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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 x-axis the real axis and
the y-axis the imaginary axis. We often
call the xy-plane 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 non-zero
complex number as the point in the xy-plane
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 x-axis 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 non-zero 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,
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(1)
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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,
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(2)
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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 x-axis, you are rotating the
point about the origin in a counter-clockwise 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,
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(5)
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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 counter-clockwise direction as
the angle increases from to .
The principle value
of the argument (sometimes called the principle
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 principle 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.
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(6)
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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.
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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 principle value so we’ll find t |