calculus - Convergence of $int_0 ^{infty}frac{cos t}{t^{alpha}} dt$ related to $Gamma$ function

I would like to show that the integral

$$\int_0 ^\infty \frac{\cos t}{t^{\alpha}} dt$$ converges for $0<\alpha <1$. I already showed that it does not converge for $\alpha\leq 0$ or $\alpha \geq 1$. Do you have any hints for me? I know that I can use the gamme function (using $\cos t=\frac{e^{it}+e^{-it}}{2}$) but I don't see why I can use the gamma function just for $\alpha \in (0,1)$.

Answer

Hint. Recall that, from the definition of the Euler $\Gamma$ function, we have

$$\begin{align} \int_{0}^{\infty} e^{-bt} \, t^{-\alpha} \, dt = \frac{\Gamma(1-\alpha)}{b^{1-\alpha}}, \quad 0<\alpha<1, \Re b>0. \tag1 \end{align}$$ Then put $b:=b_\epsilon:=\epsilon+i,\, \epsilon>0$, in $(1)$, let $\epsilon \to 0^+$ and take the real part to get

$$\begin{align} \int_{0}^{\infty} t^{-\alpha} \cos t \, dt & = \sin \left(\frac{\pi \alpha}{2}\right)\Gamma(1-\alpha), \quad 0<\alpha<1. \tag2 \end{align}$$