[BLANK_AUDIO] Today we're going to talk about saddle-point asymptotics, which is

where we turn to estimate coefficients for generating functions that have no

singularity. This is the final step in complex

asymptotics. It's the last way to get a quick

asymptotic estimate of generating function coeffients according to our

standard overview. to understand Saddle point we need to

talk about modulus surfaces. and this is a a three dimensional

extension of what we've been using in two dimensions.

and to warm up I, I just want to look at two dimensional plots of functions where

we plot x versus the absolute value of some function of x, in a Cartesian plot.

So for example the function 2x is just a line, but absolute value of 2x is a v

when it hits zero it bounces back up again.

or one over x is a familiar hyperbola. but absolute value of one over x, is a a

little inverted funnel. sign x goes up and down.

absolute value of sign x keeps bouncing off the origin.

one minus x squared is the parabola. but absolute value of one minus x squared

starts to look a bit like batman. and as we get to more complicated

functions we get more interesting shapes so this is a cubic for example.

and I just bring these up in two dimensions so that the shapes that we see

in three dimensions can be, are not quite so jarring and can be interpreted.

in terms of shapes like this. So now what we want to do is look at

three dimensional versions of the plots of analytic functions that we've been

working with. And that those things are called modular

surfaces, which is a plot of x y and absolute value of some function.

whereas somewhere points in the, in the Cartesian plane will correspond to x plus

yi. And we've been using plots like this to

help us identify poles and understand the topography of the, of the poles in

complex functions already. For example, here's a plot of 1 plus 4z

squared over 1 minus 4z squared. where black corresponds, the blacker a

point is, the higher the absolute value of the function.

And so, with a plot like this, we can see c equals plus a half and minus a half

where the denominator vanishes, as we get closer to those points.

The value of absolute value of f of z gets bigger and bigger.

we're the near 2I, I over 2 and minus I over 2 where the numerator goes to zero

then it gets to be white. but now in three-dimensions, this plot

has a much more fascinating shape. so, this is the same plot in, in 3D.

now, we'll be looking at properties of plots like this called modulus surface

ah, [COUGH] to help understand the saddle, saddle point method.

so one of the first questions come up is, can we get any shape this way?

and one interesting aspect of these kind of plots, is that no, actually the shapes

that we get are very highly constrained by by the situation.

in fact, there's only four different types of points.

on these kinds of surfaces. there's zeros where the thing actually

does get to zero the absolute value of the function actually gets to zero.

so in, in this case it's either [UNKNOWN] there's ordinary points which is pretty

much of everything else, where these things are, are very smooth and it's not

surprising they're smooth because. Analytic functions are smooth.

They're infinitely differential where, where they're defined.

And then there's poles, which we've already talked about.

and then there's a fourth type of point, called the saddle point, and that's what

we're going to focus on in this lecture. okay, so let's go through these types of

points one at a time. So the first one is, zeroes.

and, we'll first talk about simple zeroes.

So that's where the function's zero, and the derivative's, not zero.

so this is a, a typical example of a modular plot of a function, a simple

function showing off a zero. So this function is f of z equals 2z.

So in polar coordinates that's two r e to the i theta.

but the modulus is always two r. So it doesn't depend on theta.

So that means if we go out a distance r, then we're going to go up two r.

and then we'll have a circle so its a cone.

so its the same for all theta and then it comes down to a point right where at zero

zero where f of z equals zero. so that's a typical zero.

Now what's interesting is that all zeroes kind of look like this.

they have the same local behavior. that's because in analytic function we

can expand using Taylor's theorem in terms of its derivatives.

so we can write at any point f of z equals f of z 0, plus f prime of z 0

times z minus z 0 an so forth. and then if we, if its a zero then the

first term is zero and the second one is a constant that's not zero and then after

that as we get closer to, as z gets closer to z zero then it behaves just

like that constant times z minus z zero. Just a linear function.

So just like this. And they all have the same local

behavior. And that also doesn't depend on theta.

So for example, here's the function that has two zeros, 1 minus Z squared.

so you can see looking at the profile of it, it looks kind of like our batman

parable where it bounces off the axis and goes up but around the zeros it's little

points like this because it has the same local behavior as a simple zero at the

two points plus and minus one. so that's just doing the math.

So the functions that it behaves like locally.

and then so if we take one minus z q then we have three roots over 1 minus z to the

8th, we have 8 routes, and so forth. And these surfaces look quite intricate

and complicated, but, actually, all they are is collections of these points that

go down to the zeros. So now that's a so called simple zero if

you have a point where the derivatives are zero for a while and then finally

there's a place where some derivative is non-zero.

That's called a zero of order p. so like z cubed has a zero of order

three, so the point is curved a little bit more down at those at, at those

zeros. and so for typical functions you might

have multiple types of zeros. So this is a function z squared plus z

cubed and then you can see the little profile of the cubic function there.

one of the zero's is of order one, the other one is of order two.

So, it's fairly easy to understand what zero's are.

Those are the points where it touches. and if it's of order one, it's always

going to be a point. Otherwise, it'll have a, more rounded

contour where it touches. so that's what zeros are like.