Exponential integral

1. Apr 29, 2009

zetafunction

if we suppose that f(x) is Real then what condition should we impose to the integral

$$g(u) = \int_{-\infty}^{\infty}dx e^{iuf(x)}$$

in order to exists for every 'u' no matter if it gives a function or a distribution (i.e in case f(x)=x then we have a dirac delta)

also le us suppose we must calculate f(x) so $$g(u) = \int_{-\infty}^{\infty}dx e^{iuf(x)}$$ is correct for a certain given function or idstribution g(u)

my idea is that if we could expand for big 'u' the integral into a divergent series

$$g(u) = \sum _{n=0}^{\infty}a_{n}u^{-n}h(u)$$

for a certain h(u)

also if we knew the expression for f(x) and its inverse f(-1)(x) could we then make the change of variable x=f(-1)(r) in our integral equation to transform it into a Fourier integral ?