 In this video, we delve into the very important question about what does it mean to take a complex exponent? I mean, it makes sense to do things like i squared, because that's just i times i, which would be negative one in the situation. And we could even make sense out of doing something like what's i to the square root of two power, because that would be involving taking some limits and things, because we can do things like i to the one half, we can do radicals. And the square root of two as an irrational number is the limit of rational numbers. We can take, you can use calculus to make sense out of irrational exponents. So that takes care of any real exponent. But what if you wanted to do some type of imaginary exponent? Like what if you wanted to do e to the i? What would that even mean, right? Or another option I have listed here is two to the one minus i. Well, the good thing about the second one is if, if we're gonna make any sense out of it, two to the one minus i, if we're taking exponents, you know, cause let's say we've never, ever considered an imaginary exponent before, we would want whatever that new exponent rule is, it has to be compatible with the old exponent rules. So therefore, this should be the same thing as two to the first times two to the negative i, for which as this is just a real exponent, that's just gonna give us a two. And if you have a negative exponent, I mean, you should do division. So you get two divided by two to the i. So whatever is in question right here, it really just depends on if we can do two to the i, we can do the rest of it, or same thing right here. If we can do e to the i, I think we can do the rest of it. And so if we look at this more general, you take a to, so a is some number raised to a complex exponent, x plus yi. Well, whatever complex numbers do as exponents, it should be compatible with the rules we already have. If you add exponents, that means multiplication a to the x times a to the yi. And when you have multiplication, the exponents, you should be able to factor it and you get something like a to the y to the i. Oh, you see that already. But we should be able to expand that a to the iy by exponent rules, right? This should look something like a to the y, which is a real number to the i. So we really only have to deal with the i-th power of things. That's all that kind of matters. But then what we can do even better is if you have a to the iy, you can actually replace a with e to the natural log of a. Notice that e to the natural log of a is just a again because the exponential and the natural log are inverse functions of each other. So we're gonna replace a with e to the natural log of a. And then by exponent rules, this becomes e to the iy natural log of a. And then kind of factoring this thing one more time, we're gonna get e to the iy natural log of a. So what I'm trying to say here is, turns out that all that we have to do is if we can figure out what e to the i is, then we actually can make sense out of every single imaginary exponent. And instead of just doing e to the i, we're actually gonna work with e to the iy, where y could be any real number. If we can take e to a purely imaginary exponent, it turns out we can make sense out of any complex exponents whatsoever. And although this might be bizarre, right? To even think about e to the i, what does that even mean quantitatively? Whether we ought to do it or not, turns out there is actually a very natural candidate on how to handle this creature right here. And the idea is we're actually gonna use power series to make sense of this imaginary exponent. Because with power series, we can make perfect sense of what this is. And I'll show you what I mean in just a moment. And it turns out that this is really just gonna connect us to the polar form of complex numbers. And so it's not gonna seem so bizarre once we have the right perspective. So to help us do exponents, imaginary exponents for e, we're gonna introduce Euler's formula, which notice if we have a complex number z equals x plus yi, we'll call its modulus r and its argument theta. Then we can write a complex number in polar form, trigonometric form, z equals r cosine theta plus i sine theta. Now this is what Euler's theorem says. Euler's theorem says that this is the same thing as r e to the i theta. That's the same thing as this thing right here. So more specifically, e to the i theta is just cosine plus i sine theta. That's the definition of an imaginary exponent. So we have to establish this principle right here. If Euler's identity holds, then this general statement about polar forms of complex numbers will follow very immediately. And so in order to compute this, we're gonna use the Taylor series. I should say the Maclaurin series for e to the x, cosine of x and sine of x. That's how we're gonna handle this thing. Now I should mention that we've done this using real variables. So as we start moving into complex variables, things get a little bit different. But what I will tell you in this video is that we can do this in such a way that the Taylor series still hold very nicely. No problem with that. The point is when you look at the Maclaurin series of say e, we're gonna call it e to the z instead of e to the x because we're assuming that we could have a complex variable here. The Maclaurin series would still be the infinite sum where n goes from zero to infinity of z to the n over n factorial. And when you expand that out, you get one plus z plus z squared over two factorial plus z cubed over three factorial, et cetera. We could also consider the Maclaurin series for sine of z and cosine of z in this very same manner, right? Sine of z would be the infinite sum as n goes from zero to infinity. It alternates as a negative one to the n. You get odd powers of z over odd factorials. Cosine does the same thing. It's just you have even powers over even factorials like so. And when you look at those Maclaurin series, when you look at those Maclaurin series, there's nothing about those Maclaurin series that dictate you must have a real number because in order to do a Maclaurin series, what do you need? You need addition. You need subtraction. You need exponents, which you're really just taking exponents by integers, right? Which, you know, z cubed just means z times z times z. So far all of those things make sense for complex numbers. We can add, subtract, and multiply out complex numbers. We even have division going on here, but the division really is just division by a whole number, is division by an integer, which in particular we could do division by complex numbers if we had to, but that's not even necessary. And so we have to add, subtract, multiply, and divide, which are all things that we can do for complex numbers. Then the other issue, which this is the really sticky point, right, is convergence. How do we make sure that this infinite sum works, right? Even for a complex number. Well, these Maclaurin series, right? The way we proved that the radius convergence for these three Maclaurin series was infinity really comes down to the ratio test. That's what we did previously. Or maybe you could use Taylor's inequality. That's also an appropriate tool. But the thing is not going to the details of those. Those rules of convergence can be extended to the complex plane as well. And this is actually what we call it the radius of convergence because in the complex plane, you don't get intervals of convergence. You get circles of convergence. And that circle has a radius, the radius of convergence. Now for the natural exponential sign and cosine, the radius of convergence will still be infinity, which means that circle, if it has an infinite radius, is actually the entire complex plane. So without going into the details of the complex analysis going on right there, I'm saying these Maclaurin series will still converge for imaginary or complex exponents. So that's what we then define. That's then how we define e to the i theta right here. We define it to be evaluate the Maclaurin series at i to the n, okay? And there should be some parentheses right here. You're going to take e to the i theta. You're going to plug in e to the i raised to the nth power. You divide by n factorial. You take the sum as n equals zero to infinity. And if you write this out in expanded form, you're going to get one plus i theta squared over two factorial plus i theta cubed over three factorial, et cetera, et cetera, et cetera. Now as we do each and every one of these powers, I want you to simplify the power i theta to the n. Well, with theta, you always get theta, theta squared, theta cubed, theta fourth, whatever. But what about i, right? This is something to remember. When you take i to the zero, you're going to get one because everything to zero is one. If you take i to the first, you're going to get back an i. i squared is a negative one. And i cubed is a negative i. And then when you go to the fourth power, this process just repeats itself. So you always get these four numbers. It repeats in this cycle over and over and over again. And we just recognize that every four steps, you're going to start the cycle over again. So you're going to get one i negative one, negative i. One i negative one, negative i. It repeats itself over and over and over and over again. And so that's what the expansion of e to the i theta will look like. Now what if we regroup things, right? Because you're going to notice that some terms have multiples of i, so these would be imaginary terms, because theta's going to be a real number. In factorial, it'll be a real number. So you have all these imaginary terms, and then you also have all of these real terms. So if we gather the real parts together, like that, and we add up together the imaginary parts, well, since all the imaginary parts have a multiple of i factored out, we get the following. We'll notice that for the imaginary part, you get theta minus theta cubed over three factorial plus theta to the fifth over five factorial. That can be written in the following Maclaurin series, which that's just the Maclaurin series for sine of theta. And if we do the same thing for the real parts, you're going to get one minus theta squared over two factorial plus theta to the fourth over four factorial minus theta to the sixth over six factorial. If you put that together as a single Maclaurin series, we see that this is just the Maclaurin series for cosine of theta, which then establishes Euler's identity here, that e to the i theta is just cosine of theta plus i sine theta. And it's quite interesting that how the natural exponential is actually connected to these trigonometric functions, cosine and sine. It kind of blows our minds away. And this connection actually alludes to the fact why, why do we call the function e to the x minus e to the x over two? I'm sorry, that's not right. e to the negative x over two. Why do we call that cinch? Why do we call hyperbolic sine and the function e to the x plus e to the negative x over two? Why do we call that Cosh hyperbolic cosine? Why in the world do these functions deserve trigonometric analogous names? Well, that's because the natural exponential e is actually quite deeply connected with trigonometry. And it turns out that cinch and Cosh act as imaginary friends to cosine and sine. You don't see that when you look at the real line, but in the complex plane you see a duality that's between the trigonometric functions and the hyperbolic functions and they're connected by e. But let's just do some examples of this right here if you want to compute e to the i pi. Well, in this situation your angle of play would be pi itself. So e to the pi i would look like cosine of pi plus i sine of pi. Cosine of pi is negative one and sine of pi is zero, in which case this would simplify just to be negative one and that's it. It's quite a beautiful formula right there. e to the pi i equals negative one. I remember the first time I saw this fact it was told to me in an AP calculus class when I was in high school. And I couldn't believe it's like how in the world could the numbers e pi i interact in such a way that gives you something so simple as negative one? In my disbelief I turned to my graphing calculator it was a TI-89 which could do complex arithmetic. I type it in e to the pi i and then the calculator said negative one. And then I just accepted it as gospel at that moment. It's like, oh, well the calculator says it must be true. Of course I've learned since then to not put so much stock in calculations, calculators themselves. But nonetheless by Euler's identity we can see very quickly that e to the pi i is in fact negative one. A very beautiful formula right there. Let's use a slightly more complicated exponent here. If you take e to the negative one plus i pi over two. Well, since there's two parts, a real part and an imaginary part we're gonna bring in the two pieces. You have e to the negative one and then e to the i pi halves. Well e to the negative one, that's a purely real number, that'll just become one over e. And then we have e to the i pi halves that becomes cosine of pi halves plus sine of pi halves. Cosine of pi halves that should be zero. Sine of pi halves is one so you can get one over e plus times one plus i. So that just gives you i over e or one over e times i, whichever you prefer. So we can simplify this exponent to give us i over e right there. And then as a last example, this time let's say z is equal to two times e to the pi i over three. Same basic idea. We're gonna have a two sitting in front of it. We're gonna take e to the pi i over three. I should mention that in terms of complex numbers. This right here really is the polar coordinate of this thing. I should mention that if you take e to the i theta this modulus is always equal to one. You can argue that if you take cosine theta plus r sine theta, its modulus will always equal one. Because in that situation, if we kind of unwrap this a little bit, you're gonna get the square root of cosine squared theta plus sine squared theta. Which sine squared plus cosine squared we know is one. So you see that e to the i theta is always a complex number whose modulus is one. In fact, this is the general way of describing a complex number whose modulus is one. It can always be written in this form e to the i theta. So then in general, you'll see complex numbers written in the form r times e to the i theta for which you see right here the modulus and the argument of the complex number. And so this is commonly referred to as the polar form of that complex number. So continue on with the calculation. You'll get two times cosine of pi thirds plus i sine of pi thirds. Which cosine of pi thirds is one half. Sine of pi thirds is root three over two. You can distribute the two through and you recapture the complex number two plus i square root of three. And so Euler's identity is pretty awesome. It really is. And it shows you how you can actually use power series to calculate things that you might not realize power series are necessary for. Power series is a theoretical tool for extending functions from real variables into complex variables. Which helps us get this complex, I should say polar form for complex numbers here.