TITLE: First order PDE in complex variables?
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: You can start by looking at the chain rule for 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 (or $\bar\partial-$problem). As Daniele points out this Q&A is a god resource for the $\bar\partial-$problem in 1 dimension.