I'm in the process of learning elementary complex analysis at the moment, and looking at all of the interesting properties they exhibit, I could not help but wonder:
If $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is a real two-dimensional continuously differentiable function, what (minimal) conditions can we impose on $f$ such that it exhibits some of the interesting properties of holomorphic functions? And to what extent can this be done? i.e are there properties of holomorphic functions that are only guaranteed to be satisfied in trivial cases?
Essentially what I'm asking is what is the best equivalent/parallel we have in the realm of real $\mathbb{R}^2\rightarrow\mathbb{R}^2$ functions to holomorphic functions?
As for what I have so far, if we want to ensure that close path integrals (say a curve $l$ encircling an area $D$) are always $0$ inside convex domains we have, by stokes' theorem:
$$\oint_{\partial D}f(x,y)\cdot dl = \iint_{D}(\nabla\times f) \cdot dxdy=0$$ And since this equality has to be satisfied everywhere and on arbitrarily small areas $D$ we conclude that $\nabla \times f = 0$ everywhere.
Taking this further we can derive that the above condition is satisfied if and only if $$f(x,y)=(u(x,y),v(x,y)), \quad\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$$ which is similar to one of the Cauchy-Riemann equations.
If we instead require that $\nabla \cdot f=0$ we are able to derive $$\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}$$ which is similar to the other Cauchy-Riemann equation.
Are functions which satisfy both conditions a good parallel to holomorphic functions then? And are they the specific object of study of some mathematical theory?
You may be intrested in harmonic functions that is functions of the form $\nabla^2f=0$. They are the real or complex part of a complex function and exibit many properties such as being infinitely differentiable, having a maximum principle and an analogue of Liouvilles' theorem. Note that if a function satisfies both conditions you mention then it is harmonic.