Consider the Mellin transform on bounded support $(0,1).$ I computed the following:
$$\mathscr M[f;s]=F(s)=\int_0^1 x^{s-1}e^{\frac{1}{\log(x)}}~dx=\frac{2K_1(2\sqrt{s})}{\sqrt{s}} $$
Where $K_1$ is the modified Bessel function of the second kind.
Is that the correct result? I'm not sure I got it completely correct.
Why does the Mellin transform give an oscillatory function?
After performing the transform what can I use the result for?
Before posting this question I read about Mellin transforms and how they are used in analytic number theory because of Perron's formula. They can also be used to solve differential equations in some cases.