L7 1 3Review The Complex Exponential

>> As we've seen, we can represent
complex numbers in terms of

either a rectangular manifestation with a

real plus an imaginary part
or z equals a plus jb.

We can also represent it in polar form,

where we have an angle Theta and
orientation from the horizontal axis and

the magnitude of z being the distance
from the origin out to the point.

I've redrawn here the equations

that relate the polar and
rectangular coordinates.

Changing from rectangular polar,

we have the magnitude of
z is equal to that from

Pythagorean's theorem
and the angle is that.

Going from polar coordinates,

where the coordinate values
are magnitude of z and Theta,

back to the rectangular manifestation
where the coordinates are a and b.

So we go back to polar,

we have a is equal to magnitude z

cosine Theta and b is equal to
the magnitude of z sine Theta.

Now, let's write the complex
number z in real form,

with a real part plus an imaginary part,

but do it in terms of
the polar coordinates.

In other words, z then is
equal to the real part a.

Let's write it like this. a plus jb.

Which is equal to, a is equal to the
magnitude of z times the cosine of

Theta plus j times the magnitude
of z times the sine of Theta.

See what I've done, I've written it in

rectangular form with a real part
plus j times the imaginary part.

But I've written the quantities in terms of

the polar values of z are maxi and Theta.

Now let's factor out the magnitude of z,

so this then is equal to,

give myself more room,

the magnitude of z times the cosine

of Theta plus j sine of Theta.

Now we look at that in brackets,

that cosine of Theta plus j sine
Theta and we recognize that from

Euler's formula as being e to the j Theta.

Thus, another way of writing
z this complex number,

what we are going to refer to as
its Complex Exponential form,

which uses the polar coordinate values
of magnitude z and Theta.

Thus Z is equal to magnitude
of z e to the j Theta.

Why get so excited about this?

In this complex exponential form we can

use all of the properties of exponents.

In other words, say we had
let's just do it over here.

Say we had one number or
one expression six e to

the 3x and we want to multiply
it by two e to the 1x.

Well, we know that in doing this type
of thing you can be a multiply

the coefficients six times two is 12.

Because they are raised to the same base
we got e to the 3x and e to the x.

When you multiply e to
the 3x times e to the x,

you simply add the exponent
e to the 3x plus x is 4x.

Similarly, if we wanted to divide
six e to the 3x by two e to the x,

you simply divide the coefficients,

six divided by two is three and you
subtract the exponents 3x minus x is 2x,

and we would have three e to the two x.

So what we're saying then
is that we can represent

complex numbers in
either rectangular form a plus jb

or polar form z maxi and Theta but that

polar form is equivalent to or can be

represented in this
complex exponential form,

where we have the magnitude of
z times e to the j angle of z.

We're going to see that
this form right here,

this polar form makes

multiplying and dividing
complex numbers very, very easy.

That's where we're headed now.