evaluating the integral of sinc function from zero to infinity by shifting the poles instead of deforming the contour.

Question

$\int_0^{\infty} {{sin x}\over x} dx$

I Know to do this integral by deforming the contour in a complex plane around $Z=0$.

My question is can we do this integral by shifting the poles (just change the pole location to $x_0+i\epsilon $ or $x_0-i\epsilon $ and give limits $\epsilon$ tends to zero), I tried a lot but my limit diverges.

Can anyone please help me?

Answer 1

For $a \in \Bbb{R}^*$ you can evaluate $$\int_{-\infty}^\infty \frac{e^{i x}}{x+ia}dx$$ with the residue theorem

Taking the complex conjugate it means you know $\int_{-\infty}^\infty \frac{e^{-i x}}{x-ia}dx$ and $$\int_{-\infty}^\infty \frac{\sin(x+ia)}{x+ia}dx$$ letting $a\to 0$ you get your result