We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. For all $n \in \mathbb{N}$, we define $$\gamma_{n} : \begin{cases} [0,1] \rightarrow \mathbb{C} \\ t \mapsto (2n+1)\pi\, \mathrm{e}^{2\mathrm{i}\pi t} \end{cases}$$ and for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t.$$ Show that, for all $n \in \mathbb{N}$, $Q_{n} \in \mathcal{E}$.
We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. For all $n \in \mathbb{N}$, we define
$$\gamma_{n} : \begin{cases} [0,1] \rightarrow \mathbb{C} \\ t \mapsto (2n+1)\pi\, \mathrm{e}^{2\mathrm{i}\pi t} \end{cases}$$
and for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$,
$$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t.$$
Show that, for all $n \in \mathbb{N}$, $Q_{n} \in \mathcal{E}$.