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.