Welcome to the third lecture in our third meeting of the course Analysis of a Complex Kind. Today we'll learn about the complex exponential function. We started talking about this function at the end of last class. We discovered that the function f(z) defined by e to the x times cosine y plus i times e to the x sine y is an entire function. Where z=x+iy and e to the x cosine y and sine y are just the regular real-valued functions that we know from calculus. Let's look at some of the properties of this function. Suppose, for example, that y = 0. So this y right here is equal to 0. Well, cosine of 0 = 1. And so this y would also be equal to 0, but sine of 0 = 0. So in that case, this function would simplify to the e to the x times 1 + plus i e to the x times 0. Well, that's just e to the x because this whole last term gets canceled out by the 0. So when y = 0, then f(z) is simply e to the x. In other words, the function f agrees with our regular exponential function if you plug in something real that doesn't have an imaginary part. Another thing to notice is that we could rewrite f(z) a little bit. We can factor out the e to the x term right here, and are left with e to the x times parentheses cosine y + i sine y. But cosine y + i sine y we found an abbreviation for, we called that e to the iy even though we weren't quite sure why we called it e to the iy. It was an abbreviation at the time. But now that we're looking at e to x times e to iy. We would like to think of this as e to the x+iy because the rules for our regular real-valued exponential function would indeed allow us to combine these two exponents. We haven't proved any such thing right now, but this is a reasonable thought to have. And that looks like e to the z, whatever e to the z might be. So with these thoughts in mind, we actually define the complex exponential function e to the z exactly like that. It's also sometimes denoted exp(z). And it's defined by e to the z is e to the x times e to the iy, where z=x+iy. And e to the iy is still shorthand for cosine y + i sine y. This is the complex exponential function and we'll explore it in today's lecture. Again, as a reminder, e to the z is e to the x times e to the iy. What is the length of e to the z? So how far is e to the z from the origin? Well, the absolute value of e to the z can be found by simply taking the absolute value of e to the x times e to the iy. And we know the absolute value of the product of two complex numbers is the product of the absolute values. So norm of e to the x, norm of e to the iy. But e to the iy, remember what that really is? e to the iy is short for cosine y + i sine y. And it is simply a number on the circle of radius 1, such that the angle formed with a positive real axis is y. This is e to the iy, which was cosine y + i sine y. All numbers of this form have distance 1 from the origin because they're on the circle of radius 1. So the norm of e to the iy is 1 just by how this number e to the iy is defined. The norm of e to the x. e to the x is our real-valued exponential function. Remember what that looks like? e to the x looks like this. This is the regular exponential function. It's always above the x-axis, it's always positive. It can never be negative. So the norm of e to the x is simply e to the x. What's the argument of e to the z? Now that we know how far e to the z is from the origin, we would like to know what angle does it form with a positive real axis? What's the argument of e to the x times e to the iy? What is e to the x times e to the iy? That's a number in polar form, where this is its length or radius. But for a number in polar form, radius times e to the i something, the something gives you the angle that it forms with a positive real axis. So the angle this number forms with a positive real axis is y. Truthfully speaking, I should have written here +2 pi k, where k is any integer. So the way you should read this equation is an argument of e to the z is y. What is e to the z + 2 pi i? So suppose you add 2 pi i to your z and then stick that into the exponential function. Well, since 2 pi i is purely imaginary, that means that you're adding that to the imaginary part of z. So it's the same as e to the x times e to the i(y + 2 pi). We're adding the 2 pi to the imaginary part of z. But e to the i theta is a function that is periodic with period 2 pi, because we're looking at the angle that is formed with the positive real axis and it just starts repeating after 2 pi. So e to the i(y + 2 pi) is the same as e to the iy. So that this simplifies to the e to the x times e to the iy, which is e to the z. So the function e to the z is periodic. And the period is 2 pi i. When you add 2 pi i to z, the exponential function remains the same. Finally, we would like to establish some properties that are similar to the ones that we have for the real-valued exponential function. We would hope that e to the z+w is e to the z times e to the w. Let's check that this holds with our definition of e to the z which is up here. What is e to the z+w? We'll write z as x + iy and w as u + iv and plug that in. So e to the z+w becomes e to the (x+iy)+(u+iv). Let's reorganize and collect the real parts and collect the imaginary parts, because we'll need those to apply the definition of our exponential function. The real part of the sum and the exponent is x+u. And the imaginary part is y+v. Now we know e to this complex number is e to the real part, so e to the (x+u), times e to the i times the imaginary part, so e to the i(y+v). That's just by definition of the exponential function. e to the x+u, now we're talking all real values. x is real, u is real, and e is our regular real-valued exponential function, so the rules apply. This is e to the x times e to the u. e to the i(y+v), we proved that that equals e to the iy times e to the iv. We use the addition theorems for sine and cosine functions. And now we just reorganize things again. We collect e to the x, and e to the iy, and then e to the u. And since it's a product of complex numbers and multiplication is commuted if we can just reorganize things. But e to the x times e to the iy, that's simply e to the z. And e to the u, e to the iv is e to the w. And we established indeed, that the same rule, holds for complex values, as it does for real values. What is 1 over e to the z? Well, I claim this e to the negative z, and the reason is that, when I look at e to the z times e to the negative z, by the rule we just established, we can simply add the two exponents. So, that's e to the z plus negative z, so z minus z. But z- z is 0, and e to the 0 is 1. So, e to the z times e to -z is 1. Now, if we divide both sides of this equation by e to the z, you find that either the -z is 1 / e to the z. E to the z is an entire function, we already showed this. That's what set us off, you've defined this function. What's the derivative? The entire function has a derivative. Remember that, the way the function e to the z is defined, its real part is e to the x times cos y, and its imaginary part is e to the x times sin y. Again, we can find these partial derivatives that we learnt about during the last lecture. The partial derivative of u with respect to x means we differentiate e to the x cosine y with respect to x. Keeping in mind that y is fixed in a constant. The derivative of e to the x, e to the x cosine y is the constant. So ux is e to the x cosine y. Similarly vx is e to the x sine y, U y, we now keep x constants. So either the x is the constant, we need to find the derivative of cosine y which is- sine y. And vy is e to the x cosine y. We learned, that for a function that's analytic, it's derivative can be found by taking Ux plus iVx, so therefore, the derivative of our exponential function is e to the x cosine y which os ux. Plus i times e to the x sine y. That's the bx. But that's e to the z. In other words, the derivative of e of z is e to z again. Or in symbols, dbz of e to z Is e to the z. Notice that we're using not the partial derivative symbols which were d dx. We found, we used that notation for the partial derivatives. This is the complex derivative where we use the regular d for that notation. So similarly to the view value exponential function, to the derivative of the exponential function is the exponential function itself. Similarly, the derivative of e to the constant times z. How do you find that? This is actually the composition of two functions. The exponential function composed for the function that multiplies z by a. In other words, when you define the derivative of the outside function times the derivative of the inside function. The inside function is a times z. It's derivative is simply a. Therefore, we get a times e to the a-z. By the chain rule for the derivative of e to the az. What is e to the complex conjugate of z? The complex conjugate of z is x-iy, if z was x+iy. Therefore, e to the z conjugate = e to the x times e to the -iy. But we showed earlier, and we can remember this in the picture. What is e to the -iy? Even the -iy means we're taking a negative angle, -y. And get a number count here. If we reflected that with respect to the x axis, we will get this number up here, so this is Z to the minus IY and this is E to the IY and they're complex conjugates of each other as this picture shows. In other words E to the minus IY is the conjugate of E to the IY. E to the X's real value. We might just as well put a conjugate on top of it. It makes no difference because real number equals its complex conjugate. So we can extend the conjugation over the entire product. But that's a conjugate of e to the z. In other words, e to the conjugate of z equals the conjugate of e to the z When is e to the z = 1? Well, we need e to x times e to iy to be 1. This is a number in polar form. When is a number in polar form equal to 1? Well, its length needs to be 1, its angle needs to be 0. Or a 2 pi or a 4 pi. So, in other words, we need e to the x to equal 1, and we need y to be 0 or 2 pi or 4 pi. When e to the x equal to 1? That's the case when x is equal to 0. So we need x to be 0 and y if the form 2k pi, In other words we'll need z to be of the form x+iy, so 0+i2kpi. So those are the points where the exponential function takes the value 1. When is e to the z equal to e to the w? We already saw that if z = w + 2 pi i, then e to the z = e to the w. But are there other possibilities? For e to the z to equal e to the w. Let's find out. If we divide both sides of this equation by e to the w We know that one over e to the w is e to the minus w, so e to the z divided by e to the w becomes e to the z minus w. And that then needs to be = e to the w over e to the w which is one. When is e to the z-w=1? Well that is the case when z-w is of the form 2 pi ik. In other words when z is = w+a multiple of 2 pi ik. So, the cases we observed earlier were the only ways for e to the z to equal e to the w. Let's understand the mapping w equals e to the z. We did this earlier for some other functions. We look at as a mapping from the z plane to the w plane. With w being equal to e to the z. For example, let's look at images of horizontal lines. What's a horizontal line? A horizontal line is of the form x + iy0 where y0 is fixed. We're fixing the imaginary part and just varying x. For those points x + iy0, they get mapped to e to the x + iy0, which is e to the x Either the I, Y zero. But Y zero is a fixed number so, this is not going to change this is fixed. And the only thing that can change is E to the X. And E to the X if X runs from negative infinity to infinity, E to X will go from zero to infinity, it never takes the value zero but anything as close to zero as possible. And even the IY is a fixed angle from the origin. Let's look at some examples. For example, this red dotted line right there. What is its image? It's the line where y0 is equal to pi. What is e to the i pi? e to the i pi is equal to -1. Forming an angle of pi with a positive real axis. In other words, all these numbers, either the x times e to the i-y-zero lie in the negative real axis. So the image of the red dotted line right here is the negative real axis. Except in the origin. Let's look at the green line next. What is its image? In my picture looks like the imaginary part is a little bit bigger than pi/2. This angle here is just a little bit bigger than pi/2. Even the iy0 is a point on the unit circle, whose radius forms the angle y0 with a positive real axis. And therefore the image of the green line is this green line right here. Let's look at the turquoise line. It looks like its imaginary part is around pi over 4. So the image is going to be this turquoise line right there. How about the yellow line? The yellow line up here has an imaginary part just a little bit bigger than pi. So we're looking at this angle right here, and therefore the image of the yellow line is this radial segment down here. You notice that I drew another yellow line in the Z plane a little bit above- i pi. And indeed this reflects that the exponential function is periodic with periods 2 pi i. This second yellow line has distance 2 pi i from the other yellow line. In other words, its image is the same as the image of the other yellow line. So these two yellow lines both get mapped to the radial segment over in the W plane. Finally there's a purple line here and you can check that it gets mapped to something like this purple line over here. What are the images of vertical lines? A vertical line is of the form x0 + iy for now x0 is fixed. So now the real part is fixed and the imaginary part varies. Those points get mapped to either the x0 + iy, which is e to the x0 times e to the iy and so now this number is fixed. The distance from the origin is fixed for all these images and the angle varies. In other words for a fixed line that's vertical, it's image is a circle. For example the yellow line that is on the imaginary axis so x0 would be 0. Gets mapped to e to the 0 just 1 * e to the i y. But e to the iy. If y is allowed to vary from negative infinity to infinity. Gets matched to the circle. And this circle is mapped over and over and over again. Because this keeps repeating because there's 0 here. And say 2 pi i is right here. Then this little short segment all ready gets mapped to the full unit circle. And afterwards, we just keep repeating. Similarly, the red dotted line gets mapped to e to the x0 * e to the iy, this x0 is a little bigger than 0, that's what e to the x0 is bigger than 1. So it gets mapped to a circle whose radius is bigger than the previous one. And if we move over to the green line, its radius is going to be even bigger then the one with the red circle. Similarly, if I go to negative x0, e to the something negative is going to be a number less than 1. It's still positive but a number less than 1. So these radii are going to be less than 1. And the, the circles are going to be sitting inside the unit circle. Putting it all together, we can actually figure out what the image of a vertical strip is. Suppose we look at the set of all z, whose real part is between 0 and 1. Let's see, here's 1 and there's 0 and this is my vertical strip S. What is its image? We already saw that the yellow line, which is on the imaginary axis gets matched to the unit circle. The green line, for which I said the real bar is equal to 1, gets mapped to the circle of radius e. In other words, the strip from 0 to 1 gets mapped to this annulus, between the circles are radius 1 and of radius e. If I draw a horizontal line with this picture in addition, the one that I drew is maybe at pi over 4, then I cut out a portion of that annulus. So for example, this turquoise portion here of the strip gets mapped to this turquoise portion of the annulus. These kinds of pictures help you understand what the exponential function does graphically. Just a few final questions. When is e to the z = 0? Well, let's find out. e to the z is equal to 0 when e to the x times e to they iy is 0. We know e to the iy, that's for all numbers on the circle of radius 1, they can never be 0. When is a product of two numbers 0? Well, one of the factors has to be 0. The second one can't be 0 so it has to be the first one. So we need, if that's e to the x to be 0. If you remember what e to the x looks like, e to the x looks like this. It never crosses the x axis. It's never equal to 0. In other words, e to the z can never take the value 0. Just like the real value exponential function, it never takes the value 0. Let's go a little bit further. Suppose I'm giving you a point in the complex plane, not the origin but any other point. Then can I find the w such that e to the w is equal to that point z? Let's write z in polar form as absolute value of z times e to theta. And let's write w in Cartesian form because that works best for this problem. So let's write w as u + iv. Then e to the w being e to the z is the same as e to the u times e to the iv, which is e to the w, being equal to absolute value of z to the e to the i theta, which is simply z. And this was e to the w. Now these are two numbers in polar form. When are two complex numbers in polar form the same? Well they would have to agree in their distance from the origin. So e to the u would have to be the absolute value of z and then their angles have to agree modular 2pi. So either the iv would have to be e to the i theta. In other words, when does e to the u equal to the absolute value of z? This is all be of valued now. That's the case when you u is the natural logarithm of the absolute value of c. And when it's e to the iv the same as e to the i theta, well, v and theta can only differ by a multiple of 2 pi. In other words, w, which is u + iv, has to be of the form natural log of z, that's my u, plus i * an argument of z, which is what theta is. You just saw the basic ideas in how to define a complex logarithm. You see there's an issue already with the argument being here, the argument function is a multi value function, and we'd have to decide how to pick the argument. So that we could hopefully make the logarithm function an analytic function. And indeed that is possible.