Polar & Exponential Form
Most people are familiar with complex numbers in the form ,
however there are some alternate forms that are useful at times. In this section we’ll look at both of those
as well as a couple of nice facts that arise from them.
Geometric Interpretation
Before we get into the alternate forms we should first take
a very brief look at a natural geometric interpretation to a complex numbers
since this will lead us into our first alternate form.
Consider the complex number . We can think of this complex number as either
the point in the standard Cartesian coordinate system or
as the vector that starts at the origin and ends at the point . An example of this is shown in the figure below.
In this interpretation we call the xaxis the real axis and
the yaxis the imaginary axis. We often
call the xyplane in this
interpretation the complex plane.
Note as well that we can now get a geometric interpretation
of the modulus. From the image above we can see that is nothing more than the length of the
vector that we’re using to represent the complex number . This interpretation also tells us that the
inequality means that is closer to the origin (in the complex plane)
than is.
Polar Form
Let’s now take a look at the first alternate form for a
complex number. If we think of the nonzero
complex number as the point in the xyplane
we also know that we can represent this point by the polar coordinates ,
where r is the distance of the point
from the origin and is the angle, in radians, from the positive xaxis to the ray connecting the origin
to the point.
When working with complex numbers we assume that r is positive and that can be any of the possible (both positive and
negative) angles that end at the ray.
Note that this means that there are literally an infinite number of
choices for .
We excluded since is not defined for the point (0,0). We will therefore only consider the polar
form of nonzero complex numbers.
We have the following conversion
formulas for converting the polar coordinates into the corresponding Cartesian coordinates
of the point, .
If we substitute these into and factor an r out we arrive at the polar
form of the complex number,


(1)

Note as well that we also have the following formula from
polar coordinates relating r to a and b.
but, the right side is nothing more than the definition of
the modulus and so we see that,


(2)

So, sometimes the polar form will be written as,
The angle is called the argument of z and is
denoted by,
The argument of z
can be any of the infinite possible values of each of which can be found by solving
and making sure that is in the correct quadrant.
Note as well that any two values of the argument will differ
from each other by an integer multiple of . This makes sense when you consider the
following.
For a given complex number z pick any of the possible values of the argument, say . If you now increase the value of ,
which is really just increasing the angle that the point makes with the
positive xaxis, you are rotating the
point about the origin in a counterclockwise manner. Since it takes radians to make one complete revolution you
will be back at your initial starting point when you reach and so have a new value of the argument. See the figure below.
If you keep increasing the angle you will again be back at
the starting point when you reach ,
which is again a new value of the argument.
Continuing in this fashion we can see that every time we reach a new
value of the argument we will simply be adding multiples of onto the original value of the argument.
Likewise, if you start at and decrease the angle you will be rotating
the point about the origin in a clockwise manner and will return to your
original starting point when you reach . Continuing in this fashion and we can again
see that each new value of the argument will be found by subtracting a multiple
of from the original value of the argument.
So we can see that if and are two values of arg z then for some integer k
we will have,


(5)

Note that we’ve also shown here that is a parametric representation for a circle of
radius r centered at the origin and
that it will trace out a complete circle in a counterclockwise direction as
the angle increases from to .
The principal value
of the argument (sometimes called the principal
argument) is the unique value of the argument that is in the range and is denoted by . Note that the inequalities at either end of
the range tells that a negative real number will have a principal value of the
argument of .
Recalling that we noted above that any two values of the
argument will differ from each other by a multiple of leads us to the following fact.


(6)

We should probably do a couple of quick numerical examples
at this point before we move on to look the second alternate form of a complex
number.
Example 1 Write
down the polar form of each of the following complex numbers.
(a)
(b)
(c)
Solution
(a) Let’s first
get r.
Now let’s find the argument of z. This can be any angle
that satisfies (4),
but it’s usually easiest to find the principal value so we’ll find that
one. The principal value of the
argument will be the value of that is in the range ,
satisfies,
and is in the second quadrant since that is the location
the complex number in the complex plane.
If you’re using a calculator to find the value of this
inverse tangent make sure that you understand that your calculator will only
return values in the range and so you may get the incorrect value. Recall that if your calculator returns a
value of then the second value that will also satisfy
the equation will be . So, if you’re using a calculator be
careful. You will need to compute both
and the determine which falls into the correct quadrant to match the complex
number we have because only one of them will be in the correct quadrant.
In our case the two values are,
The first one is in quadrant four and the second one is in
quadrant two and so is the one that we’re after. Therefore, the principal value of the
argument is,
and all possible values of the argument are then
Now, let’s actually do what we were originally asked to do. Here is the polar form of .
Now, for the sake of completeness we should acknowledge
that there are many more equally valid polar forms for this complex
number. To get any of the other forms
we just need to compute a different value of the argument by picking n.
Here are a couple of other possible polar forms.
(b) In this
case we’ve already noted that the principal value of a negative real number
is so we don’t need to compute that. For completeness sake here are all possible
values of the argument of any negative number.
Now, r is,
The polar form (using the principal value) is,
Note that if we’d had a positive real number the principal
value would be
(c) This
another special case much like real numbers.
If we were to use (4) to find the argument we
would run into problems since the real part is zero and this would give
division by zero. However, all we need
to do to get the argument is think about where this complex number is in the
complex plane. In the complex plane
purely imaginary numbers are either on the positive yaxis or the negative yaxis
depending on the sign of the imaginary part.
For our case the imaginary part is positive and so this
complex number will be on the positive yaxis. Therefore, the principal value and the
general argument for this complex number is,
Also, in this case r
= 12 and so the polar form (again using the principal value) is,

Exponential Form
Now that we’ve discussed the polar form of a complex number
we can introduce the second alternate form of a complex number. First, we’ll need Euler’s formula,


(7)

With Euler’s formula we can rewrite the polar form of a
complex number into its exponential form
as follows.
where and so we can see that, much like the polar
form, there are an infinite number of possible exponential forms for a given
complex number. Also, because any two
arguments for a give complex number differ by an integer multiple of we will sometimes write the exponential form
as,
where is any value of the argument although it is
more often than not the principal value of the argument.
To get the value of r
we can either use (3)
to write the exponential form or we can take a more direct approach. Let’s take the direct approach. Take the modulus of both sides and then do a
little simplification as follows,
and so we see that .
Note as well that because we can consider as a parametric representation of a circle of
radius r and the exponential form of
a complex number is really another way of writing the polar form we can also
consider a parametric representation of a circle of
radius r.
Now that we’ve got the exponential form of a complex number
out of the way we can use this along with basic exponent properties to derive
some nice facts about complex numbers and their arguments.
First, let’s start with the nonzero complex number . In the arithmetic section we gave a fairly
complex formula for the multiplicative
inverse, however, with the exponential form of the complex number we can
get a much nicer formula for the multiplicative inverse.
Note that since r
is a nonzero real number we know that . So, putting this together the exponential
form of the multiplicative inverse is,


(8)

and the polar form of the multiplicative inverse is,


(9)

We can also get some nice formulas for the product or
quotient of complex numbers. Given two
complex numbers and ,
where is any value of and is any value of ,
we have
The polar forms for both of these are,


(12)



(13)

We can also use (10)
and (11)
to get some nice facts about the arguments of a product and a quotient of
complex numbers. Since is any value of and is any value of we can see that,
Note that (14) and (15)
may or may not work if you use the principal value of the argument, Arg z.
For example, consider and . In this case we have and the principal value of the argument for
each is,
However,
and so (14) doesn’t hold if we use
the principal value of the argument.
Note however, if we use,
then,
is valid since is a possible argument for i, it just isn’t the principal value of
the argument.
As an interesting side note, (15)
actually does work for this example if we use the principal arguments. That won’t always happen, but it does in this
case so be careful!
We will close this section with a nice fact about the
equality of two complex numbers that we will make heavy use of in the next
section. Suppose that we have two
complex numbers given by their exponential forms, and . Also suppose that we know that . In this case we have,
Then we will have if and only if,
Note that the phrase “if and only if” is a fancy
mathematical phrase that means that if is true then so is (16)
and likewise, if (16)
is true then we’ll have .
This may seem like a silly fact, but we are going to use
this in the next section to help us find the powers and roots of complex
numbers.