Consider the equation


$$f'(x)+ g(x)f(x)=0$$


This equation is an ODE and has a solution $$ f(x)=C e^{ \int\_1^x g(x) \ dx}.$$


Similarly, we can look at complex variables and consider the equation and Wirtinger derivatives


$$ (\partial\_{\bar z} f)(z) +g(z) f(z)=0.$$


Can one still write down an explicit solution?



First order PDE in complex variables?
You can start by looking at the [chain rule for wirtinger derivatives](https://en.wikipedia.org/wiki/Wirtinger_derivatives), from which you deduce that


$$
\partial\_{\bar z} \exp(h(z)) = \exp(h(z)) \cdot \partial\_{\bar z} h(z)
$$


Therefore, if you find a function $h$ such that $\partial\_{\bar z} h = - g(z)$ (I think you forgot a "$-$" sign in your solution for the real case!) taking $f(z) = \exp(h(z)) $ will solve your problem. In general, this is known as the [d-bar problem](https://en.wikipedia.org/wiki/DBAR_problem) (or $\bar\partial-$problem). As Daniele points out this [Q&A](https://math.stackexchange.com/questions/3125765/let-h-mathbbc-to-mathbbc-in-ck-mathbbc-with-compact-support-find) is a god resource for the $\bar\partial-$problem in 1 dimension.


