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May 23rd

May 23rd

# A Remarkable Formula, and Its Applications

The purpose of this document is to tie together a few threads from this
course. Trigonometric functions , exponential functions, complex numbers and
polynomials are all intimately related to one another. Let me take you on a
quick tour.

1 The Keystone

There is the remarkable formula

Why is this true? I cannot answer that right now. If you continue studying
math, you will find out in calculus II.

But this formula really is neat. Check this out:

That's right. . That's unexpected and maybe cool, but it is
probably not so useful. Are there applications of this formula that are useful?

2 The Master Trigonometric Identity

We will be embarking on a study of trigonometric identities. A couple of things
to remember about such identities: First, there are infinitely many of them.
Second, all of the most important ones come straight from the keystone above.

Example: Calculate .

Solution : On the one hand we expect that by addition of expo -
nents. Keep this in mind. On the other hand:

Therefore,

How about another example. Let's try

Example: Calculate .

Solution On the one hand we expect that

On the other hand:

Thus,

Now, two complex numbers are equal if and only if their real and imaginary
parts are equal. Therefore we pick up two identities:

and

All of the other formulas given in sections 7.3-7.6 of the textbook can be
found by similar arguments.

So, this is the master trigonometric identity. This is cool and maybe a
little useful (but do we really want to re-derive these identities all the time?).
Nonetheless, perhaps we were hoping for a bigger payoff . There is some payoff
coming. First we need a little detour.

3 Visualizing the Complex Numbers

As we have seen, we can invent a number i such that i2 = -1. The question
arises: where does this strange number i appear on the number line? Perhaps
we would feel better about i if we could "locate" it.

The answer is that i is not on the number line at all. Instead, i lies one unit
above the number line. More specifically, we create the "number plane" (more
usually called the complex plane), where the x-axis measures the real part of
the number and the y-axis measures the imaginary part. For example, we can
plot 2 - 3i on the number plane as follows:

This provides us with a visualization of complex numbers ana logous to rect -
angular coordinates. To wit: we can convert a point on the plane to a complex
number, or vice versa. For example, the number (-1,-5) could be interpreted as
the complex number -1 - 5i.

In this formulation, addition ( and subtraction ) has an intuitive geometric
meaning. Given 2 complex numbers = a + bi and = c + di, plot the two
points
and draw arrows to them. Then make the parallelogram with corners at
0, a + bi and c + di. The fourth corner is = (a + c) + (b + d)i. The two
figures be low show this process for the numbers -1 - 2i and 5 + i. Notice that
the sum of these two is 4 - i.

Figure 1: Visualizing the Addition of Two Complex Numbers

What about multiplication? Multiplication isn 't so easy if we're thinking in
rectangular terms, but it does have a nice geometric interpretation if we reorient
our thinking...

4 Polar Coordinates

If we have a point in the plane, we are used to describing it by telling the
horizontal distance from the origin (the x-coordinate) followed be the vertical
distance from the origin (the y-coordinate). There is another way: instead, we
could measure the total distance from the point to the origin. Let's call this r.
We then have

If we just know r, we can't identify the point. The other coordinate we need
is θ, which measures the angle the point makes with respect to the x-axis. (If
this seems strange, think about how pilots talk: "bogey 500 feet away at 3:00."
Here, r = 500 ft and 3:00 identifies the direction.) This system is called polar
coordinates.

Then, we can convert from polar to rectangular by the formulas

That is, if we know r and θ, the point we're talking about in rectangular
coordinates is (r cosθ ; r sinθ ). As a complex number, this is

It will also be very useful to convert the other way. If we know x and y, can
we find r and θ? We can. The formulas are

If x = 0, then θ is either  (if y > 0) or (if y < 0).

5 Multiplication

Now, about complex multiplication: Let , and . Then

In other words: When we multiply two complex numbers together, the new
length is the product of the old lengths and the new angle is the sum of the old
angles.

Example: Calculate

Solution: Let and . We'll start by converting these
to polar form.

Now, multiply.

In other words, the result has a length of and makes an angle of
. It would usually be good to convert this to rectangular coordinates
(that is, make it of the form a + bi), but taking sin and cos is not so
easy (it requires trigonometric identities like perhaps the half angle formulas).
Nonetheless, look at the geometry of this situation, particularly the angles:

Figure 2: Visualizing the Multiplication of Two Complex Numbers

As a consequence of this viewpoint, notice that if we have a complex number
in polar form , then we have

and generally,

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