Complex functions that is not quite analytic A complex-valued function $f(z)=u(x,y)+iv(x,y)$ is analytic if it satisfies the Cauchy-Riemann equations:
$\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}=0$
$\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}=0$
Is there a class of functions where only one of these relations holds?
$\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \ne 0$
$\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} = 0$
The real derivatives are no longer equal to zero, but the imaginaries cancel. These of course will not be analytic, but I am curious if this generalization has been the focus of study.
 A: Given a complex function, $f(z)=u(x,y)+iv(x,y)$, the Pólya vector field is based on the conjugate $\overline{f(z)}$: $\langle u(x,y),-v(x,y)\rangle$. The Cauchy-Riemann equations amount to saying that the "scalar curl" and divergence of the Pólya vector field are both $0$. Individually, these quantities being zero have names and significant consequences on their own:
The (scalar) curl being zero makes the vector field "irrotational". And it is often called conservative, at least in nice situations with continuous second-order partial derivatives and a simply-connected domain (the exact definition/usage of "conservative" may depend on the source).
The divergence being zero makes the vector field "incompressible" or "solenoidal", which has its own nice properties.
However, I doubt that these properties of the original complex function are often studied, as these irrotational/incompressible vector fields don't typically arise naturally as the Pólya vector field of an easy-to-write complex function. For example, if $f(z)=\overline{z}^2$ or $f(z)=z*\overline{z}$, then neither of the Cauchy-Riemann equations holds.
