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COMPLEX ANALYSIS

A Short Course

M.Thamban Nair

Department of Mathematics

Indian Institute of Technology Madras

January-May 2011

• 1 Complex Plane Preface v
  • 1.1 Complex Numbers
  • 1.2 Some Definitions and Properties
    • 1.2.1 Metric on C
    • 1.2.2 Polar representation and nth-roots
    • 1.2.3 Steriographic projection
  • 1.3 Problems
• 2 Analytic Functions
  • 2.1 Differentiation
  • 2.2 Holomorphic or Analytic Functions
    • 2.2.1 Analytic extension
    • 2.2.2 Geometric representations
    • 2.2.3 Curves in the complex plane
  • 2.3 Fractional linear transformations
    • 2.3.1 The map z 7 → 1 /z
    • 2.3.2 Extended plane
    • 2.3.3 Fractional linear transformations
    • 2.3.4 Image of inverse points
  • 2.4 Problems
• 3 Elementary Functions
  • 3.1 Exponential Function
  • 3.2 Hyperbolic and Trigonometric Functions
  • 3.3 Logarithms
  • 3.4 Branches of arg(z) and log(z)
  • 3.5 Problems
• 4 Power Series iv Contents
  • 4.1 Convergence
  • 4.2 Problems
• 5 Integration
  • 5.1 Integrals along Piecewise Smooth Curves
  • 5.2 Cauchy’s Theorem
  • 5.3 Cauchy’s Integral Formulas - 5.3.1 Appendix
  • 5.4 Zeros of analytic functions - 5.4.1 Identity theorem - 5.4.2 Maximum modulus principle - 5.4.3 Schwarz’s lemma - 5.4.4 On harmonic functions
  • 5.5 Problems
• 6 Laurent Series and Isolated Singularities
  • 6.1 Laurent Series
  • 6.2 Isolated Singularities
  • 6.3 Problems
• 7 Residues and Real Integrals
  • 7.1 Residue theorem
  • 7.2 Calculation of Residues
  • 7.3 Evaluation of Improper Integrals
  • 7.4 Problems
    • Index

Preface

• This book is primarily for the students and teachers of IIT Madras.
• This is based on a Core Course that I have given for the sec- ond semester students of M.Sc. (Mathematics) at IIT Madras. The audience included some B.Tech. students and a faculty member (Dr. Parag Ravindran) from Mechanical Engineering department.
• The contents of the book is in the line of the well-written, small book Complex Function Theory^1 by Donald Sarason. I fondly acknowledge some e-mail discussions that I had with Prof. Sarason during the time of giving the course.

IIT Madras M. Thamban Nair June 2011

(^1) Second Editin, Hindustan Book Agency (‘trim’ series), New Delhi, 2008.

v

Complex Plane

1.1 Complex Numbers

After having the real field R, it is natural to look for a bigger field in which algebraic equations such as

x^2 + 1 = 0 (∗)

has a solution. Of course, the + sign here must be the symbol for addition in the bigger field. Since two fields can be considered to be identical if there is a surjective isomorphism between then, it is enough to have a field which contains an isomorphic image of R and having required properties such as solution to algebraic equations. We shall define such a field with the intention of having a solution to the equation (∗).

Definition 1.1.1 The set C of complex numbers is the set of all ordered pairs (x, y) of real numbers with the following operations of addition and multiplication:

(x 1 , y 1 ) + (x 2 , y 2 ) = (x 1 + x 2 , y 1 + y 2 ),

(x 1 , y 1 ).(x 2 , y 2 ) = (x 1 x 2 − y 1 y 2 , x 1 y 2 + x 2 y 1 ). ♦ The proof of the following theorem is left to the reader.

Theorem 1.1.1 The following hold. (i) C is a field with additive identity (0, 0) and multiplicative identity (1, 0). (ii) The map ϕ : R → C defined by ϕ(x) = (x, 0), x ∈ R,

is a field isomorphism.

1

2 Complex Plane

We observe that the multiplicative inverse of a nonzero complex number z = (x, y) is given by ( x x^2 + y^2

, − y x^2 + y^2

Writing i = (0, 1)

and for x ∈ R, x˜ = (x, 0),

we observe that i^2 = −˜ 1 ,

and C = {x˜ + i˜y : x, y ∈ R}.

With the above notations, the addition and multiplication in C can be written as

(˜x 1 + iy˜ 1 ) + (˜x 2 + iy˜ 2 ) = (˜x 1 + ˜y 2 ) + i(˜y 1 + ˜y 2 ) = (x˜ 1 + y 2 ) + i (y˜ 1 + y 2 ),

(˜x 1 + iy˜ 1 ).(˜x 2 + iy˜ 2 ) = (˜x 1 x˜ 2 − ˜y 1 ˜y 2 ) + i(˜x 1 y˜ 2 + ˜x 2 ˜y 1 ) = (x 1 x˜ 2 − y 1 y 2 ) + i (x 1 y˜ 2 + x 2 y 1 ).

Throughout this course, we identify ˜x with x for every x ∈ R, so that C is the set of all numbers of the form

a + ib with a, b ∈ R.

Thus, a + i0 = a, 0 + i1 = i and i^2 = − 1 ,

and for nonzero z = x + iy,

1 z

:= z−^1 = x x^2 + y^2

− i y x^2 + y^2

One of the important properties of this field is that, not only equa- tion (∗) has a solution in C, but every algebraic equation also has a solution. This is the so called fundamental theorem of algebra which we shall prove in the due course.

4 Complex Plane

• A subset G of C is open in C if and only if every point in G is an interior point of G.
• A point z 0 ∈ C is call a boundary point of a set A ⊆ C if every open ball containing z 0 contains some point of A and some point of its compliment, i.e., for every r > 0, B(z 0 , r) ∩ A 6 = ∅ and B(z 0 , r) ∩ Ac^6 = ∅.
• A subset S of C is closed in C it contains all its boundary points. It can be shown that a set S ⊆ C is closed if and only if Sc^ := C \ S is open if and only if for every sequence (zn) in S,

zn → z =⇒ z ∈ S.

• A function f : A → C defined on a subset A of C is continuous at a point z 0 ∈ A if and only if for every ε > 0, there exists δ > 0 such that

z ∈ A, |z − z 0 | < δ =⇒ |f (z) − f (z 0 )| < ε.

Equivalently, f is continuous at a point z 0 ∈ A if and only if for every ε > 0, there exists δ > 0 such that

z ∈ A ∩ B(z 0 , δ) =⇒ f (z) ∈ B(f (z 0 ), ε).

• A function f : A → C defined on a subset A of C has the limit ζ ∈ C at a point z 0 ∈ C if and only if for every ε > 0, there exists δ > 0 such that

z ∈ A, 0 < |z − z 0 | < δ =⇒ |f (z) − ζ| < ε,

and in that case we write

zlim→z 0

f (z) = ζ.

• A subset A of C is bounded if and only if there exists α > 0 such that |z| ≤ α ∀z ∈ A.
• A subset A of C is compact if and only if every sequence in A has a convergent subsequence.

Some Definitions and Properties 5

Theorem 1.2.2 The set C is a complete metric space.

Proof. Let (zn) be a Cauchy sequence in C. Writing zn = xn +iyn with xn, yn ∈ R, we have

|xn − xm| ≤ |zn − zm|, |yn − ym| ≤ |zn − zm|

for all n, m ∈ N. Hence, (xn) and yn) are Cauchy sequences in R. Since R is a complete metric space with respect to the absolute-value metric, there exist x, y ∈ R such that xn → x and yn → y. Then, writing z = x + iy, we have

|zn − z|^2 = (xn − x)^2 + (yn − y)^2 → 0.

Thus, (zn) converges to z.

1.2.2 Polar representation and nth-roots

Let z be a nonzero complex number and let θ be the angle which the line segment joining 0 to z makes with the positive real axis, and r = |z|, the length of the line segment. Then it is clear from the geometry that z = r(cos θ + i sin θ). (∗)

Definition 1.2.2 The representation (∗) of a nonzero z ∈ C is called its polar representation, and an angle θ for which (∗) holds is called an argumet of z, denoted by arg(z). ♦

Note that each (r, θ) with r > 0 and θ ∈ R represents a unique nonzero z ∈ C with the representation (∗), but a non zero z ∈ C has many polar representations, namely,

z = r(cos θ + i sin θ), θ ∈ {θ 0 + 2πk : k ∈ Z},

where θ 0 is one of the angles for which (∗) holds. We note that if z 1 = r 1 (cos θ 1 + i sin θ 1 ) and z 2 = r 2 (cos θ 2 + i sin θ 2 ), then

z 1 z 2 = r 1 r 2 [cos(θ 1 + θ 2 ) + i sin(θ 1 + θ 2 )].

Thus, if z = r(cos θ + i sin θ) and n ∈ N, then

zn^ = rn(cos nθ+i sin nθ) = rn[cos(nθ+2kπ)+i sin(nθ+2kπ)], k ∈ Z.

Problems 7

i.e., uλ = (λx, λy, 1 − λ), λ ∈ R.

Clearly, for every u = (x, y, 0) ∈ X there exists one and only λ := λu such that uλ ∈ S^2. We consider the map

u := (x, y, 0) 7 → (λux, λuy, 1 − λ)

from X to S^2 \ {(0, 0 , 1)}. Note that

uλ ∈ S^2 ⇐⇒ λ^2 x^2 + λ^2 y^2 + (1 − λ)^2 = 1

if and only if λ = 0 or λ(x^2 + y^2 + 1) − 2 = 0. The point λ = 0 correspond to u 0. Hence,

λu =

x^2 + y^2 + 1.

Thus the map

z := x + iy 7 →

2 x 1 + |z|^2 ,^

2 x 1 + |z|^2 ,^

|z|^2 − 1 1 + |z|^2

is a bijective continuous function from C onto S^2 \ {(1, 0 , 0)} with its inverse (α, β, γ) 7 → α^ +^ iβ 1 − γ

which is also continuous.

1.3 Problems

1. Show that C is a field under the addition and multiplication defined for complex numbers.
2. Show that the map f : R → C defined by f (x) = (x, 0) is a field isomorphism.
3. For a nonzero complex number x, show that z−^1 = ¯z/|z|.
4. Show that for z 1 , z 2 in C, |z 1 + z 2 | ≤ |z 1 | + |z 2 |.
5. Show that d(z 1 , z 2 ) := |z 1 − z 2 | defines a metric on C, and it is a complete metric.

8 Complex Plane

6. Show that |z 1 − z 2 | ≥ |z 1 | − |z 2 | for all z 1 , z 2 ∈ C.
7. Suppose α, β, γ are nonzero complex numbers such that |α| = |β| = |γ|. Show that α + β + γ = 0 ⇐⇒ 1 α

+^1

β

+^1

γ

8. Suppose z 1 , z 2 , z 3 are vertices of an equilateral triangle. Show that z^21 + z 22 + z^23 + = z 1 z 2 + z 2 z 3 + z 3 z 1.
9. Show that the equation of a straight line using complex variable z is given by ¯αz + α¯z + γ = 0 for some α ∈ C and γ ∈ R.
10. If |z| = 1 and z 6 = 1, then show that

1 + z 1 − z =^ ib^ for some^ b^ ∈^ R.

11. For n ∈ N, derive a formula for the nth^ root of a complex number z using its polar representation.
12. Let S^2 be the unit sphere in R^3 with centre at the origin, i.e., S^2 := {(α, β, γ) ∈ R^3 : α^2 + β^2 + γ^2 = 1}. Show that the steriographic projection

z := x + iy 7 →

2 x 1 + |z|^2

, 2 x 1 + |z|^2

, |z|

1 + |z|^2

is a bijective continuous function from C onto S^2 \ {(1, 0 , 0)} with its inverse (α, β, γ) 7 → α^ +^ iβ 1 − γ

which is also continuous.

13. Show that the functions z 7 → Re(z), z 7 → Im(z), z 7 → |z| are continuous functions on C.
14. Show that lim z→ 0 z |z|

does not exist.

10 Analytic Functions

whenever z is in some neighbourhood of z 0.

The following theorem can be proved (exercise) using arguments similar to the real case of real valued functions of a real variable.

Theorem 2.1.1 Let z 0 be an interior point of Ω ⊆ C. Then the following holds.

(i) If f differentiable at z 0 ∈ Ω, then f is continuous at z 0. (ii) If f and g are differentiable at z 0 ∈ Ω, then f + g and f g are differentiable at z 0 , and (f +g)′(z 0 ) = f ′(z 0 )+g(z 0 ), (f g)′(z 0 ) = f ′(z 0 )g(z 0 )+f (z 0 )g′(z 0 ).

(iii) If f and g are differentiable at z 0 ∈ Ω and if g(z 0 ) 6 = 0, then f /g is differentiable at z 0 , and ( f g

(z 0 ) =

g(z 0 )f ′(z 0 ) − g′(z 0 )f (z 0 ) [g(z 0 )]^2.

(iv) If f is differentiable at z 0 ∈ Ω and g is differentiable in a neighbourhood of f (z 0 ), then g ◦ f is differentiable at z 0 and (g ◦ f )′(z 0 ) = g′(f (z 0 )f ′(z 0 ).

Now, let us write f (z) as u(z) + iv(z), where u(z) = Ref (z) and v(z) = Imf (z). Recall that f is differentiable at z 0 ∈ Ω if and only if there exists c ∈ C such that

R(z) |z − z 0 |

→ 0 as z → z 0 ,

where R(z) = f (z) − f (z 0 ) − c(z − z 0 ). Writing

z = x + iy, z 0 = x 0 + iy 0 , c = a + ib,

we have

R(z) = f (z) − f (z 0 ) − c(z − z 0 ) = [u(z) − u(z 0 )] + i[v(z) − v(z 0 )] − (a + ib)[(x − x 0 ) + i(y − y 0 )] = [u(z) − u(z 0 ) − a(x − x 0 ) + b(y − y 0 )] +i[v(z) − v(z 0 ) − b(x − x 0 ) − a(y − y 0 )] = R 1 (z) + iR 2 (z),

Differentiation 11

where R 1 (z) = u(z) − u(z 0 ) − [a(x − x 0 ) − b(y − y 0 )], R 2 (z) = v(z) − v(z 0 ) − [b(x − x 0 ) + a(y − y 0 )].

Thus,

R(z) |z − z 0 | →^0 ⇐⇒^

R 1 (z) |z − z 0 | →^0 &^

R 2 (z) |z − z 0 | →^0

if and only if u and v are differentiable as a functions of two real variables at (x 0 , y 0 ), and

a = ∂u ∂x

(x 0 , y 0 ), −b = ∂u ∂y

(x 0 , y 0 ),

b = ∂v ∂x

(x 0 , y 0 ), a = ∂v ∂y

(x 0 , y 0 ),

i.e.,

∂u ∂x (x^0 , y^0 ) =^

∂v ∂y (x^0 , y^0 )^ &^

∂u ∂y (x^0 , y^0 ) =^ −^

∂v ∂x (x^0 , y^0 ),

and in that case

f ′(z 0 ) = a + ib =

∂u ∂x (x^0 , y^0 ) +^ i

∂v ∂x (x^0 , y^0 ) = ∂v∂y (x 0 , y 0 ) − i ∂u∂y (x 0 , y 0 ).

Thus, we have proved the following theorem.

Theorem 2.1.2 The function f is differentiable at z 0 ∈ Ω if and only if its real part u and imaginary part v are differentiable at (x 0 , y 0 ) and ux, uy, vx, vy satisfy the equations

ux(z 0 ) = vy(z 0 ), uy(z 0 ) = −vx(z 0 ), (∗)

and in that case

f ′(z 0 ) = ux(z 0 ) + ivx(z 0 ) = vy(z 0 ) − iuy(z 0 ). Equations in (∗) are called the Cauchy-Riemann equations, or in short CR-equations.

Now, recalling from a sufficient condition for differentiability of a real valued function of two variables, we have the following sufficient condition of differentiability of f at z 0 ∈ Ω.

Differentiation 13

Now, r^2 = x^2 + y^2 and tan θ = y/x so that

2 r ∂x∂r = 2x, 2 r ∂r∂y = 2y,

sec^2 θ ∂x∂θ = − (^) xy 2 , secθ^ ∂θ∂y =^1 x ,

i.e., ∂r ∂x =^

x r ,^

∂r ∂y =^

y r , x^2 + y^2 x^2

∂θ ∂x

= − y x^2

, x

(^2) + y 2 x^2

∂θ ∂y

=^1

x

i.e.,

∂r ∂x =^

x r ,^

∂r ∂y =^

y r ,^

∂θ ∂x =^ −^

y r^2 ,^

∂θ ∂y =^

x r^2.

Now,

∂u ∂x =^

∂u ∂r

∂r ∂x +^

∂u ∂θ

∂θ ∂x ,^

∂u ∂y =^

∂u ∂r

∂r ∂y +^

∂u ∂θ

∂θ ∂y.

Thus, ∂u ∂x =^

x r

∂u ∂r −^

y r^2

∂u ∂θ ,^

∂u ∂y =^

y r

∂u ∂r +^

x r^2

∂u ∂θ.^ (1) Similarly,

∂v ∂x

= x r

∂v ∂r

− y r^2

∂v ∂θ

, ∂v ∂y

= y r

∂v ∂r

• x r^2

∂v ∂θ

Recall that the CR-equations in Cartesian coordinates are

∂u ∂x

= ∂v ∂y

, ∂u ∂y

= − ∂v ∂x

Hence, (1) − (2) give

x r

∂u ∂r −^

y r^2

∂u ∂θ =^

y r

∂v ∂r +^

x r^2

∂v ∂θ ,^ (5) y r

∂u ∂r +^

x r^2

∂u ∂θ =^

x r

∂v ∂r −^

y r^2

∂v ∂θ.^ (6) The equations (3) − (4) imply that

r ∂u ∂r

= ∂v ∂θ

, ∂u ∂θ

= −r ∂v ∂r

These are the CR-equations in polar coordinates.

14 Analytic Functions

2.2 Holomorphic or Analytic Functions

Definition 2.2.1 Let Ω ⊆ C.

(i) A function f : Ω → C is said to be analytic at a point z 0 ∈ Ω if there exists r > 0 such that B(z 0 , r) ⊆ Ω and f is differentiable at every point in B(z 0 , r).

(ii) A function f : Ω → C is said to be holomorphic or analytic on Ω 0 ⊆ Ω if f is analytic at every point in Ω 0.

♦

• If f : Ω → C is analytic on Ω 0 ⊆ Ω, then Ω 0 is an open set.
• f : Ω → C is not analytic at a point z 0 ∈ Ω if and only if for every r > 0, there exists ζ ∈ B(z 0 , r) ∩ Ω such that f is not differentiable at ζ.

Definition 2.2.2 A complex valued function defined and analytic on the entire complex plane is called an entire function. ♦

• f : Ω → C is analytic at a point z 0 ∈ Ω if it is analytic on some open set containing z 0.
• f : Ω → C is analytic on Ω if and only if u := Re(f ) and v := Im(f ) have continuous first partial derivatives in Ω and they satisfy the CR-equations at every point in Ω.

Remark 2.2.1 In the subject of complex analysis, it is very common to say a function

f is analytic at a point z 0 ∈ C

to mean that f is defined in an open neigbourhood of z 0 and f is analytic at z 0. Usually, a function is given in terms of certain expression, and in that case, the domain of definition of f is taken to be the largest sub- set of C in which the expression makes sense. For example, consider the expression

f (z) =

z.