 Hi, I'm Zor. Welcome to Inizor Education. We will continue talking about Euler's formula, which basically defines a complex exponential. Now, I was explaining in the previous lecture certain considerations, basically, certain reasoning behind this formula. I didn't prove this. So basically, I would like to take this formula as a definition of complex exponent. And to prove that this particular definition makes sense, I would like to go through regular properties of exponent and basically prove that all these properties are held for this definition. That would kind of confirm the validity of all this. So let's just go through the properties of exponents and see if everything is really reasonable according to the rules of exponential functions. All right, now the first is, remember, we had this rule. Now, what if the exponent is complex? Well, in our case, we are talking about the exponent, about complex exponent, but only imaginary part of it, because the real part is just a factor in this particular case. So let's take e to the power of e times zero. Now, what that would be according to this definition, it's a cosine of an angle of zero radians plus i sine. Now, this is zero, this is one, so we have one. So this rule is held. Okay, next. A little bit less trivial. Now, less trivial rule, which is basically one of the fundamental rules of exponents, is this one. Some of the exponent is basically a product of separate expressions for the same base and each exponent is used separately, and then we'll multiply the results. All right, now, is this held for this particular definition? Well, let's try. e to the power i x plus y equals to cosine of x plus y plus i sine of x plus y, according to the definition, right? Now, e to the power i x times e to the power i y, so that's basically the product of this, right? It's equal to this times the same with y, right? So it's cosine x plus i sine x times cosine y plus i sine y equals cosine to cosine. Now, that's the real part. Now, another real component is sine by sine, because there is an i by i, which is i squared, which is minus one. So it's minus sine x sine y. Now, the imaginary component is i sine by cosine sine x cosine y plus cosine x sine y. So let's compare this expression and this expression. Well, I mean, obviously they are identical because we know that the cosine of a sum of two angles is cosine by cosine minus sine by sine. So this is exactly equals to this and the sum of two angles is sine cosine plus cosine sine. So these are identical formulas. That's why this is equal to this, which proves actually that this rule is held for complex exponents. Next. So, so far this definition seems to be reasonable. That's actually my purpose to prove that this is reasonable. Now another property. Well, this is basically a definition of the negative exponent, right? So the negative exponent is such a number, which if multiplied by the positive exponent would give one, right? That's what it is. So that's how we define negative exponent. It's one over the positive one, right? Well, let's check if this actually is true. Now, e to the power minus i x, I would like to prove that this is one over e to the power of i x, which means multiplied by e to the power of i x, it should be equal to one. Now, is this true? Well, it is because we have already we have already proven that you can add the exponents and if you add them you will get zero and e to the power of zero would be one. So that that rule is also held. What else? The multiplication of the exponents. So e to the power of i x to the power of y is equal to e x y. Okay. Now, how to prove this? Well, this is a little bit more involved and here's the way how I would approach it. First, if y is a natural number n, is this true? Well, let's just think about it. e to the power i x to the power of n is basically a multiplication of e to the power of i x n times, right? n times. Now, we know that if you are multiplying this, you can basically say this is the e and the power will be the exponent will be a sum of these, which means i x plus i x plus etc plus i x n times which is equal to e to the power of i x n, right? So for integer, positive integer n, the formula is true. Well, actually for non-negative because for n is equal to zero, it's also true. Now, for negative n it's also true because we have already proven this. So, if you multiply this to the power of n, that's this to the power of n and obviously n goes to denominator which means that this is equal to e to the power minus i x n, which is necessary to prove, right? So, basically for any integer n, it's true. Now, let's talk about rational n, rational y. That's why we equals to p over q and I'm talking about positive just for sake of the time. Now, what does it mean? Now, you remember that we have actually defined this as such a number which if raised into the power of q gives a to the power of t, right? So, it's a, in other words, it's q's root of a to the power of t. That's how we defined it, basically, right? So, it means that by definition of this rational exponent, I have to basically prove that this raised to the q's degree is equal to e to the i x to the power of t. That's what I have to prove, right? Well, this is definition of I don't have to prove this. This is a definition of p over q used as an exponent. But what I have to prove is that e to the power of i x to the power of p over q equals to e to the power i x p over q. Now, what this means? This means that e to the power i x p over q to the power of q equals e to the power i x p. So, this is definition, this is definition of this thing and this is definition of this thing, right? And I basically have to prove that this is equal to this because I'm raising to the power of q the left part and I'm raising to the power of q this thing. This thing, right? This thing. So, considering I have raised in the power of q and I have exactly the same results because I have already proven that for any integer exponent, this is true. Which means that whatever I was raising into the power of q, which is this thing in this case and this thing in that case, they are equal. So, e to the power i x to the power of p over q is equal to e to the power i x p over q. That actually have been proven because if I'm raising both into the power of q, I get the same results. And these are the same because I have just already proven it for an integer p. So, that proves actually that this is actually true. Again, it follows from the definition of the fractional exponent. Okay, next. Next is something which is not exactly related to properties of the exponential functions, but nevertheless, it's very important for complex exponents. The modular is equal to one. Well, this modulus, or absolute value if you wish, what is this? Well, if you have a complex number represented as a plus i b, this is a, this is b, the length of this is actually the modulus. Let's call it r. And if this is an angle phi, then this is the same as r times cosine phi, because the cosine is a, right? r times cosine a plus i sine phi, right? So, lo and behold, our definition, our definition of e to the power of i x is exactly this graphical representation on the coordinate plane of the complex number. And obviously, r is equal to one in this particular case, because what is r? r squared is equal to a squared plus b squared, right? So, cosine squared x plus sine squared x, which is one. So, that proves that for any x, the modulus or absolute value of e to the power of i x is equal to one. Which means that for any x, all these numbers are on the unit circle. And x is just an angle in regions. So, that's the geometric representation of this. And finally, what I would like to point out is that you remember that whenever you multiply any complex number, wherever it is, z, by number which is represented in polar form, in coordinate form with a modulus and an angle, this is a combination of two different actions. Number one, you multiply the modulus of z by the modulus of this number. And in this case, the modulus is one, so we don't really multiply it on anything. So, z basically retains its length. But then, multiplying by this actually means a rotation of this angle by this. So, the new complex number, which is the result of multiplication of z by this, is just a rotation of this particular vector by an angle x in this case, or whatever the letter is used. So, that's my last property basically. z times e to the i x is a rotation by x regions. Well, that's it. That's all the properties I wanted to talk about when defining my complex exponent. So, these properties are just a confirmation that whatever we have defined is a reasonable definition. Well, that's it. Thanks very much and good luck.