the radius and convergence is bigger than a rho and if a rho is not equal to zero.

So, it's just supplying this to zero of a rho and

plugging in zero of a rho in this theorem, then we get slightly more general.

We could do one over one minus zero of a rho to the alpha, and that pulls out a rho

to the n in the in the asymptotics.

So that's just the elementary calculation to see where that comes from.

So that corollary's the one that we'll really be interested in.

So, that's the theorem.

If we have a generating function of that form, we know the asymptotics.

And that applies to the two major problems that we've talked about in this

lecture for catalan numbers.

So T of z equals one over 2 of z one minus square root of one minus four z.

So there is a tiny complication, because the first term has to cancel out.

But ignoring that, what we're using is alpha equals one half in

this alpha equals minus one half, so

that's one minus z over rho to a minus one-half in the denominator,

that's square root, and rho equals a fourth, so that's square root of 1- 4z.

And then f in this case is just a constant, just the half out front.

And so we wind up needing gamma of minus one half,

and then that's 2 times gamma of one half, just because gamma of

alpha plus 1 equals alpha gamma of alpha minus 2 squared of pi.

Now you plug it all in.

And then maybe it's easiest to think about it by just multiplying both sides by z.

And then apply these exactly.

It is not hard to finish the calculation to show that this transfer

theorem immediately gives us the asymptotic of the catalan numbers.

So one transfer theorem to get the generating function.

Another transfer theorem,

the analytic one, to get the asymptotics of the coefficients.

And the same theorem, works for the derangements problem.