Complex spectral Fourier transform ($Q(k)={\int }_{\mathbb{C}}{e}^{\bar{k}z-k\bar{z}}q(z)dA(z)$)

A complex-coordinate parametrization of the two-dimensional Fourier transform. With $z=x+iy$, $k=k_1+i{k}_2$, and $({\xi }_1,{\xi }_2)=(2k_2,-2k_1)$, its exponent is $-i\xi \cdot (x,y)$. Since $d\xi =4dA(k)$, its inverse is $q(z)={\pi }^{-2}\int e^{k\bar{z}-\bar{k}z}Q(k)dA(k)$. The spectral equation $({\partial }_{\bar{z}}-k)F=q$ gives ${\partial }_{\bar{k}}F=-e^{k\bar{z}-\bar{k}z}Q(k)/\pi$, linking Fourier inversion to a spectral dbar equation.