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Complex Numbers

December 6, 2005

1 Introduction of Complex Numbers
1.1 Definition
A complex number z is a pair of real numbers

z = (a, b) (1)

Definition of addition:

(a, b) + (c, d) = (a + b, c + d) (2)

Definition of multiplication:

(a, b) · (c, d) = (ac − bd, ad + bc) (3)

Definition of multiplication by a scalar

(a, b) · c = (ac, bc) (4)

Note that multiplying by a scalar c yeilds the same result as muliplying by the
complex number (c, 0).

2 Notation
2.1 The zero
Adding (0, 0) to a complex number leaves the latter unchanged, so (0, 0) is the
"0" of addition.
The additive inverse of z = (a, b) is (−a, −b) since

(a, b) + (−a, −b) = (0, 0) (5)

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2.2 The unity
Multiplying a complex number by (1, 0) leaves the former unchanged, so (1, 0)
is the "1" of multiplication. ³ ´
The multiplicative inverse of (a, b) is a2 +b2 , a2−b 2 since (try it!)
a
+b

µ ¶
a −b
(a, b) · , = (1, 0) (6)
a2 + b2 a2 + b2

2.3 The complex conjugate
The complex conjugate of z = (a, b) denoted by z = (a, b), is defined to be
¯
(a, −b):
z = (a, −b)
A complex number times its complex conjugate has 0 for the second compo-
nent ¡ ¢
(a, b) · (a − b) = a2 + b2 , 0 (7)

2.4 Real and Imaginary parts
Two real valued functions, Re and Im, are defined on the field of complex
numbers

Re (a, b) = a (8)
Im (a, b) = b (9)

2.5 Absolute value
Another real valued function, the absolute value – denoted by |(a, b)|, is defined
for complex numbers: p
|(a, b)| = a2 + b2 (10)
This is the length of the vector (a, b) in the 2D sense.

2.6 Argument
The argument of the comlex number also comes from the vector intepreation.
It is the angle formed by the vector and the positive x-axis quoted so that it
falls in (−π, π]. The function is denote by α = Arg (a, b).

2.7 The i notation
In order to remember the complicated and seemingly arbitrary definition of
multiplication, use the following pneumonic rule. Write

(a, b) = a + ib (11)

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and, when multiplying, pretend that it is real addition and multiplication by i
and that i2 = −1. Then

(a, b) · (c, d) = (a + ib) (c + id) (12)
= ac + i2 bd + i (ad + bc) (13)
= ac − bd + i (ad + bc) (14)
= (ac − bd, ad + bc) (15)

so it works.

3 Connection to trigonometry
3.1 The basic relationship
Recall that

sin (α + β) = sin α cos β + cos α sin β (16)
cos (α + β) = cos α cos β − sin α sin β (17)

Therefore,

(cos α + i sin α) (cos β + i sin β) = cos (α + β) + i sin (α + β) (18)

3.2 Euler’s notation
Deﬁne eiα according to
eiα = cos α + i sin α (19)
Then
eiα eiβ = ei(α+β) (20)
so the exponentiation formally works.
Everything that you might guess works actually works:
¡ iα ¢−1
e = e−iα (21)
iα
de
= ieiα (22)
dα

3.3 A pretty relation
The equation
eiπ = −1 (23)
expresses a relationship among the most fundamental constant e, π, 1 and, if
you believe that i is a number, i.

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3.4 Roots of unity
The equation
xN = 1 (24)
always has exactly N roots in complex numbers. They can be expressed as
2πn
x = ei N , 0 ≤ n < N. (25)

The equation
xN = eiα (26)
α
also has exactly N roots. They are the same roots as 1 except "turned" by ei N
α+2πn
x = ei N (27)

4 Polar form of the complex number
4.1 Deﬁnition
A compex number z = a + ib can be rewritten as
p µ ¶
a b
z = a + ib = a2 + b2 √ + i√ (28)
a2 + b2 a2 + b2
√
Let r = a2 + b2 and α be the angle between −π and π such that
a
cos α = √ (29)
a2 + b2

b
sin α = √ (30)
a2 + b2
You recognize that r = |z| and α = Arg (z). Then

z = reiα (31)

This is the polar form of the complex number.

4.2 Multiplication
Let z1 = r1 eiα1 and z2 = r2 eiα2 . Then

z1 z2 = r1 eiα1 r2 eiα2 (32)
= r1 r2 ei(α1 +α2 ) (33)

Multiplying two complex numbers is equivalent to multiplying their absolute
values and adding their agruments.
If z = reiα , then 1/z = (1/r) e−iα .

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5 Analytic functions
5.1 Introduction
For the sake of this writeup, a complex function f of z is called analytic if it
is a "valid" expression purely in terms of z. By "valid" we mean (let a be a
complex constant)

az n (34)
eaz (35)
ln z (36)
cos z, sin z (37)

and sums, products, and compozition thereof. "Invalid" are the following ex-
pressions

z
¯ (38)
Arg z (39)
|z| (40)
Re z, Im z (41)

Analytic functions will have a number of very many attractive and useful prop-
erties.

5.2 Deﬁnitions
Letting z = x + iy or, in polar form, z = reiα . Then

zn = rn (cos nα + i sin nα) (42)
n
zn = (x + iy) (43)
ez = ex (cos y + i sin y) (44)
ln z = ln r + iα (45)
eiz + e−iz
cos z = (46)
2
eiz − e−iz
sin z = (47)
2
cz = ez ln c (48)

6 Cauchy-Riemann Equations
Suppose that z = x + iy and that f (z) is analytic and

f (z) = u (x, y) + iv (x, y) (49)

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Then
∂u ∂v
= (50)
∂x ∂y
∂u ∂u
= − (51)
∂y ∂x
It follows (see Exercises) that both u and v are harmonic:
uxx + uyy = 0 (52)
vxx + vyy = 0 (53)

7 Exercises
1. Show that eln z = z and that ln ez = z. In other words, ez and ln z are, in
fact, the inverses of each other.
2. Show that if u (x, y) and v (x, y) satisfy the Cauchy-Riemann equations,
then both u (x, y) and v (x, y) are harmonic.
3. Show that all basic analytic functions z, ez , and ln z satisfy the Cauchy-
Rieman equations.
4. Suppose that f1 (z) and f2 (z) are analytic according to the formal deﬁn-
ition. Show that
a. f1 (z) + f2 (z) is analytic (easy)
b. cf1 (z) is analytic, where c = a + ib is a complex constant (easy)
c. f1 (z) f2 (z) is analytic (more diﬃcult). Note that b. is a special case of c.
d. f1 (f2 (z)) is analytic.
You have now shown that any "valid" expression of z is an analytic function.
¡ ¢
For example, cos ez + z 2 is analytic and its real an imaginary parts (both
exeedingly cumbersome) are harmonic.
5. Reduce to the form a + bi:
π
ei 2 (54)
eiπ (55)
eπ/6 (56)
i5 (57)
5i (58)
ii (59)
i
ii (60)
6.
a. Derive an expression for arccos z.
b.
arccos i (61)
arccos 1 (62)
arccos 2 (63)

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Complex numbers

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