Improper integral involving trigonometric function

Question

I was wondering what happens when evaluating an improper integral involving a trigonometric function where the denominator is a rational function with a zero at $x=0$.

The example I have in mind is $$\int_{-\infty}^{\infty}\frac{sin(ax)}{x(x-i)(x+i)} dx$$

If I rewrite the sine in terms of the exponential and then evaluate two integrals, one for the upper half-plane and one for the lower I have a problem because of the singularity at $z=0$ which seems to be in both halves of the plane. Do I treat it in both integrals then? Any suggestions/comments are welcome.

Answer 1

One way of avoiding to deal with the (redundant) singularity after spliting the integrals is to first differentiate w.r.t. $a$ and use Eulers identity

$$ I'(a)=\Re \int_{\mathbb{R}} \frac{e^{i a x}}{(x-i)(x+i)} $$

Assuming that $a>0$ we have to close our contour of integration, which is a big semicircle , in the upper halfplane(for $a<0$ it is the other way round)

We get

$$ I'(a)=\Re(2\pi i \text{Res}(x=i))=\pi e^{-a} $$

integrating back w.r.t. $a$ yields $$ I(a)=-\pi e^{-a}+C $$

and the constant of integration is fixed by the requirement that $I(0)=0$

which brings us to our final result

$$ I(a)=\pi(1-e^{-a}) $$

I leave the case $a<0$ to you

Answer 2

Another approach is to use partial fraction expansion to write

$$\int_{-\infty}^\infty\frac{\sin(ax)}{x(x^2+1)}\,dx=\int_{-\infty}^\infty\frac{\sin(ax)}{x}\,dx-\int_{-\infty}^\infty\frac{x\sin(ax)}{x^2+1}\,dx \tag 1$$

Now, the first integral on the right-hand side of $(1)$ can be evaluated by using Cauchy's Integral Theorem and writing

$$\begin{align} \int_{-\infty}^\infty\frac{\sin(ax)}{x}\,dx&=\lim_{\epsilon \to 0^+}\left(\text{Im}\left(\int_{-\infty}^{-\epsilon}\frac{e^{iax}}{x}\,dx+\int_\epsilon^\infty \frac{e^{iax}}{x}\,dx\right)\right)\\\\ &=\lim_{\epsilon \to 0^+}\left(-\text{sgn}(a)\text{Im}\left(\int_\pi^0 \frac{e^{ia\epsilon e^{i\theta}}}{\epsilon e^{^{i\theta}}}\,i\epsilon e^{i\theta}\,d\theta\right)\right)\\\\ &=\pi\, \text{sgn}(a) \end{align}$$

where we have tacitly used Jordan's Lemma to find that the contribution from integration over the infinite semi-circular contour integral that encloses the contour in a half plane is zero.

More simply, using the Residue Theorem, the second integral on the right-hand side of $(1)$ becomes

$$\int_{-\infty}^\infty \frac{x\sin(ax)}{x^2+1}\,dx=2\pi i \,\text{sgn}(a)\,\text{Im}\left(\text{Res}\left(\frac{ze^{iaz}}{z^2+1},z= i \text{sgn}(a)\right)\right)=\text{sgn}(a)\pi e^{-|a|}$$

where again, one can use Jordan's Lemma to show that the contribution from integration over the infinite semi-circle is zero.

Putting everything together yields

$$\bbox[5px,border:2px solid #C0A000]{\int_{-\infty}^\infty\frac{\sin(ax)}{x(x^2+1)}\,dx=\pi\, \text{sgn}(a)\left(1-e^{-|a|}\right)}$$