Integration involving "partial bar" operator I am not able to deal with the following problem. Giving function
$$G(x-\xi,y-\eta,k)=\frac{1}{4\pi^2i}\int_{-\infty}^\infty\int_{-\infty}^\infty\frac{e^{i\mu(x-\xi)-i\nu(y-\eta)}}{\nu-i\mu(\mu+2k)}d\nu d\mu$$
and to make clear, let $k=k_1+ik_2$, how to prove
$$\frac{\partial G(x-\xi,y-\eta,k)}{\partial \bar k}=\frac{1}{2\pi}\text{sgn}(-k_1)e^{-2ik_1[(x-\xi)-2k_2(y-\eta)]}?$$
It seems to me that $\bar \partial \frac{1}{\pi(z-\zeta)}=\delta(z-\zeta)$ should be made use of, but I am not sure how to carry out that.

Sketch: Just take the differentiation inside. 
$$ \bar{\partial}_k \frac{1}{\nu - i \mu^2 - 2i\mu k} = -2\pi i \mu \delta(\nu - i \mu^2 - 2i\mu k) $$
the delta function can be decomposed
$$ \delta(\nu - i \mu^2 - 2i\mu k) = \delta(\nu + 2\mu k_2) \delta(\mu(\mu + 2k_1)) $$
the rest follows from the usual change of variable formula involving the $\delta$ distribution. (Integrate first in $\mu$. Because of the $\mu$ outside already, the only root of $\mu(\mu+2k_1)$ that contributes is the one $\mu = -2k_1$.)
