Consider the fractional differential equation \begin{align} D_{|x|}^\alpha u(x) +bu(x)=f(x) \end{align} with $0<\alpha<2$ on an unbounded domain. Instead of $D_{|x|}^\alpha$ one also often sees $\Delta^{\alpha/2}$, meaning the fractional (symmetric) Laplacian with Fourier transform \begin{align} \mathcal{F}\left( D_{|x|}^\alpha u(x)\right) = |\kappa |^\alpha \tilde u(\kappa). \end{align} To compute the Green's function $G(x)$ of the above problem, I need the inverse Fourier transform of $g(\kappa)$, given by \begin{align} g(\kappa) = \frac{1}{|\kappa|^\alpha +b}. \end{align} I know, that for \begin{align} h(\kappa) = \frac{1}{\kappa^\alpha +b} \end{align} the inverse Fourier transform is given by \begin{align} H(x) = x^{\alpha-1} E_{\alpha,\alpha}(-bx^\alpha) \end{align} where $E_{\alpha,\beta}(x)$ is the Mittag-Leffler function. Are there any references regarding this topic? I have also found numerous results for the Green's function on a domain that is a ball of size $R$ with $u(x)=u(|x|)$, meaning that we have a radial problem. However, this is not the case in my problem.
Any hint is very much appreciated.