I am attempting to compute the vacuum expectation value for the energy density of a particular system (Quantum Field Theory). I come across the following integral $$\int_0^\infty \mathrm{d}k \: e^{-i(t - x)k}k^{-i\omega/a} $$ where $t,x,k,\omega,a\in\mathbb{R}$. Mathematica gives me a conditional expression with constraints on the imaginary parts of $t,z,\omega,a$ which are not satisfied because they are all real parameters. I have tried using identities such as Proof of $\Gamma(z) e^{i \pi z/2} = \int_0^\infty t^{z-1} e^{it}\, dt$ but then realised that they also require the same constraints. I also tried the substitution $\rho = ln(k)$ (here $k$ is a positive frequency) but encountered a similar issue.
Does anyone have suggestions to analytically evaluate this? My contour integration/complex analysis knowledge is fairly fundamental so would prefer a physicist's solution.
The integral $$\int_0^\infty k^{i p} e^{i x k} dk, \quad p, x \in \mathbb R$$ diverges. If that helps, the Fourier transform exists in the distributional sense, with the image being a singular functional: $$\mathcal F[k^{i p} H(k)] = i e^{-\pi p /2} \Gamma(i p + 1) (x + i0)^{-i p - 1}, \\ (x + i0)^\lambda = x_+^\lambda + e^{i \pi \lambda} x_-^\lambda, \\ (x_\pm^\lambda, \phi) = \int_0^\infty x^\lambda (\phi(\pm x) - \phi(0) H(1 - x)) dx + \frac {\phi(0)} {\lambda + 1}, \\ \operatorname{Re} \lambda = -1, \operatorname{Im} \lambda \neq 0.$$