I am studying Young's Introduction to nonharmonic Fourier series and I am currently stucked in the proof of Theorem 4 from Chapter 3 (after Levinson). The question is: why can we state that, for $n \in \mathbb{N} $ and $\alpha < 0$, $$\int_{-\pi}^{\pi} (1+e^{it})^{\alpha}e^{int} \,dt = \lim_{r \to 1^-} \int_{-\pi}^{\pi} (1+re^{it})^{\alpha}e^{int} \,dt \,?$$ If $\alpha \geq 0$, it is pretty clear that it is possible to do so, as the function would be bounded. In this case I am struggling to find a proper bound for the function in order to apply the dominated convergence theorem, since $$\lvert 1 + re^{it} \rvert^2 = \left( 1 + r\cos t \right)^2 + \left( r\sin t \right)^2 = 1 + 2r\cos t + r^2$$ can be arbitrarily close to $0$ when $t = -\pi,\pi$ .
Note: in the original text $\alpha = 2(\frac{1}{2p} + \varepsilon) - 1$, where $\varepsilon >0$ and $p \in (1,\infty)$.