Greens function for Helmholtz equation I'm having trouble deriving the Greens function for the Helmholtz equation. I happen to know what the answer is, but I'm struggling to actually compute it using typical tools for computing Greens functions.
In particular, I'm solving this equation:
$$ (-\nabla_x^2 + k^2) G(x,x') = \delta(x-x') \quad\quad\quad x\in\mathbb{R}^3 $$
I know that the solution is 
$$ G(x,x') = \frac{e^{-i k |x-x'|}}{4\pi|x-x'|} $$
but I would like to understand how to obtain this answer (so that I can find Greens functions for other, more complicated, equations). The typical approach seems to be to solve in Fourier space and then transform back into real space, but I'm having a lot of trouble with the transformation back into real space. In particular, evaluating this integral is hard:
$$ G(x,x') = \int d^3p \frac{e^{-ip(x-x')}}{p^2+k^2}$$
(where $p$ is my Fourier space variable, and $k$ is the same $k$ as in the original equation above). Is there an easier way? Am I doing something wrong? 
 A: The integral is not hard!  The measure $d^3p$ is equal to $|p|^2d|p|\, d(\cos \theta) d\phi$ and the exponetial $\exp\{-i{\bf p}\cdot ({\bf x}-{\bf x}')\}$  is $\exp\{-i |p| |{\bf x}-{\bf x}'|\cos\theta\}$, so everything is a straightforward integral over decoupled scalars. 
We are to show that 
$$
G(|x|)= \int_{{\mathbb R}^3}\frac{d^3 p}{(2\pi)^3} \frac{e^{i{\bf p}\cdot {\bf x}}}{|{\bf p}|^2+m^2 } = \frac 1{4\pi|{\bf x}|} e^{-m{|\bf x}|}
$$
(note that the OP's equation is not quite correct: there is no $i$ in the exponent on the RHS when his $k^2$ (my $m^2$) is positive).
We use the measure I gave above to do the integrals over the angles $\theta$ and $\phi$ to  see that this is equal to
$$
G(|{\bf x}|)=C \frac 1{|{\bf x}|}\int_0^\infty  d|p| \frac{|{\bf p}| \sin |{\bf p}||{\bf x}|}{|{\bf p}|^2+m^2}
$$
for some constant $C$.
We do not need to use Jordan's lemma to do the remaining integral. We observe that the elementary integral
$$
\int_0^\infty   e^{-mx}\sin (px) \, dx= \frac p{p^2+m^2}
$$
is a half-line Fourier transform, and the  desired $|{\bf p}|$ integral is its  inverse Fourier transform.
When the OP's $k^2$ becomes negative we have a wave equation.  This  does not have a unique Green function and it  is in this case that there are singularities on the integration contour of the final $|{\bf p}|$ integral. How we route the contour around them by adding $\pm i\epsilon$'s then determines whether we  replace $m$ by $i|k|$ or $-i|k|$, and physical refers to outgoing  waves (and a causal Green function) or incoming waves (and an anti-causal Green function). In this case some knowledge of complex variable techniques  is useful, but still not necessary as we can still invert a Fourier transform. 
