= Complex spectral Fourier transform
{title2=$Q(k)=\int_{\mathbb C}e^{\bar kz-k\bar z}q(z)dA(z)$}
A complex-coordinate parametrization of the two-dimensional <Fourier transform>. With $z=x+iy$, $k=k_1+ik_2$, 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 $q(z)=\pi^{-2}\int e^{k\bar z-\bar kz}Q(k)dA(k)$. The spectral equation $(\partial_{\bar z}-k)F=q$ gives $\partial_{\bar k}F=-e^{k\bar z-\bar kz}Q(k)/\pi$, linking <Fourier inversion> to a <spectral dbar equation>.
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