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A First Course in
Complex Analysis
Version 1.
Matthias Beck, Gerald Marchesi, and Dennis Pixton
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] [email protected]
Copyright 2002–2007 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. Special thanks go to Collin Bleak, Jon Clauss, Sharma Pallekonda, and Joshua Palmatier for comments after teaching from this book.
- 1 Complex Numbers
- 1.1 Definition and Algebraic Properties
- 1.2 Geometric Properties
- 1.3 Elementary Topology of the Plane
- 1.4 Theorems from Calculus
- Exercises
- 2 Differentiation
- 2.1 First Steps
- 2.2 Differentiability and Analyticity
- 2.3 The Cauchy–Riemann Equations
- 2.4 Constants and Connectivity
- Exercises
- 3 Examples of Functions
- 3.1 M¨obius Transformations
- 3.2 Infinity and the Cross Ratio
- 3.3 Exponential and Trigonometric Functions
- 3.4 The Logarithm and Complex Exponentials
- Exercises
- 4 Integration
- 4.1 Definition and Basic Properties
- 4.2 Antiderivatives
- 4.3 Cauchy’s Theorem
- 4.4 Cauchy’s Integral Formula
- Exercises
- 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 Revisited and Morera’s Theorem
- Exercises
- CONTENTS
- 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 Analytic 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 Rouch´e’s Theorem
- Exercises
- 10 Discreet 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
- Index
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)
1.1 Definition and Algebraic Properties
The complex numbers can be defined as pairs of real numbers,
C = {(x, y) : x, y ∈ R} ,
equipped with the addition (x, y) + (a, b) = (x + a, y + b)
and the multiplication (x, y) · (a, b) = (xa − yb, xb + ya).
One reason to believe that the definitions of these binary operations are “good” is that C is an extension of R, in the sense that the complex numbers of the form (x, 0) behave just like real numbers; that is, (x, 0) + (y, 0) = (x + y, 0) and (x, 0) · (y, 0) = (x · y, 0). So we can think of the real numbers being embedded in C as those complex numbers whose second coordinate is zero. The following basic theorem states the algebraic structure that we established with our defini- tions. Its proof is straightforward but nevertheless a good exercise.
Theorem 1.1. (C, +, ·) is a field; that is:
∀ (x, y), (a, b) ∈ C : (x, y) + (a, b) ∈ C (1.1) ∀ (x, y), (a, b), (c, d) ∈ C :
(x, y) + (a, b)
(a, b) + (c, d)
∀ (x, y), (a, b) ∈ C : (x, y) + (a, b) = (a, b) + (x, y) (1.3) ∀ (x, y) ∈ C : (x, y) + (0, 0) = (x, y) (1.4) ∀ (x, y) ∈ C : (x, y) + (−x, −y) = (0, 0) (1.5)
CHAPTER 1. COMPLEX NUMBERS 3
D D
WWWWWWWWWWWW k k
// // // // // // /
W W z 1
z 2
z 1 + z 2
Figure 1.1: Addition of complex numbers.
that gives another vector—much less so if we additionally demand 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) of x + iy is
r = |x + iy| =
x^2 + y^2 ,
and an argument of x + iy is a number φ such that
x = r cos φ and y = r sin φ.
This means, naturally, that any complex number has many arguments; more precisely, all of them differ by a multiple of 2π. The absolute value of the difference of two vectors has a nice geometric interpretation: it is the distance of the (end points of the) two vectors (see Figure 1.2). It is very useful to keep this geometric interpretation in mind when thinking about the absolute value of the difference of two complex numbers.
D D
WWWWWWWWWWWW k k
jjjj^ j jjjjj jjjjj jjjj
44 z^1
z 2
z 1 − z 2
Figure 1.2: Geometry behind the “distance” between two complex numbers.
The first hint that 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 complex numbers. Let’s say we have two complex numbers, x 1 + iy 1 with absolute value r 1 and argument φ 1 , and x 2 + iy 2 with absolute value r 2 and argument φ 2. This means, we can write
(^1) The name has historical reasons: people thought of complex numbers as unreal, imagined.
CHAPTER 1. COMPLEX NUMBERS 4
x 1 + iy 1 = (r 1 cos φ 1 ) + i(r 1 sin φ 1 ) and x 2 + iy 2 = (r 2 cos φ 2 ) + i(r 2 sin φ 2 ) To compute the product, we make use of some classic trigonometric identities:
(x 1 + iy 1 )(x 2 + iy 2 ) =
(r 1 cos φ 1 ) + i(r 1 sin φ 1 )
(r 2 cos φ 2 ) + i(r 2 sin φ 2 )
= (r 1 r 2 cos φ 1 cos φ 2 − r 1 r 2 sin φ 1 sin φ 2 ) + i(r 1 r 2 cos φ 1 sin φ 2 + r 1 r 2 sin φ 1 cos φ 2 ) = r 1 r 2
(cos φ 1 cos φ 2 − sin φ 1 sin φ 2 ) + i(cos φ 1 sin φ 2 + sin φ 1 cos φ 2 )
= r 1 r 2
cos(φ 1 + φ 2 ) + i sin(φ 1 + φ 2 )
So the absolute value of the product is r 1 r 2 and (one of) its argument is φ 1 + φ 2. Geometrically, we are multiplying the lengths of the two vectors representing our two complex numbers, and adding their angles measured with respect to the positive x-axis.^2
MMMMMMMM f f F F
rrrrrrrrrrrrrrrrr x x
z 2 z 1
z 1 z 2
φ 1
φ 2
φ 1 + φ 2
Figure 1.3: Multiplication of complex numbers.
In view of the above calculation, it should come as no surprise that we will have to deal with quantities of the form cos φ + i sin φ (where φ is some real number) quite a bit. To save space, bytes, ink, etc., (and because “Mathematics is for lazy people”^3 ) we introduce a shortcut notation and define eiφ^ = cos φ + i sin φ. At this point, this exponential notation is indeed purely a notation. We will later see 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 1.2. For any φ, φ 1 , φ 2 ∈ R,
(a) eiφ^1 eiφ^2 = ei(φ^1 +φ^2 )
(b) 1/eiφ^ = e−iφ
(c) ei(φ+2π)^ = eiφ
(d)
eiφ
(^2) One should convince oneself that there is no problem with the fact that there are many possible arguments for complex numbers, as both cosine and sine are periodic functions with period 2π. (^3) Peter Hilton (Invited address, Hudson River Undergraduate Mathematics Conference 2000)
CHAPTER 1. COMPLEX NUMBERS 6
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 1.3:
|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.
Finally by (1.12)
|z 1 + z 2 |^2 ≤ |z 1 |^2 + 2 |z 1 z 2 | + |z 2 |^2 = |z 1 |^2 + 2 |z 1 | |z 2 | + |z 2 |^2 = |z 1 |^2 + 2 |z 1 | |z 2 | + |z 2 |^2 = (|z 1 | + |z 2 |)^2 ,
which is equivalent to our claim. For future reference we list several variants of the triangle inequality:
Lemma 1.4. For z 1 , z 2 , · · · ∈ C, we have the following identities:
(a) The triangle inequality: |±z 1 ± z 2 | ≤ |z 1 | + |z 2 |.
(b) The reverse triangle inequality: |±z 1 ± z 2 | ≥ |z 1 | − |z 2 |.
(c) The triangle inequality for sums:
∑^ n
k=
zk
∑^ n
k=
|zk|.
The first inequality is just a rewrite of the original triangle inequality, using the fact that |±z| = |z|, and the last follows by induction. The reverse triangle inequality is proved in Exercise 15.
1.3 Elementary Topology of the Plane
In Section 1.2 we saw that the complex numbers C, which were initially defined algebraically, can be identified with the points in the Euclidean plane R^2. In this section we collect some definitions and results concerning the topology of the plane. While the definitions are essential and will be used frequently, we will need the following theorems only at a limited number of places in the remainder of the book; the reader who is willing to accept the topological arguments in later proofs on faith may skip the theorems in this section. Recall that if z, w ∈ C, then |z − w| is the distance between z and w as points in the plane. So if we fix a complex number a and a positive real number r then the set of z satisfying |z − a| = r is the set of points at distance r from a; that is, this is the circle with center a and radius r. The inside of this circle is called the open disk with center a and radius r, and is written Dr(a). That is, Dr(a) = {z ∈ C : |z − a| < r}. Notice that this does not include the circle itself. We need some terminology for talking about subsets of C.
CHAPTER 1. COMPLEX NUMBERS 7
Definition 1.1. Suppose E is any subset of C. (a) A point a is an interior point of E if some open disk with center a lies in E.
(b) A point b is a boundary point of E if every open disk centered at b contains a point in E and also a point that is not in E.
(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 1.2. A set is open if all its points are interior points. A set is closed if it contains all its boundary points. Example 1.1. 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 1.2. C and the empty set ∅ are open. They are also closed! Definition 1.3. 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 1.3. 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 1.4. 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.
CHAPTER 1. COMPLEX NUMBERS 9
Now suppose that G is a connected open set. Choose a point z 0 ∈ G and define two sets: A is the set of all points a so that there is a chain of segments in G connecting z 0 to a, and B is the set of points in G that are not in A. Suppose a is in A. Since a ∈ G there is an open disk D with center a that is contained in G. We can connect z 0 to any point z in D by following a chain of segments from z 0 to a, and then adding at most two segments in D that connect a to z. That is, each point of D is in A, so we have shown that A is open. Now suppose b is in B. Since b ∈ G there is an open disk D centered at b that lies in G. If z 0 could be connected to any point in D by a chain of segments in G then, extending this chain by at most two more segments, we could connect z 0 to b, and this is impossible. Hence z 0 cannot connect to any point of D by a chain of segments in G, so D ⊆ B. So we have shown that B is open. Now G is the disjoint union of the two open sets A and B. If these are both non-empty then they form a separation of G, which is impossible. But z 0 is in A so A is not empty, and so B must be empty. That is, G = A, so z 0 can be connected to any point of G by a sequence of segments in G. Since z 0 could be any point in G, this finishes the proof.
1.4 Theorems from Calculus
Here are a few theorems from real calculus that we will make use of in the course of the text.
Theorem 1.7 (Extreme-Value Theorem). Any continuous real-valued function defined on a closed and bounded subset of Rn^ has a minimum value and a maximum value.
Theorem 1.8 (Mean-Value Theorem). Suppose I ⊆ R is an interval, f : I → R is differentiable, and x, x + ∆x ∈ I. Then there is 0 < a < 1 such that
f (x + ∆x) − f (x) ∆x = f ′(x + a∆x).
Many of the most important results of analysis concern combinations of limit operations. The most important of all calculus theorems combines differentiation and integration (in two ways):
Theorem 1.9 (Fundamental Theorem of Calculus). Suppose f : [a, b] → R is continuous. Then
(a) If F is defined by F (x) =
∫ (^) x a f^ (t)^ dt^ then^ F^ is differentiable and^ F^ ′(x) = f (x).
(b) If F is any antiderivative of f (that is, F ′^ = f ) then
∫ (^) b a f^ (x)^ dx^ =^ F^ (b)^ −^ F^ (a). For functions of several variables we can perform differentiation operations, or integration op- erations, in any order, if we have sufficient continuity:
Theorem 1.10 (Equality of mixed partials). If the mixed partials ∂
(^2) f ∂x∂y and^
∂^2 f ∂y∂x are defined on an open set G and are continuous at a point (x 0 , y 0 ) in G then they are equal at (x 0 , y 0 ).
Theorem 1.11 (Equality of iterated integrals). If f is continuous on the rectangle given by a ≤ x ≤ b and c ≤ y ≤ d then the iterated integrals
∫ (^) b a
∫ (^) d c f^ (x, y)^ dy dx^ and^
∫ (^) d c
∫ (^) b a f^ (x, y)^ dx dy^ are equal. Finally, we can apply differentiation and integration with respect to different variables in either order:
CHAPTER 1. COMPLEX NUMBERS 10
Theorem 1.12 (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 ∂f∂x exists and is continuous on R. Then
d dx
∫ (^) d
c
f (x, y) dy =
∫ (^) d
c
∂f ∂x (x, y) dy.
Exercises
- Find the real and imaginary parts of each of the following:
(a) zz−+aa (a ∈ R). (b) 3+5 7 i+1i.
(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 + 3i). (c) √^3 2+3−ii. (d) (1 + i)^6.
- Write in polar form:
(a) 2i. (b) 1 + i. (c) −3 +
3 i.
- Write in rectangular form:
(a)
2 ei^3 π/^4. (b) 34 eiπ/^2. (c) −ei^250 π.
- 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 (^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.
CHAPTER 1. COMPLEX NUMBERS 12
(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.
- 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 ∂f∂x , interchange the order of integrations, and then differentiate using the Fundamental Theorem of Calculus.
Chapter 2
Differentiation
Mathematical study and research are very suggestive of mountaineering. Whymper made several efforts before he climbed the Matterhorn in the 1860’s and even then it cost the life of four of his party. Now, however, any tourist can be hauled up for a small cost, and perhaps does not appreciate the difficulty of the original ascent. So in mathematics, it may be found hard to realise the great initial difficulty of making a little step which now seems so natural and obvious, and it may not be surprising if such a step has been found and lost again. Louis Joel Mordell (1888–1972)
2.1 First Steps
A (complex) function f is a mapping from a subset G ⊆ C to C (in this situation we will write f : G → C and call G the domain of f ). This means that each element z ∈ G gets mapped to exactly one complex number, called the image of z and usually denoted by f (z). So far there is nothing that makes complex functions any more special than, say, functions from Rm^ to Rn. In fact, we can construct many familiar looking functions from the standard calculus repertoire, such as f (z) = z (the identity map), f (z) = 2z + i, f (z) = z^3 , or f (z) = (^1) z. The former three could be defined on all of C, whereas for the latter we have to exclude the origin z = 0. On the other hand, we could construct some functions which make use of a certain representation of z, for example, f (x, y) = x − 2 iy, f (x, y) = y^2 − ix, or f (r, φ) = 2rei(φ+π). Maybe the fundamental principle of analysis is that of a limit. The philosophy of the following definition is not restricted to complex functions, but for sake of simplicity we only state it for those functions.
Definition 2.1. Suppose f is a complex function with domain G and z 0 is an accumulation point of G. Suppose there is a complex number w 0 such that for every > 0, we can find δ > 0 so that for all z ∈ G satisfying 0 < |z − z 0 | < δ we have |f (z) − w 0 | < . Then w 0 is the limit of f as z approaches z 0 , in short lim z→z 0 f (z) = w 0.
This definition is the same as is found in most calculus texts. The reason we require that z 0 is an accumulation point of the domain is just that we need to be sure that there are points z of the domain which are arbitrarily close to z 0. Just as in the real case, the definition does not require
CHAPTER 2. DIFFERENTIATION 15
Just as in the real case, we can “take the limit inside” a continuous function:
Lemma 2.3. If f is continuous at w 0 and limz→z 0 g(z) = w 0 then limz→z 0 f (g(z)) = f (w 0 ). In other words,
lim z→z 0 f (g(z)) = f
lim z→z 0 g(z)
2.2 Differentiability and Analyticity
The fact that limits such as limz→ 0 ¯zz do not exist points to something special about complex numbers which has no parallel in the reals—we can express a function in a very compact way in one variable, yet it shows some peculiar behavior “in the limit.” We will repeatedly notice this kind of behavior; one reason is that when trying to compute a limit of a function as, say, z → 0, we have to allow z to approach the point 0 in any way. On the real line there are only two directions to approach 0—from the left or from the right (or some combination of those two). In the complex plane, we have an additional dimension to play with. This means that the statement “A complex function has a limit...” is in many senses stronger than the statement “A real function has a limit...” This difference becomes apparent most baldly when studying derivatives.
Definition 2.3. Suppose f : G → C is a complex function and z 0 is an interior point of G. The derivative of f at z 0 is defined as
f ′(z 0 ) = lim z→z 0
f (z) − f (z 0 ) z − z 0
provided this limit exists. In this case, f is called differentiable at z 0. If f is differentiable for all points in an open disk centered at z 0 then f is called analytic at z 0. The function f is analytic on the open set G ⊆ C if it is differentiable (and hence analytic) at every point in G. Functions which are differentiable (and hence analytic) in the whole complex plane C are called entire.
The difference quotient limit which defines f ′(z 0 ) can be rewritten as
f ′(z 0 ) = lim h→ 0
f (z 0 + h) − f (z 0 ) h
This equivalent definition is sometimes easier to handle. Note that h is not a real number but can rather approach zero from anywhere in the complex plane. The fact that the notions of differentiability and analyticity are actually different is seen in the following examples.
Example 2.2. The function f (z) = z^3 is entire, that is, analytic in C: For any z 0 ∈ C,
lim z→z 0
f (z) − f (z 0 ) z − z 0
= lim z→z 0
z^3 − z^30 z − z 0
= lim z→z 0
(z^2 + zz 0 + z 02 )(z − z 0 ) z − z 0
= lim z→z 0 z^2 + zz 0 + z^20 = 3z^20.
Example 2.3. The function f (z) = z^2 is differentiable at 0 and nowhere else (in particular, f is not analytic at 0): Let’s write z = z 0 + reiφ. Then
z^2 − z 02 z − z 0
z 0 + reiφ
− z 02 z 0 + reiφ^ − z 0
z 0 + re−iφ
z 02 reiφ^
z 02 + 2z 0 re−iφ^ + r^2 e−^2 iφ^ − z 02 reiφ
= 2 z 0 re−iφ^ + r^2 e−^2 iφ reiφ^
= 2z 0 e−^2 iφ^ + re−^3 iφ.
CHAPTER 2. DIFFERENTIATION 16
If z 0 6 = 0 then the limit of the right-hand side as z → z 0 does not exist since r → 0 and we get different answers for horizontal approach (φ = 0) and for vertical approach (φ = π/2). (A more entertaining way to see this is to use, for example, z(t) = z 0 + (^1) t eit, which approaches z 0 as t → ∞.) On the other hand, if z 0 = 0 then the right-hand side equals re−^3 iφ^ = |z|e−^3 iφ. Hence
lim z→ 0
z^2 z
∣ = lim z→ 0
∣|z|e−^3 iφ
∣ = lim z→ 0 |z|^ = 0^ ,
which implies that lim z→ 0
z^2 z
Example 2.4. The function f (z) = z is nowhere differentiable:
lim z→z 0
z − z 0 z − z 0
= lim z→z 0
z − z 0 z − z 0
= lim z→ 0
z z does not exist, as discussed earlier. The basic properties for derivatives are similar to those we know from real calculus. In fact, one should convince oneself that the following rules follow mostly from properties of the limit. (The ‘chain rule’ needs a little care to be worked out.) Lemma 2.4. Suppose f and g are differentiable at z ∈ C, and that c ∈ C, n ∈ Z, and h is differentiable at g(z). (a)
f (z) + c g(z)
= f ′(z) + c g′(z)
(b)
f (z) · g(z)
= f ′(z)g(z) + f (z)g′(z)
(c)
f (z)/g(z)
f ′(z)g(z) − f (z)g′(z) g(z)^2
(d)
zn
= nzn−^1
(e)
h(g(z))
= h′(g(z))g′(z). In the third identity we have to be aware of division by zero. We end this section with yet another differentiation rule, that for inverse functions. As in the real case, this rule is only defined for functions which are bijections. A function f : G → H is one-to-one if for every image w ∈ H there is a unique z ∈ G such that f (z) = w. The function is onto if every w ∈ H has a preimage z ∈ G (that is, there exists a z ∈ G such that f (z) = w). A bijection is a function which is both one-to-one and onto. If f : G → H is a bijection then g is the inverse of f if for all z ∈ H, f (g(z)) = z. Lemma 2.5. Suppose G and H are open sets in C, f : G → H is a bijection, g : H → G is the inverse function of f , and z 0 ∈ H. If f is differentiable at g(z 0 ), f ′(g(z 0 )) 6 = 0, and g is continuous at z 0 then g is differentiable at z 0 with
g′(z 0 ) =
f ′^ (g(z 0 ))
