I'm trying to solve assignment (1.5) in Bellan's "Fundamentals of Plasma Physics" using Fourier transforms, but I'm stuck integrating the Laplacian. Here's the problem:
Equation (1.5) is the following:
$$\nabla^{2} \Phi - \frac{1}{\lambda_D^2} \Phi = \frac{q_T}{\epsilon_0} \delta(\vec r).\tag{1.5}$$
Now if I multiply this with $e^{-i \vec k \cdot \vec r }$ and integrate over $\vec r$ I get simple results for the right-hand side ($-\frac{q_T}{\epsilon_0}$) and the second term on the left-hand side ($\frac{1}{\lambda_D^2} \tilde \Phi (\vec k)$), but can't wrap my head around the first integral. Integrating by parts gives:
$\int \nabla^{2} \Phi e^{-i \vec k \cdot \vec r } d \vec r = \nabla \Phi e^{-i \vec k \cdot \vec r } - \int \nabla \Phi (-i \vec k) e^{-i \vec k \cdot \vec r } d \vec r $
And using integration by parts on the second term on the right-hand side again gives:
$\int \nabla^{2} \Phi e^{-i \vec k \cdot \vec r } d \vec r = \nabla \Phi e^{-i \vec k \cdot \vec r } - [ \Phi (-i \vec k) e^{-i \vec k \cdot \vec r } - \int \Phi (-i \vec k)^2 e^{-i \vec k \cdot \vec r } d \vec r ] = \nabla \Phi e^{-i \vec k \cdot \vec r } + \Phi i \vec k e^{-i \vec k \cdot \vec r } + k^2 \tilde \Phi (\vec k)$
However, to obtain the correct result (1.43) for the potential in the assignment, I figure that the first two terms should vanish, but I don't know why. I've never integrated the Laplacian, so I could be terribly mistaken in my approach; any help is greatly appreciated!