MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 2 Changed an example.

Using Cauchy's integral formula one very easily obtains

The precise asymptotics of the inequality partition function $n! \ge \frac{n^n}{e^n}$, but p(n)$ are, as far as I don't knowof a hands-on proof, totally inaccessible by finitary tools. More generally the methods of complex analysis are a great way to obtain incredibly precise asymptotics on many finitary sequences of concrete interest, i.e. trees satisfying certain properties or sequences related to the running times of algorithms, etc.

A related example is comparing the elementary and non-elementary proofs of the prime number theorem; my recollection is that the elementary proof is quite a bit more involved.
show/hide this revision's text 1

Using Cauchy's integral formula one very easily obtains the inequality $n! \ge \frac{n^n}{e^n}$, but I don't know of a hands-on proof. More generally the methods of complex analysis are a great way to obtain incredibly precise asymptotics on many finitary sequences of concrete interest, i.e. trees satisfying certain properties or sequences related to the running times of algorithms, etc.

A related example is comparing the elementary and non-elementary proofs of the prime number theorem; my recollection is that the elementary proof is quite a bit more involved.