Complex Analysis Mario Bonk, Study notes of Algebra

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Complex Analysis

Mario Bonk

Course notes for Math 246A and 246B

University of California, Los Angeles

• Preface
• 1 Algebraic properties of complex numbers
• 2 Topological properties of C
• 3 Differentiation
• 4 Path integrals
• 5 Power series
• 6 Local Cauchy theorems
• 7 Power series representations
• 8 Zeros of holomorphic functions
• 9 The Open Mapping Theorem
• 10 Elementary functions
• 11 The Riemann sphere
• 12 M¨obius transformations
• 13 Schwarz’s Lemma
• 14 Winding numbers
• 15 Global Cauchy theorems
• 16 Isolated singularities
• 17 The Residue Theorem
• 18 Normal families
• 19 The Riemann Mapping Theorem
• CONTENTS
• 20 The Cauchy transform
• 21 Runge’s Approximation Theorem
• References

1 Algebraic properties of complex numbers

1.1. Intuitive idea. Complex numbers are expressions of the form a + bi with real numbers a and b. Here i is the imaginary unit. One computes (i.e., adds and multiplies) with complex numbers as usual, but sets i^2 = i·i := −1. For example,

(3 + 5i)(2 + i) = 6 + 10i + 3i + 5i^2 = 6 + 13i − 5 = 1 + 13i.

1.2. Definition of the complex numbers. For a rigorous definition we let the set of complex numbers be

C := {(a, b) : a, b ∈ R}

with the correspondence (a, b) ∼= a + bi in mind. Addition and multiplication are defined accordingly:

(a, b) + (c, d) := (a + c, b + d), (a, b) · (c, d) := (ac − bd, ad + bc).

One often omits the multiplication sign and writes zw := z · w for z, w ∈ C. One also uses the convention that multiplication binds stronger than addition. So u + vw = u + (v · w) for u, v, w ∈ C, etc. That one computes with complex numbers “as usual” is mathematically expressed by the following fact.

Theorem 1.3. (C, +, ·) is a field, that is:

1. (C, +) is an abelian group, which means that

1.1 the addition + is associative,

1.2 there exists a neutral element 0 := (0, 0) ∈ C with respect to addition,

1.3 every element (a, b) ∈ C has an (additive) inverse (−a, −b) ∈ C,

1.4 the addition + is commutative.

2. (C, +, ·) is a commutative ring, which means that

1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 10

Definition 1.5. Let z = a + bi ∈ C, where a, b ∈ R. We define

(i) Re(z) := a (the real part of z),

(ii) Im(z) := b (the imaginary part of z),

(iii) ¯z := a − bi (the complex conjugate of z),

(iv) |z| :=

a^2 + b^2 (the absolute value of z).

1.6. Geometric interpretations. These concepts and also addition of complex numbers have obvious geometric interpretations if one identifies z = a + bi with the point (a, b) in the plane R^2.

1.7. Subtraction and division. As in every field one can define a notion of subtraction and division of complex numbers. Namely, if z ∈ C, one denotes the additive inverse of z by −z and defines

w − z := w + (−z)

for z, w ∈ C. If z = a + bi, w = c + di, then −z = (−a) + (−b)i, and so

w − z = (c − a) + (d − b)i.

If z 6 = 0, then we denote by z−^1 the (multiplicative) inverse of z. For z = a + bi 6 = 0 we have

z−^1 =

a a^2 + b^2

b a^2 + b^2

i.

One defines z/w =

z w

:= w · z−^1.

One can compute with fractions of complex numbers as usual (as in any field). Using the fact that z z¯ = |z|^2 , one can simplify fractions of complex numbers by multiplying in numerator and denominator by the complex conjugate of the denominator. For example,

2 + i 3 + 5i

(2 + i)(3 − 5 i) (3 + 5i)(3 − 5 i)

=

(6 + 5) + (−10 + 3)i 32 + 5^2 =

11 − 7 i 34

i.

1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 11

Theorem 1.8. Let z, w ∈ C. Then

(i) Re(z) = 12 (z + ¯z),

(ii) Im(z) = (^21) i (z − ¯z),

(iii) Re(z + w) = Re(z) + Re(w),

(iv) Im(z + w) = Im(z) + Im(w),

(v) z ∈ R iff Im(z) = 0 iff z = ¯z,

(vi) ¯z = z,

(vii) z + w = ¯z + ¯w, z − w = ¯z − w¯,

(viii) z · w = ¯z · w¯,

(ix)

( (^) z w

¯z w ¯

(x) z · ¯z = |z|^2 ,

(xi) |z| = 0 iff z = 0,

(xii) |z · w| = |z| · |w|,

(xiii)

z w

∣ =^

|z| |w|

(xiv) |z + w| ≤ |z| + |w| (triangle inequality).

Proof. The proofs of these facts are straightforward, often tedious, and we omit the details; as an example, we will prove (xii). If z = a + bi and w = c + di, where a, b, c, d ∈ R, then

z · w = (ac − bd) + (ad + bc)i,

1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 13

(v) ew^ = ez^ iff w = z + 2πik with k ∈ Z.

Proof. (i) Obvious from the definition.

(ii) If z = x + iy and w = u + iv with x, y, u, v ∈ R, then

z + w = (x + u) + i(y + v).

Note that

cos(y + v) = cos y cos v − sin y sin v, sin(y + v) = sin y cos v + cos y sin v.

Hence

ez^ · ew^ = ex(cos y + i sin y)eu(cos v + i sin v), = ex+u

(cos y cos v − sin y sin v) + i(sin y cos v + cos y sin v)

= ex+u(cos(y + v) + i sin(y + v)) = e(x+u)+i(y+v)^ = ez+w.

(iv) Let z = x + iy, x, y ∈ R. Then

ez^ = 1 ⇔ ex(cos y + i sin y) = 1 ⇔ ex^ cos y = 1 and ex^ sin y = 0, ⇔ ex^ cos y = 1 and sin y = 0, ⇔ ex^ cos y = 1 and y = nπ for n ∈ Z, ⇔ ex(−1)n^ = 1 and y = nπ for n ∈ Z, ⇔ ex^ = 1 and y = nπ for some n ∈ Z even, ⇔ x = 0 and y = nπ for some n ∈ Z even, ⇔ z = x + iy = 2πik for k ∈ Z.

(iii) Using (ii) and (iv) we have

ez+2πi^ = ez^ · e^2 πi^ = ez^ · 1 = ez

for z ∈ C.

1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 14

(v) Note that ez^6 = 0 for z ∈ C, because ez^ · e−z^ = e^0 = 1 6 = 0. So for z, w ∈ C we have by (iv),

ew^ = ez^ ⇔ ew−z^ = ew^ · e−z^ = ez^ · e−z^ = e^0 = 1, ⇔ w − z = 2πik for some k ∈ Z, ⇔ w = z + 2πik for some k ∈ Z.

1.11. Mapping properties of exp. The exponential function maps lines parallel to the real axis to rays starting at 0; lines parallel to the imaginary axis are mapped to circles centered at 0. The exponential function maps the strip

S = {x + iy : x ∈ R, 0 < y < 2 π}

bijectively onto C \ [0, ∞).

1.12. Polar coordinates. If z = x + iy, x, y ∈ R, then by using polar coordinates we can write

x = r cos ϕ, y = r sin ϕ,

where r ≥ 0 and ϕ ∈ R. Hence every complex number can be written as

z = x + iy = r(cos ϕ + i sin ϕ) = reiϕ,

where r ≥ 0 and ϕ ∈ R. Note that r = |z| is the absolute value of z. The angle ϕ is called the argument of z, written ϕ = arg(z). It is only determined up to integer multiples of 2π. If Re(z) 6 = 0, then

tan ϕ =

Im(z) Re(z)

This formular allows the computation of ϕ for given z.

1.13. Geometric interpretation of multiplication and division of complex numbers. Let z = reiα^ and w = seiβ^ , where r, s ≥ 0 and α, β ∈ R. Then z · w = reiαseiβ^ = rsei(α+β),

1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 16

We conclude that a complex number a = reiϕ, a 6 = 0, has n distinct nth roots zk = n

reiαk^ ,

where

αk =

ϕ n

2 π n

k with k ∈ { 0 ,... , n − 1 }.

1.15. Examples. (a) Third roots of a = −8: We have a = −8 = 8eiπ; so

zk =

8 eiαk^ = 2eiαk^ ,

αk = π 3 + 23 π k with k ∈ { 0 , 1 , 2 }.

Hence α 0 = π 3 , α 1 = π, α 2 = 53 π ,

and

z 0 = 2 eiπ/^3 = 2(cos π 3 + i sin π 3 ) = 2(^12 + i

√ 3 2 ) = 1 +^ i

z 1 = 2 eiπ^ = − 2 , z 2 = 2 ei^5 π/^3 = 2(cos 53 π + i sin 53 π ) = 2(−^12 − i

√ 3 2 ) =^ −^1 −^ i

(b) Computation of square roots by a different method: To solve the equation z^2 = −3 + 4i,

for example, we use the ansatz z = a + bi with a, b ∈ R and solve for a and b:

z^2 = (a + bi)^2 = a^2 − b^2 + 2abi = −3 + 4i.

Hence a^2 − b^2 = − 3 and 2 ab = 4. (1)

Squaring both equations and adding leads to

(a^2 − b^2 )^2 + 4a^2 b^2 = a^4 + 2a^2 b^2 + b^4 = (a^2 + b^2 )^2 = (−3)^2 + 4^2 = 25.

Thus, a^2 + b^2 = 5.

Combining this with (1) gives

2 a^2 = 2 ⇒ a^2 = 1 ⇒ a 1 = 1, a 2 = − 1

1 ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS 17

and b 1 = 2, b 2 = − 2.

So we get the solutions

z 1 = 1 + 2i and z 2 = − 1 − 2 i.

Note that z 2 = −z 1 as it should be.

(c) Solutions of quadratic equations can be computed as usual by complet- ing the square, etc. A more general fact is true: Every polynomial equation

zn^ + an− 1 zn−^1 + · · · + a 1 z + a 0 = 0,

where a 0 ,... , an− 1 ∈ C has a solution z ∈ C. This Fundamental Theorem of Algebra will be proved later in this course.

2 TOPOLOGICAL PROPERTIES OF C 19

Remark 2.4. Every point in X is an interior point, an exterior point, or a boundary point of M , and these cases are mutually exclusive. An interior point of M always belongs to M , while an exterior point always lies in the complement of M in X. A boundary point of M may or may not belong to M.

Definition 2.5. Let (X, d) be a metric space, and M ⊆ X. Then M is called open if it has only interior points, i.e., for all x ∈ M there exists  > 0 such that B(x, ) ⊆ M. The set M is called closed if it contains all of its boundary points.

Example 2.6. D := {z ∈ C : |z| < 1 } is an open set in C, called the open unit disk. The set D := {z ∈ C : |z| ≤ 1 } is a closed set in C, called the closed unit disk. The set M = D ∪ { 1 } is neither open nor closed.

Definition 2.7. Let (X, d) be a metric space, and M ⊆ X. Then M := M ∪ ∂M is called the closure of M. One can show that M is the smallest closed set in X containing M. Our notation for the closed unit disk D was motivated by the fact that this set is the closure of D.

Theorem 2.8. Let (X, d) be a metric space. Then the following statements are true:

(i) a set in X is

open closed

if its complement is

closed open

(ii)

a union an intersection

of a family of

open closed

sets in X is

open closed

(iii)

an intersection a union

of a finite family of

open closed

sets in X is

open closed

Remark 2.9. Suppose X is a set together with a family O of its subsets called open sets. If ∅, X ∈ O and if the properties (ii) and (iii) in the previous theorem are satisfied, then (X, O) is called a topological space and the system O a topology on X. By what we have seen, every metric d on a set X determines a natural system O of open sets that form a topology on X. One calls this the topology induced by d.

2 TOPOLOGICAL PROPERTIES OF C 20

Definition 2.10. Let (X, d) be a metric space, and {xn} be a sequence of points in X.

(i) The sequence {xn} is called convergent if there exists a point x ∈ X (the limit of {xn}) such that for all  > 0 there exists N ∈ N such that for all n ∈ N with n ≥ N we have d(x, xn) < . One can show that if the limit x exists, then it is unique, and one writes x = lim n→∞ xn or simply xn → x.

(ii) The sequence {xn} is called a Cauchy sequence if for all  > 0 there exists N ∈ N such that for all n, k ∈ N with n, k ≥ N we have d(xn, xk) < .

(iii) A point x ∈ X is called a sublimit of {xn} if there exists a subsequence {xnk } of {xn} such that lim k→∞

xnk = x. In this case, we say that {xn} subconverges to x.

Proposition 2.11. A sequence {zn} in C converges if and only if the se- quences {Re(zn)} and {Im(zn)} of real numbers converge. In case of convergence we have

lim n→∞ zn = lim n→∞ Re(zn) + i lim n→∞ Im(zn).

Proof. ⇒: Suppose {zn} converges, and let z := lim n→∞

zn. Note that | Re(w)| ≤

|w| for w ∈ C. Hence

| Re(zn) − Re(z)| ≤ |zn − z|

for n ∈ N. This implies that lim n→∞

Re(zn) = Re(z). Similarly, lim n→∞

Im(zn) =

Im(z).

⇐: Suppose {Re(zn)} and {Im(zn)} converge. Let a := lim n→∞ Re(zn),

b := lim n→∞ Im(zn), and z := a + bi. Note that for all w ∈ C we have

|w| =

Re(w)^2 + Im(w)^2 ≤ | Re(w)| + | Im(w)|.

Hence |zn − z| ≤ | Re(zn) − a| + | Im(zn) − b|.

It follows that lim n→∞ zn = z.