1.2 Polar Form of Complex Numbers
1.2.1 Vector Representation
A complex number
has two real components: the real part
and the imaginary part
. Let us write them in the form of a 2D vector, cf.
Fig.
1.1
.
Polar Form: Any complex number
can be written in polar form, (remember that
in polar coordinates)
1.2.2 Argument and Modulus
The length of the vector
representing the complex number
,
is called modulus of
.
The angle
,
is called the argument of
. The angle
is usually restricted to
, even though any interval of length
will do, and we sometimes use
.
1.2.3 Manipulations in Vector/Polar
Form
Addition:
Therefore, we have to add the vectors
representing the complex numbers.
Complex Conjugate:
The angle becomes negative: check what this
means geometrically!
Multiplication:
This means that
Try to sketch an example for this in the
diagram for yourself!
1.2.4 Complex exponential
Let us differentiate a complex number
of unit modulus,
w.r.t.
.
|
(1.24) |
In short,
This suggests that we can write
as
. We shall construct further proof the consistency of
this suggestion (which can also be used to define the
complex exponent) below.
1.2.5 De Moivre’s Theorem
We can easily generalise the multiplication
of two complex numbers in polar form to calculcate an arbitrary
power of
,
(integer
):
Since
, this means
which is a useful equation for proving
trigonometric identities; it is also useful for doing many
integrals that occur in physics.
Example:
Clearly, this is entirely consistent with the
properties of the exponent:
with
and
is
.