1 Additional material for MAS381: 1. Revision
of complex numbers and complex functions
The purpose of this short introductory part is to review the basics of complex
numbers and complex arithmetic.
A complex number is an expression of the form z = x + iy, where x, y ∈ R
are real and i =
√−1. The set of all complex numbers is denoted by C. We call
x = Re{z} the real part of z and y = Im{z} the imaginary part of z = x+ iy.
(Note: The imaginary part is the real number y, not iy.) A real number x is
merely a complex number with zero imaginary part, Im{z} = 0, and so we may
regard R ∈ C.
The complex conjugate of z = x+ iy is z¯ = x− iy. Complex conjugation is
compatible with complex arithmetic, i.e.
z + w = z¯ + w¯, zw = z¯w¯. (1)
In particular, the product between a complex number and its conjugate is a real
and positive number, i.e.
zz = (x+ iy)(x− iy) = x2 + y2 (2)
The square-root of the right-hand side expression is often called the modulus or
norm of the complex number z = x+ iy, and is written
|z| =
√
x2 + y2. (3)
Equation (2) can be written in terms of the modulus as
zz = |z|2. (4)
Rearranging the factors, we can deduce the formula for the reciprocal of a non-
zero complex number as
1
z
=
z
|z|2 , z 6= 0,
or equivalently
1
x+ iy
=
x− iy
x2 + y2
. (5)
The general formula for complex division can be written as
w
z
=
wz
|z|2 =
u+ iv
x+ iy
=
(xu+ yv) + i(xv − yu)
x2 + y2
. (6)
The modulus of a complex number,
r = |z| =
√
x2 + y2
is one component of its polar coordinate representation
x = r cos θ, y = r sin θ, z = r(cos θ + i sin θ). (7)
The polar angle, θ, which measures the angle that the line connecting z to the
origin makes with the horizontal axis is known as the phase or argument and it
can be written as
θ = arg(z). (8)
As such, the argument is only defined up to an integer multiple of 2pi. The
unique principal value of the argument is restricted to −pi < arg(z) < pi (please
note that this argument should not be confounded with the argument z of a
function f(z)).
Euler formula for the complex exponential reads
eiθ = cos θ + i sin θ (9)
and can be used to compactly rewrite the polar form (7) as
z = reiθ, r = |z|, θ = arg(z). (10)
We note that the modulus and the argument of a product of complex numbers
can be readily computed as
|zw| = |z| |w|, arg(zw) = arg(z) + arg(w), (11)
the latter formula requiring that we allow multiple-values arguments; the for-
mula does not hold as stated for all z, w when the principal value of the argument
is used. Similarly, the modulus and argument of the reciprocal of a non-zero
complex number are ∣∣∣∣1z
∣∣∣∣ = 1|z| , arg
(
1
z
)
= −arg(z). (12)
On the other hand, complex conjugation preserves the modulus, but negates
the argument, i.e.
|z| = |z|, arg(z) = −arg(z). (13)
The latter formula is not valid for the principal value of the argument when z
lies on the negative real axis.
The power of a complex number can be easily calculated using De Moivre’s
theorem. If the polar form of a complex number is z = r(cos θ + i sin θ), then
zn = rn(cosnθ + i sinnθ)
As a consequence of the above relation, we can write that if z1 = r1(cos θ1 +
i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then
z1z2 = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]
z1
z2
=
r1
r2
[cos(θ1 − θ2) + i sin(θ1 − θ2)]
2
A number w is called an nth root of a complex number z if wn = z, and write
w = z1/n. From the Moivre’s theorem we can show that if n is a positive integer,
z1/n = [r(cos θ + i sin θ)]1/n = r1/n
[
cos
(
θ + 2kpi
n
)
+ i sin
(
θ + 2kpi
n
)]
where k = 0, 1, 2, 3 . . . . The above relation also says that there are n different
values of z1/n for z1/n, i.e. n different nth roots of z, provided z 6= 0.
A way of representing complex numbers as points on a coordinate plane, also
known as the Argand plane or the complex plane, using the x-axis as the real
axis and the y-axis as the imaginary axis. In the diagram shown here (see Fig.
Figure 1: The representation of a complex number in the complex (or Argand)
plane
1), a complex number z is shown in terms of both Cartesian (x, y) and polar (r,
θ) coordinates. To each complex number there corresponds one and only one
point in the plane, and conversely to each point in the plane there corresponds
one and only one complex number.
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