0:01

Okay. Now we have the mathematical background

to be able to discuss the complex exponential function and to derive

Euler's Formula. So now to do this, we have to use

something that we're not going to be able to derive here, which is the Taylor

Series Expansions for functions. Now there's a nice Wikipedia article on

this. So if you're not familiar with Taylor

series expansions from perhaps the first calculus course, go take a look at that.

And that will explain what's going on. But the idea of a Taylor Series

Expansion, this is actually a Maclaurin Series Expansion, which is a Taylor

Series around x equals 0. But what this is, is I can take a

function like sin x and I can approximate it with a polynomial function in

increasing powers of x. Now, this is the particular Taylor Series

Expansion for sin x. Sin x starts off for small x looking like

x. And then the first correction to that

would be an x cubed over 3 factorial. And remember that n factorial is just n

times n minus 1 times n minus 2, all the way down to 1.

So three factorial is 3 times 2 times 1 is 6.

And then I add to that x to the fifth term and x to the seventh term.

And so as I add more and more terms, this becomes a better approximation of the

value of sin of x. And so there's a way to find this the

expansion for a particular function by taking successive derivatives and it's

fairly straight forward to, to come up with these formulas.

Now, at the same time, there's a Taylor Series expansion for cos x.

So you know that for small values of x, cos x is about equal to 1.

And then as x starts to deviate from zero, I have to add these correction

terms. And in the cosine expansion, I only have

even powers of x and it's always, whatever the exponent is up here, I have

that factorial downstairs. And, notice that the signs alternate in

both of these expansions. Now, the last Taylor Series Expansion

that we need in the proof is that for e to the x.

Now, e to the x is all positive signs, and it's also all powers of x.

So that's the third Taylor Series Expansion that we need.

Now, what I want to do is consider the complex exponential, e to the j phi, or

phi. so I'm going to just plug this in to the

Taylor Series Expansion, where x is j times phi, and write that out.

So this is 1 plus j phi to the first plus j phi squared over 2 factorial, and just

write all of those out. Now I have to raise j to successive

powers. So I have j squared.

Well, that j squared is negative 1. And so I'm going to have this term is

going to be negative 1 times phi squared over two factorial, and it's real.

So I'm going to group all of the real terms together.

So the first one I have is 1, then I have this one, phi squared over 2 factorial

with a minus sign, so that's from that term.

Now the next one that's going to be real is j to the fourth.

j to the fourth is plus 1. So this is phi to the fourth over 4

factorial, with a plus sign. And so all the even powers here are going

to contribute terms, real terms. And the odd terms, the first one I have

is just j times phi over 1 factorial. So there's the first one.

The next one is j cubed. Well, j cubed is minus j.

So I have a minus 1 times j, and then phi cubed over 3 factorial.

And then the fifth order term is going to give me a plus j, again.

And so I pick up these alternating signs and all of the terms have odd powers,

here. But everything is multiplied by j.

So here's all of the real terms, come from all of the even powers and here are

all of the odd terms that come from the odd powers.

And this is, I'm sorry, this is all of the imaginary terms that come from the

odd powers. Now, if you look at this, this is the

Taylor series expansion for cosine. And this is, is the Taylor Series

Expansion for sine. So you can identify that e to the j phi

is just cosine phi plus j sine phi. So, by using the Taylor Series Expansions

for all three functions in plugging in j phi, where x is, I can prove that this

identity holds. And that's called Euler's formula.

5:36

Okay. Now, let's take a look at Euler's formula

and there's a remarkable equation that you can derive from that.

So here's Euler's formula, e to the j phi is cosine phi plus j sine phi.

I'm going to plug phi equals pi into this equation.

So it's cos pi plus j sin pi. Well cos pi is negative 1.

sin pi is 0. And so I get e to the j pi equals

negative 1. Or I can rewrite that as e to the j pi

plus 1 equals 0. Now this really is a remarkable equation

because it combines that a variety of numbers that you had no idea that they

were linked together in such in a nice, simple, elegant way.

So, you have 1 and 0 the fundamental numbers in, in arithmetic and counting.

pi, the ratio of the circumference to the diameter of the circle.

That you're you would've encountered in grade school somewhere.

And then j, the square root of negative 1, and e, this number that pops out of a

number of calculus problems. So it turns out that all of these numbers

are somehow affiliated together in this equation.

And they, they seem to come from very disparate areas of mathematics, but the,

they're all linked together by that equation.

7:07

Now, what we want to do is we're going to be using the, the complex exponential

form a lot in the analysis that's going to follow.

So, r e to the j phi is just this. It's just r times e to the j phi.

So it's the cos phi plus j sin phi, both terms multiplied by r.

Now, it's always important to keep, I think it's important to keep a geometric

picture in your mind of what's going on. It's, it's easier for me to think in

terms of pictures. so a particular point in the complex

plane or a value of this function for a given value of r and phi is that

corresponds to some point. And I have it's a distance r from the

origin at an angle phi. in, it, in a direction that's an angle

phi off of the x axis. And the real part of this is the

projection back onto the real axis. And that's just r cos fi.

The imaginary part of this complex number here is the projection onto the imaginary

axis and that's r sin phi. Now, let's take a look at one last thing

to set up a discussion of phasers. If I plug in this equation.

I'll just make r1. And let's plug in a value of phi equal to

0. So the first for that value, I just get

cos 0 plus j sin 0. Well I know that cos 0 is 1, sin 0 is 0.

So this is just the real number 1. Now, what I want to do is plot this on

the complex plane. So the real and imaginary axis, here's

one unit over the real axis. Here's one unit up the imaginary axis.

That's, I'll label that j and that first point, e to the j0 is right here on the

real axis at position 1. Now, if I increment this by pi over 2',

now you think of this as multiplying e to the j0 by e to the j pi over 2,' and I

add the exponents when I multiply, so that brings me to e to the j pi over 2.

If I plug pi over 2 into Euler's Formula, I get cos pi over 2, which, you know is

0. And sin pi over 2, sin of 90 degrees is

1. So this whole expression becomes j.

Now that's up here. So this is useful to keep in mind,

multiplying by this factor, e to the j pi over 2, is like rotating 90 degrees in

the complex plane. So you can think of this, the operation

of multiplying this factor as sort of a rotation, in the complex plane.

Now, if I take and apply another rotation by multiplying by e to j pi over 2 again,

so I multiply this by e to the j pi over 2.

And if I add the exponents I, I'm here now at e to the j pi.

And if I plug in pi into Euler's Formula, I get cos pi as negative 1, sin pi is 0.

This is the same formula that's up here. And If I plot that one, I'm over here.

So I'm 1 unit out on the negative, in the negative direction on the real axis.

So that's like an additional rotation by pi over 2.

And if I do it one more time, then that brings me around to this point.

So if you plug in look up cos 3 pi over 2 and sin 3 pi over 2, this is negative 1.

That's 0. And that brings me here.

And then one additional rotation or multiplying this one more time by e to

the j pi over 2, brings me to e to the j 2 pi.

And I'm back where I started. Cos of 2 pi is 1.

Sin of 2 pi is 0. And so this is the same point again.

So, every time I multiply by e to the j pi it's like a rotation by 90 degrees in

the complex plain.