Complex numbers
J. R. Chasnov
Department of Mathematics
Hong Kong University of Science and Technology
Fall 2007
We define the imaginary number i to be one of the two numbers that√
satisfies
the rule (i)2 = −1, the other number being −i. Formally, we write i = −1. A
complex number z is written as
z = x + iy,
where x and y are real numbers. We call x the real part of z and y the imaginary
part and write
x = Rez, y = Imz.
Two complex numbers are equal if and only if their real and imaginary parts
are equal.
The complex conjugate of z = x + iy, denoted as z̄, is defined as
z̄ = x − iy.
We have
= (x + iy)(x − iy)
zz̄
= x2 − (i)2 y2
= x2 + y 2 ;
and we define the absolute value of z, also called the modulus of z, by
|z| =
=
(zz̄)1/2
p
x2 + y 2 .
We can add, subtract, multiply and divide complex numbers to get new
complex numbers. With z = x + iy and w = s + it, and x, y, s, t real numbers,
we have
z + w = (x + s) + i(y + t);
zw
=
=
z − w = (x − s) + i(y − t);
(x + iy)(s + it)
(xs − yt) + i(xt + ys);
1
z
w
=
=
=
z w̄
w w̄
(x + iy)(s − it)
s2 + t2
(xs + yt)
(ys − xt)
+i 2
.
s2 + t2
s + t2
Furthermore,
|zw| =
=
=
p
(xs − yt)2 + (xt + ys)2
p
(x2 + y2 )(s2 + t2 )
|z||w|;
and
zw
=
=
=
(xs − yt) − i(xt + ys)
(x − iy)(s − it)
z̄ w̄.
Similarly
z
|z|
,
=
w
|w|
z
z̄
( )= .
w
w̄
It is especially interesting to consider the exponential function of an imaginary argument. Let f(θ) = eiθ with θ a real number. The function f takes a
real number and returns a complex number. We write
f(θ) = g(θ) + ih(θ),
where g(θ) and h(θ) are real functions. Taking derivatives, and using
f ′ (θ)
=
=
=
ieiθ
if(θ)
−h(θ) + ig(θ),
we have
−h(θ) + ig(θ) = g′ (θ) + ih′ (θ).
Equating real and imaginary parts of both sides, we obtain the differential
equations
g′ (θ) = −h(θ), h′ (θ) = g(θ).
We have already met the two functions sin and cos whose derivatives behave in
this fashion. Indeed, the only functions that satisfy these two equations are
g(θ) = A cos θ,
h(θ) = A sin θ,
with A any constant. Since f(0) = e0 = 1 = g(0) + ih(0) = A, we find A = 1.
Therefore,
eiθ = cos θ + i sin θ.
2
Since cos π = −1 and sin π = 0, we derive the celebrated Euler’s identity
eiπ + 1 = 0,
that links five fundamental numbers: 0, 1, i, e and π using three basic mathematical operations only once: addition, multiplication and exponentiation.
The complex number z can be represented in a complex plane with the xaxis being Rez and the y-axis being Imz. This leads to the polar representation
of z:
z = |z|eiθ .
Useful trigonometric relations can be derived using eiθ and properties of the
exponential function. The addition law can be derived from
ei(x+y) = eix eiy .
We have
cos(x + y) + i sin(x + y)
= (cos x + i sin x)(cos y + i sin y)
= (cos x cos y − sin x sin y) + i(sin x cos y + cos x sin y);
yielding
cos(x + y) = cos x cos y − sin x sin y,
sin(x + y) = sin x cos y + cos x sin y.
De Moivere’s Theorem derives from einθ = (eiθ )n , yielding the identity
cos(nθ) + i sin(nθ) = (cos θ + i sin θ)n .
For example, if n = 2, we derive
cos 2θ + i sin 2θ
=
=
(cos θ + i sin θ)2
(cos2 θ − sin2 θ) + 2i cos θ sin θ.
Therefore,
cos 2θ = cos2 θ − sin2 θ,
sin 2θ = 2 cos θ sin θ.
With a little more manipulation using cos θ + sin2 θ = 1, we can derive
cos2 θ =
1 + cos 2θ
,
2
2
sin2 θ =
1 − cos 2θ
;
2
which are useful formulas for determining
Z
Z
1
1
2
cos θdθ = (2θ + sin 2θ),
sin2 θdθ = (2θ − sin 2θ),
4
4
from which one obtains the identities
Z 2π
Z
sin2 θdθ =
0
0
3
2π
cos2 θdθ = π.