Solve for $f(y)$ in $\int_0^{\infty} f (y) \cos{(y x)}\,\mathrm{d}y = \dfrac{1}{\sqrt{x}}$

Question

I' have been having trouble with the given example. I've tried to solve it by considering that the right side is the Fourijer transform of the function $f(y)$, and that the function $f(y)$ is even, but things get complicated when solving the integral to find the inverse Fourijer transform.

Answer 1

There is a Laplace transform pair $f(s)=\frac{1}{\surd s}=\int_0^\infty e^{-st}F(t)dt$, $F(t)=\frac{1}{\sqrt{\pi t}}$ , formula 29.3.4 in the book by Abramowitz and Stegun https://en.wikipedia.org/wiki/Abramowitz_and_Stegun. Setting $s=i\omega$ should basically lead to the answer.

An even more explicit solution is on page 400 of the 3rd edigion of the book by Magnus, Oberhetting and Soni (Formulas and Theorems for the special Functions of Mathematical Physics) https://dx.doi.org/10.1007/978-3-662-1176-3 which gives the pair $$ f(x)=\left\{\begin{array}{ll}0 ; & x<a\\ 1/\surd x;& x> a \end{array}\right. $$ and $$ \sqrt{2/\pi}\int_0^\infty f(x)\cos(xy)dx = \frac{2}{y^{1/2}}[\frac12 -C(ay)], $$ where $C(x)\equiv\int_0^x\cos(\pi t^2/2)dt$, so evidently $C(0)=0$.