We have $P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$ for all $z \in D$, where $p_n$ denotes the number of partitions of $n$.
Let $n \in \mathbf { N }$. Show that for all real $t > 0$,
$$p _ { n } = \frac { e ^ { n t } P \left( e ^ { - t } \right) } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta \tag{1}$$