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Next, we'll look at Complex Analysis. Now not everybody has had a full course

Â on complex analyses, but most people have some familiarity.

Â And the things that I'm going to talk about today are beautiful mathematics.

Â really everybody should understand that does mathematics should understand

Â complex analysis at this level. so I'm not going to go slowly, I'm not

Â going to go quickly. I'm just going to try to cover the

Â concepts that are really important for analytic combinatorial complex analysis

Â is really for, for people with interest in computer science.

Â it's the quintessential example of the power of abstraction.

Â it's just an idea that we build on. Really all of mathematics is like that,

Â but, but complex analysis is really a perfect example.

Â And the whole idea is that we've got minus one.

Â What happens if we want to take the square root of minus one.

Â No real number who's square is minus one. So what we'll do is we'll define a number

Â that's going to be the square root of minus 1 and we'll call that i squared is

Â minus 1. does i occur in the real world?

Â Well, no, it's an abstraction. but it's an abstraction that helps us

Â understand really a lot about the real world.

Â so then what we're going to do and all of these things are pretty simple.

Â they get more complicated as we go on. But, just starting with that idea then

Â we're going to want to do things with these numbers that involve this imaginary

Â number i, like we're going to want to add a multiple end of item.

Â We're going to want to do exponentiation, we're going to want to define functions.

Â We're going to want to be able to differentiate them and integrate them.

Â this leads to a whole theory that not only is beautiful in it's own right, but

Â also turns out to have many, many applications in science and mathematics.

Â And in particular analytic combinatorics. so there's many standard conventions and

Â again I'm going to go quickly through these things and but many of them are

Â elementary. It's really usually a matter of two

Â things. One is the representation of complex

Â numbers. is by correspondence with points in the

Â plane. So we're going to the point x y is

Â going to represent the complex number z equals x plus i y.

Â we refer to the real part of the complex number or of z, and that's the x part.

Â and we refer to the imaginary part, so that's that's the y part.

Â Then the distance from the origin so that's squared of x squared plus y

Â squared that's called the absolute value of the complex number.

Â so how far it is from the origin its sometimes how big it is?

Â And then there is a thing called the conjugate, if you have x plus iy then

Â that z then z bar is x minus iy. and so that's point flipped down on the

Â plane. and I haven't defined multiplication yet

Â but when you do you do a quick exercise to show that z time z bar equals absolute

Â value of z squared. So those are standard conventions of how

Â we refer to complex numbers. Everyone's got a real part and an

Â imaginary part. [COUGH] And we represent them by points

Â on the plain. Now to define the basic operations.

Â The natural approach. Is to just use algebra.

Â But, every time you see i squared. You just convert it to minus one.

Â So if you want to add to complex numbers. Well that's just algebra.

Â Add the real parts. Add the imagineary parts.

Â If you want to multiply, multiply em, use the distributed term law.

Â i squared to minus 1, and then collect term.

Â So, bdi squared, which is bi times di, becomes minus bd, that goes into the real

Â part. Uh,[COUGH], and then bci, and adi, those

Â go to the imaginary part. So that's the definition of

Â multiplication of two complex numbers. Uh,[COUGH], and this has all the right

Â properties it turns out that you'd expect.

Â The vision 1 over a plus bi to to make sense of that multiply top and bottom by

Â a minus bi. So then you have a minus vi, so then you

Â have a minus vi over a squared plus b squared.

Â That's the denominator is the number times its conjugate which is the square

Â of the absolute value. There's addition, multiplication, and

Â division, and if you're not comfortable with those you can try some examples with

Â the points on the plane, and so forth. What about something like, eh,

Â exponentiation well you can multiply a lot of times but now it gets complicated,

Â so now, we'll skip right to one of the basic ideas of complex analysis.

Â 5:21

And that's the idea of an analytic function.

Â So and the analytic function is one that we can represent with a power series

Â expansion and we have the same concept with the reels with the tailored series

Â and so forth. So So we talk about omega being a region

Â in the plane. We talk about a function being defined in

Â a region of the plane, and it's analytic at some point in that region.

Â If for some little circle around that point you can write a power series

Â expansion of the function And, sure enough, this familiar power

Â series from the reals are effective for familiar functions on the complex.

Â So when z is a complex number, 1 over 1 minus z equals that power series where

Â the products are defined as products of complex.

Â and that's a valid series for as long as absolute z is less than 1, or e to the z

Â actually is valid except at infinity. and [COUGH] now, let's a bit of a jump

Â from defining multiplication to this. and I don't have the time to do the the

Â entire story leading up to this but the main point is that our familiar series do

Â translate through. And it's through the concept of the

Â analytic function. When I get, in just a few minutes, to

Â some more concepts I'll try to talk a little bit about Why we know all of this

Â to be true. It's, it's really part of deep and

Â beautiful theory that you'd need to take a course in complex analysis.

Â For the purposes of this course, we're going to be using these kinds of series.

Â And in this the idea of complex differentiation.

Â Well, we define that in the same way symbolically that we'd do for real

Â number. so it's complex differentiable a function

Â is complex differentiable, or there's a special word called holomorphic.

Â at a point, if the this limit, where f of z plus delta minus, of z not plus delta

Â minus of z not divided by delta as delta goes to 0.

Â If that limit exists for delta, a complex number and, and, and z, not complex, and

Â that's the complex function. so if that limit exists, it's holomorphic

Â or complex differentiable. now this is the same notation as for

Â reals, but it's a much stronger statement because the value is independent of the

Â way that delta approaches 0. It's a complex, it can come from any

Â direction. It's a much, much stronger concept,

Â complex differentiability. Now, there's a basic equivalence theorem,

Â that says that, a function is, that these concepts are the same, analyticity, and

Â complex differentiability are the same concept.

Â A function is analytic if and only if its complex differentiable.

Â 9:03

if it's complex diferenciatable in it's derivitive of any order because you can

Â always differenciate turn by turn. That's not really true in the real.

Â In particular, we can use the tailor series representations that I just talked

Â about. Of familiar functions e and 1 over 1

Â minus z and polynomials and so forth. so Taylor's theorem in terms of

Â derivatives and so forth just immediately applies and gives us the power series

Â that we're going to want. So that[COUGH] those are the kinds of

Â power series that we're going to be extracting coefficients from in order to

Â do our counting. there's a really famous particular

Â example of this called Euler's formula. so that you take the definition of the

Â exponential an evaluate at[UNKNOWN] So what's e to the i theta?

Â Well, it's 1 plus i theta over 1 factorial plus i theta squared over 2

Â factorial. And so forth.

Â That's just right from the definition. exponentials, analytic gets defined at

Â any point, and so it's defined at i theta.

Â now if you just convert I squared to minus 1, and I to the fourth to 1, then

Â you can get rid of every other term that has an I in it.

Â Every other term does not have an I in it, and then the terms alternate in sign.

Â So, i squared equals minus 1, so it's minus theta squared over two.

Â i 4th equals i squared squared, equals plus 1, so it's plus theta to the 4th

Â over 4, and so forth. So that's just getting, that's just

Â algebra getting rid of the i squared. and then if you just separate the terms

Â out, from the previous slide the first series is cosin theta, and the second

Â series is i sin theta. That's called Euler's formula.

Â E to the i theta, equals cosin theta plus i sin theta.

Â and this an amazingly useful formula in many forms of mathematics.

Â this quote from Feynman expresses what people think.

Â And he calls it our jewel, one of the most remarkable, almost astounding

Â formulas in all of mathematics. and again, this is just an example of

Â just starting with that idea, of square root of minus 1 we can get to

Â amazing constructs like, like this. that's what complex analysis is about.

Â Now that's a quick introduction. next, we'll start to apply, some of these

Â things, to the combinatorial problems that we've been working with.

Â one thing that Euler's formula does is it gives us another way to find a

Â correspondence between complex numbers and points in the plane.

Â so e to the i theta equals cosign theta plus i sin theta, so re to the i theta

Â equals r cosign theta plus ir sin theta. And so we can use polar coordinates to

Â refer to a point on the plane. If it's at angle theta and distance of r.

Â then we can write it as r e to the i theta and we can sometimes its more

Â convienent to use that form of a complex number and if we definitely often do so.

Â So for any complex number we can use these ways to convert between the polar

Â coordinates and the Cartesian coordinates, the same we, same way as we

Â do for points in the plane. That's an introduction to complex

Â functions. And next we'll take a look at how to use

Â this information to extract coefficients from combinatorial generating functions.

Â