Proving the Hardy-Ramanujan-Rademacher series for $p(n)$

Question

How to prove the series of the Hardy-Ramanujan-Rademacher for the partition for an integer n using the Cauchy residue theorem?

Answer 1

Just search for "rademacher partition formula".

A discussion with a number of references is here:

https://en.wikipedia.org/wiki/Partition_(number_theory)#Partition_function_formulas

Answer 2

One of the references I can recommend is the book of Tom Apostol (1990) on Modular functions and Dirichlet series in number theory. In chapter $5$ the proof is explained in all detail, together with a "plan of the proof", which is very helpful. On $17$ pages we can follow an amazing proof, illustrated with several highly interesting drawings (related to Farey fractions, Ford circles, Rademacher path etc.).