Computing a Fourier transform with an Exponential

Question

If one has the following wave function in 'real' space $$ \phi(r) = e^{-a\sqrt{b^2+r^2}} $$ with $a,b\in\mathbb{R}$, is it possible to obtain an analytical form of the Fourier transformed expression $\widehat{\phi}(k)$. Computing $$ \widehat{\phi}(k)=\int^\infty_{-\infty} e^{-a\sqrt{b^2+r^2}}e^{-ikr}\,\mathrm dr $$ offers no help to a direct solution. Is there information in the literature I am missing on this. I've seen Wolfram math world has information on a FT of the function in the form $e^{-k_0|x|}$, but cannot see directly how this can be adapted to the above.

Answer 1

If you actually intend to integrate over all $ \bf \vec r$ in three-dimensional space then try using spherical coordinates.

I'll use the pure mathematicians's conventions regarding spherical coordinate notation:

$|{\bf \vec r}|= \rho$ and $ dV=d( {\bf \vec r})_3 =\rho^2 \sin \phi \ d\phi \ d\theta \ d\rho$.

Write $ \bf{\vec k}\cdot \bf { \vec r} = \kappa \rho \cos \phi$. Then integrate out in $\phi $ first.

1. Note that $\int _{\phi=0}^{\phi=\pi} e^{- i\kappa \rho \cos \phi } \sin \phi d\phi$ can be evaluated explicitly using the substitution $u= \cos \phi$. You get $2 \frac{\sin (\kappa \rho)}{ \kappa \rho}$.

2. The integral in $\theta$ is trivial and contributes a factor of $2\pi$ to the total answer.

3. Then you must evaluate $\frac{4\pi}{\kappa} \int_{\rho=0}^{\rho=\infty} \rho {\sin (\kappa \rho)} e^{- a\sqrt{b^2 +\rho^2}} d \rho$. This is apparently not explicitly computable in elementary functions (even with Mathematica), but asymptotic approximations might be found using the Steepest Descent Method.