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

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

• I have the inverse Laplace from a look-up table but the answer given is an integral of the Bessel function with J_0((t^2-u^2)^1/2)du argument; more or less, but it's close. Is this, with completing the square and such, close enough? I can probably extend it to the inverse Fourier but that would take a little longer. The book is: amazon.com/Table-Laplace-transforms-Kaufman-Roberts/dp/… Dec 25, 2020 at 22:12
• Hi @rrogers, Thank you so much, unfortunately, I don't have access to the book. I appreciate if you can provide me with the integral you mentioned.
– Bita
Dec 25, 2020 at 22:26
• My partial solution/indication is posted but I am having problems with your formula; as I mentioned in the Answer at the end. Let me know if you want the (s^2+a^2) form which is more complicated but gives similar problems. Dec 26, 2020 at 3:37

$$\mathcal{L_{\mathtt{t\rightarrow s}}}\left(f\left(t\right)\right)=g\left(s\right),\mathcal{L_{\mathtt{t\rightarrow s}}^{-\mathtt{1}}}\left(g\left(s\right)\right)=f\left(t\right)$$
$$\frac{e^{-\left(b\left(s^{2}-a^{2}\right)^{\frac{1}{2}}\right)}}{\left(s^{2}-a^{2}\right)^{\frac{1}{2}}}\rightarrow\begin{cases} 0 & 00,Re\left(s\right)>\left|Re\left(a\right)\right|$$
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.