Eulers formula

So in this video, we're going to talk about Oilers formula. One of the things I want to start out with is why. Why do we want to talk about this rather oddly looking formula? What's the big deal about this? And there is a big deal, and the big deal is e. We love e, and I'll underline that twice.

Now the reason is because when we take a derivative of e, dDT of e to the x equals e to the x; and dDT of e to the ax, where a is anything, equals a e to the ax. The property is that when you take a derivative of the function, the same function comes out; or if you take a derivative of the function, a scaled version of the same function comes out. We love this because, why? Because when we do differential equations, e to the x is the solution almost every time.

Whenever we did a circuit, e to the x was the answer. If you recall from when we were solving circuits, simple circuits with differential equations, we always said something like, "Well, we're going to guess that V of T is some constant times e to the St." That was a proposed solution; this turned out to work every time.

So there's something else we love too, and that is sinusoids or sines and cosines. Okay, we love these and that gets two lines now. Why do we love these? It's because they happen in nature. If you whistle, the air pressure looks like a sine wave. If you ring a bell, the bell moves in a sine wave. Any kind of music, if you look at the notes and music, the sound they make, the pressure waves look like sine waves.

Circuits make sine waves. Remember if we analyze this circuit in great detail, it was the LC circuit. We looked at the natural response of this, and that was a sine wave. Okay, so electric circuits make sine waves; all these things make sine waves. They occur in nature and we want to be able to analyze things that happen when sine waves are present.

So we have two things we love, and we want to relate these two things. These are going to be related through that Oilers formula. That's how we connect these two separate ideas, and let me, let's go do that.

So Euler's formula says that e to the Jx equals cosine x plus J times sin x. So S has a really nice video where he actually proves that this is true. He does it by taking the Maclaurin series expansions of e, cosine, and sine, and showing that this expression is true by comparing those series expansions.

Now I'm not going to repeat that here; we're just going to state that as fact. Now we're going to look at this equation a little bit more. So this is the expression that relates exponentials that we love to sines and cosines that we love.

Part of the price of doing that is we introduce complex numbers into our world. Here's two complex numbers. Okay, this is where complex numbers come into electrical engineering. So I have to mention the other form of this formula, which is e to the -Jx, and that equals cosine x minus J sin x.

So these two expressions together are Euler's formula or Euler's formulas. We're going to exploit this by taking; we'll be able to take the cosines and sines that we find in nature, and we're going to be able to fashion them into exponentials.

These exponentials then go into our differential equations and give us solutions. We're going to come back and pull out the cosines and sines—that's the rhythm of how we're going to use this equation to help us solve circuits.

One small point I want to share: notice that in both these equations, the cosine comes first and the sine is over here on the imaginary side. So the cosine is on the real side. This is the reason that we have a preference in the future; we're going to have a preference for talking about our real-world signals in terms of the cosine function. It's because in this Euler's formula, the cosine comes first in both cases.

So what I want to do now is take a second, and I want to see if we had our signal expressed in these exponentials, how do we recover the cosine and the sine term? How do we flip these equations around so we can solve for the cosine and the sine? It's a simple bit of algebra here; it's good to see.

Alright, so if I want to isolate the cosine term, let me get rid of these guys here. So now, to isolate the cosine term, what I'm going to do is add these two equations together, and that plus and minus are going to cancel out this second term here. That's what I'm going for.

So if I add, I'll get e to the Jx plus e to the -Jx equals cosine doubled, 2 cosine x, and the two J terms cancel out. Right, so then I can write, I'll write it over here: I can write cosine of x equals e to the Jx plus e to the -Jx over 2.

Alright, so that's the expression—that's the expression for cosine in terms of complex exponentials. Okay, let's go back and see if we can get sine. So what I'm going to do with sine is I'm going to subtract, and that gives me, what that'll do is it'll get the cosine terms to fall away, and then I get e to the Jx minus e to the -Jx equals cosine terms, and they subtract out and I get 2J sine x.

Alright, and that means I can write sine x equals e to the Jx minus e to the -Jx over 2J. And it's really easy; that J term is down there; that's easy to forget sometimes. So let me put a square around these guys because that's important—important.

So there's the two expressions; that's the two expressions for if you have complex exponentials and you want to extract the cosine, this is how you do it. If you want to extract the sine, this is how you do it. Okay, and you can either— you know, you can put these in your head, or probably easier for me, if you just remember Euler's formula. This is a pretty straightforward, quick derivation.

So the other thing we put a square around is this guy here. So we have Euler's formula and basically the cosine and sine extracted from Euler's formula.