Fourier sine transform of $\displaystyle \frac{1}{x}$ is ..(fill in the blanks)..
My thoughts: By Definition, $\displaystyle F_s(s)=\int_{0}^{\infty}f(x)\sin sxdx $
$\displaystyle F_s(s)=\int_{0}^{\infty}\frac{\sin sx}{x}dx $
How do I integrate this? This just keeps blowing up if I integrate by parts.
An online search says this is a special function called sine integral.
$\displaystyle \int\frac{\sin ax}{x} \mathrm{d}x = \sum_{n=0}^\infty (-1)^n\frac{(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!} +C$
This does not seem like the right path…. How do I solve this?
The given answer is $s^2/2$
Best Answer
Notice that if $s>0$, $y=sx$ changes the integral $\int_0^\infty \frac{\mathbb{sin}x}{x} \mathbb{d}x$ ($s$ vanishes). There are two possibilities: one is to use contour integration in the complex plane. I am going to assume that you do not master such techniques yet, so let us examine a more elementary approach.
Consider the function $f(t)=\int_0^\infty \mathbb{e}^{-tx} \frac{\mathbb{sin}x}{x} \mathbb{d}x$ for $t \geq 0$. Notice that $f'(t)=-\int_0^\infty \mathbb{e}^{-tx} \mathbb{sin}x \mathbb{d}x$. Integrate this twice by parts and you will get a linear equation in $f'(t)$ which will give you $f'(t)=-\frac{1}{1+t^2}$, hence $f(t)=-\mathbb{arctan}t+C$, with $C$ a real constant. Notice that for $t \rightarrow \infty$ you get $f(t) \rightarrow 0$ because of the exponential, thus $C=\frac{\pi}{2}$. Then $f(0)=\int_0^\infty \frac{\mathbb{sin}x}{x} \mathbb{d}x = C = \frac{\pi}{2}$.
The final answer is $\frac{\pi}{2} \mathbb{sign}x$, not the one that you claim.