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

on complex analyses, but most people have some familiarity.

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

really everybody should understand that does mathematics should understand

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

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

concepts that are really important for analytic combinatorial complex analysis

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

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

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

but, but complex analysis is really a perfect example.

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

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

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

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

minus 1. does i occur in the real world?

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

understand really a lot about the real world.

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

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

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

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

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

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

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

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

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

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

elementary. It's really usually a matter of two

things. One is the representation of complex

numbers. is by correspondence with points in the

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

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

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

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

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

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

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

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

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

plane. and I haven't defined multiplication yet

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

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

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

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

on the plain. Now to define the basic operations.

The natural approach. Is to just use algebra.

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

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

Add the real parts. Add the imagineary parts.

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

i squared to minus 1, and then collect term.

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

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

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

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

properties it turns out that you'd expect.

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

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

have a minus vi over a squared plus b squared.

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

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

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

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

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

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