and as with many of our transfer theorems that we considered earlier on for

singularity analysis and other approaches these technical conditions you know have,

have to be checked. so, if there is a multiple saddle-point,

then maybe there will be some other part of the condition to worry about, but I'm

not going to worry about that, intellectually.

and also tails have to be negligible in the first place, and then however this

estimate works. It has to be negligible, to bring the

pails back. And we'll look at a detailed, example

but, this kind of a technical detail is more appropriate to read about that, in

the book. So, what this is going to lead to is the

so-called saddle point transfer theorem. and that, all it says if you have a

contour integral. and, we write it as f of z equals e, to

the f of z. e to the little f of z just to simplify

the calculation, a little bit. it's a singularity, there's no problem

with doing that. So if that's susceptible to this sort of

approximation, then the, integral is approximated by f of z over that square

root of 2pi factor. It tells us exactly how to compute that

missing factor. this is a general technique for contour

integration, it's not just for asymptotics, that's about any contour

interval. [COUGH] it, gives a, a way to approximate

it. so, for, and the proof is very similar to

the proof that we did for the, saddle point bound.

but, so, that's the theorem, but what we want to use it for is a transfer method

for generating function. so it just translating that over to, our

use of, cauchy's theorem, where we're looking for a coefficient of z to the m

in a generating function. Then what we're going to do is just apply

this theorem with g of z with f of g equals g of z or z to the n plus 1.

So in that, and so that all that do is, is plugging into the general theorem, f

of z equals g of z over z to the n plus 1.

So our, our saddle point equation is just modified slightly and the main point is

that we evaluate our function at the saddle point

And then we have this corrective factor which is just square root of 2 pi times

the second derivative. so it's a plug and chug method of getting

coefficients of z to the n out of entire functions that have no singularities.

now there certainly is technical detail in proving suceptability and I'll try to

illustrate some of that in an example. But in this lecture what I really want to

focus in on is the idea that this kind of approximation is available.

I can't claim that this is as clean as many of the other transfer theorems that

we've looked at because checking those technical conditions can be more

challenging. so anyway, the summary is that for our

generating function g of z, and this is just doing the math backwards,

We're going to use the saddle point equation.

G prime is z over g is z equals m plus 1 over z.

So that's, theta is the solution to that. And then following that, that's the

approximation where little g is a log or big g minus m plus 1 log z.

that's a relative simple calculation for getting asymptotic estimates of

coefficients. So let's go back to our examples.

So estimating 1 over n factorial. it's a saddle point approximation.

Coefficient is z again and then e to the z.

So, g of z equals e to the z. And so, un then taking logs, our saddle

point equation is 1 minus n plus 1 over z equals 0.

so that's zeta equals n plus 1. and then the saddle point bound is just

plug in zeta second derivative us 1 over n plus 1.

so the approximation that we get is e to the n plus 1 over n plus 1 to the n plus

1 squared of 2 pi over n plus 1. and again that one goes to 1 over e and

just simple asymptotics. gives us exactly stirling's

approximation, e, e to the n over n to the n over square root of 2 pi n.

So the, this extra factor in the saddle point approximation gives us the extra

factor that we were missing before. so the choice that we always have when

using saddle point is you can do all the detailed calculations to check that the

interval is suceptible and so forth. or you always have the option of just

using the bound and sacrificing that factor which certainly in many

applications, people will choose as an option.

Now if it becomes important to get the factor, we can can get it, but it takes

some work. and again, checking susceptability, you

have to check that the tails are negligible, that the central

approximation is quadratic. And then we can complete the tails back

in, in a reasonable way. And that's really doing the calculation

for a particular function that we have. Uh, [COUGH] so, at this level, we don't

necessarily have the, the general scheme of, but, this is an important starting

point for, limiting distributions that is something, advanced that we don't study

in this course. so, let's just look at,

And, and again, this is the last lecture in a long course.

So I'm not going to do these calculations in detail.

other to, other than to just, exhibit 'em.

And this is the kind of calculation that's needed to show that it's

acceptable, for this problem. so, what we're doing is, trying to do a

contour integral for either the z or the z over the n plus 1.

or just putting it all in 1 function, z minus n plus 1 log z.

around this this circle, z equals ne to the i theta.

so first thing is to, just switch to polar coordinates z equal to n e to theta

and just plug that in there and most of it comes up.

E to the n over n comes out. and all that's left is interval 0 2pi e

to the n e to the i theta minus 1 minus i theta that's that's just simple

subsitution. so now what these saddle point method

requires is that we're able to split it into 2 contours.

1 for a little part close to the saddle point and then the rest for the tail

which is all the rest of the circle. and so those are the 2 contours and the

point is. For we have to be able to prove that for

the tales gets exponntially small. and that you can look at the text for

that proof, but when we're away from the saddle point, remember, in a 3D diagram

that's when the curve drops all the way down and so exponentially small.

And then in the part close to the so we're going to just negelect the tails

and work with the part close to the saddle point.

and another thing to notice is that if we can slightly shift the saddle point.

If we want, it simplifies the calculation quite a bit.

As long as we have that quadratic equation approximation near where we are.