Inverse Fourier Transform for $\frac{e^{-A\sqrt{(\omega^2+Bi\omega+C)}}}{\sqrt{(\omega^2+Bi\omega+C)}}$

Question

I am trying to find the inverse Fourier transform of below: $$\frac{e^{-A\sqrt{(\omega^2+Bi\omega+C)}}}{\sqrt{(\omega^2+Bi\omega+C)}}$$ Where A, B, and C are all positive and real. Please let me know if you need further information ($\omega$ is the Fourier transform variable). Thanks and Merry Christmas

Answer 1

From the lookup I have Let:
$\mathcal{L_{\mathtt{t\rightarrow s}}}\left(f\left(t\right)\right)=g\left(s\right),\mathcal{L_{\mathtt{t\rightarrow s}}^{-\mathtt{1}}}\left(g\left(s\right)\right)=f\left(t\right)$

$\frac{e^{-\left(b\left(s^{2}-a^{2}\right)^{\frac{1}{2}}\right)}}{\left(s^{2}-a^{2}\right)^{\frac{1}{2}}}\rightarrow\begin{cases} 0 & 0<t<b\\ I_{0}\left(a\left(t-b^{2}\right)^{\frac{1}{2}}\right) & b<t \end{cases}\,\,\,\,\,\,\,\,\,b>0,Re\left(s\right)>\left|Re\left(a\right)\right|$

I am having problems with your $i,\omega$; can you check the signs and consistency? In particular, your formula seems to give a diverging (t->s) transform. During completing the square. Also, I have a very strong suspicion that t should be t^2 in the Bessel function argument.