How do you integrate this function (not by parts)?

[zheng89120]

Homework Statement

integrate: r²*exp(i*k*r - r²/a²) from -infinity to +infinity (in terms of r)

Homework Equations

relevant integration table

The Attempt at a Solution

not sure what this function or the method to solve this function is called

 

[Mute]

I would start by completing the square in the exponential, and then changing variables to y = r - b, where b is whatever you get when you complete the square. Then, use know results for the integrals of Gaussian functions.

 

[CompuChip]

Without going through the whole exercise, the thing that comes to mind is splitting off a square. I think that eventually you will find a substitution like z = r/a - \tfrac12 i a k which will allow you to write it in terms of integrals like
\int e^{z^2} and \int z e^{z^2}
which you can do without partial integration.