How to find $f$? (I suppose I need to use the Laplace or the Fourier cosine transform)

Question

$$\int_{0}^\infty{f(x)\cos(xy)\,dx=e^{-y}}$$ I try to use Laplace transform for $p=0$. Can I use Fourier cosine transform?

Answer 1

Fourier and Laplace are two people: if they were alive, I guess they would like their name to be stated with a capital letter. Besides that, you may use whatever you like, but such identity cannot hold for every $y\in\mathbb{R}$, since the LHS is invariant with respect to $y\mapsto -y$, but the RHS is not.

However: $$ \int_{0}^{+\infty}\color{red}{\frac{\frac{2}{\pi}}{1+x^2}}\,\cos(xy)\,dx = \color{red}{e^{-|y|}}$$ by the properties of the Cauchy distribution.