determine the boundary points of E. (d) Determine the isolated points of E.
- The set E in the previous exercise can be written in three different ways as the union of two disjoint nonempty separated subsets. Describe them, and in each case say briefly why the subsets are separated.
- Show that the union of two regions with nonempty intersection is itself a region.
- Show that if A ⊂ B and B is closed, then ∂ A ⊂ B. Similarly, if A ⊂ B and A is open, show A is contained in the interior of B.
- Let G be the annulus determined by the conditions 2 < |z| < 3. This is a connected open set. Find the maximum number of horizontal and vertical segments in G needed to connect two points of G.
- Prove Leibniz’s Rule: Define F(x) =
∫ (^) d c f^ (x,^ y)^ dy, get an expression for^ F(x)^ −^ F(a)^ as an iterated integral by writing f (x, y) − f (a, y) as the integral of ∂ ∂^ xf , interchange the order of integrations, and then differentiate using the Fundamental Theorem of Calculus.
Optional Lab
Open your favorite web browser and go to http://www.math.ucla.edu/∼tao/java/Plane.html.
- Convert the following complex numbers into their polar representation, i.e., give the abso- lute value and the argument of the number.
34 = i = − π = 2 + 2 i = − 12 (+
3 + i) =
After you have finished computing these numbers, check your answers with the program. You may play with the > and < buttons to see what effect it has to change these quantities slightly.
- Convert the following complex numbers given in polar representation into their ‘rectangu- lar’ representation.
2 ei^0 = 3 ei π /2^ = 1 2 e
i π (^) =
e−i3/2 π^ = 2 ei2/3 π^ =
After you have finished computing these numbers, check your answers with the program. You may play with the > and < buttons to see what effect it has to change these quantities slightly.
- Pick your favorite five numbers from the ones that you’ve played around with and put them in the table, both in rectangular and polar form. Apply the functions listed to your numbers. Think about which representation is more helpful in each instance. rect. polar z + 1 z + 2 − i 2 z −z z/2 iz z¯ z^2 Rez Imz Imz i |z| 1/z
- Play with other examples until you get a “feel" for these functions. Then go to the next applet: elementary complex maps (link on the bottom of the page). With this applet, there are a lot of questions on the web page. Think about them!
