Watson's lemma

Consider

Suppose

with and
Suppose also either:

• for all and some
  OR
• 
  Then we have the Asymptotic Approximation

Proof / Method

No elegance, just suffering

Split the problem

We expect to be the main contribution, and small.

is small

Case 1

Assuming we can integrate this and get
ie #

Case 2

So

In both cases, is small beyond all orders.

contribution

Want to show

Using a property of The Gamma Function
we find that the numerator is equal to:

Now estimate

as .
Let and introduce s.t.
Get:

for some constant, so as
So as

---

For we note:

So

So

So as for all .
Finally:

but