1
$\begingroup$

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

$\endgroup$
3
  • 2
    $\begingroup$ 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/… $\endgroup$
    – rrogers
    Dec 25, 2020 at 22:12
  • 1
    $\begingroup$ 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. $\endgroup$
    – Bita
    Dec 25, 2020 at 22:26
  • $\begingroup$ 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. $\endgroup$
    – rrogers
    Dec 26, 2020 at 3:37

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged .