Need help calculating $\int_{0}^{\infty} \frac{1- \cos (t)}{t^\alpha} {\rm d}t$

Question

As part of an larger assignment I need to calculate the following integral $$ \int_{-\infty}^{\infty} \frac{1-\cos(\lambda x)}{|\lambda|^\alpha} {\rm d}\lambda \quad x \in \mathbb{R}, \,1 < \alpha \leq 2 $$ I substituted $t=\lambda|x|$ to get $$ \int_{0}^{\infty} \frac{1- \cos (t)}{t^\alpha} {\rm d}t $$ But I have no idea how to continue. Any hint would be appreciated. Thanks

Answer 1

Hint: Use the following relation for the gamma function: $$\int_0^\infty \frac{e^{-bt}}{t^a}dt = \frac{\Gamma(1-a)}{b^{1-a}}$$ This gives: $$\int_0^\infty \frac{\cos(t)}{t^a}dt = \int_0^\infty \frac{e^{it}+e^{-it}}{2t^a}dt = \Gamma(1-a)\frac{i^{1-a}+(-i)^{1-a}}{2}=\Gamma(1-a)\sin(a\pi /2)$$