# First order PDE in complex variables?

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?

• Is $f(z)$ analytic? If so then the derivative of $f(z)$ with respect to $\bar{z}$ does not exist unless $f(z)$ is a constant. – Kevin Sep 23 at 13:59
• No, it is not analytic... – Sascha Sep 23 at 14:06

## 1 Answer

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.

• @Sascha, Jaume for a solution of the $1$-dim $\bar\partial$-problem this Q&A may be relevant. – Daniele Tampieri Sep 23 at 14:49
• Indeed @daniele-tampieri! That's a way better link to the $\bar \partial$ problem than the wikipedia page that has (surprisingly?) almost no reference – Jaume Sep 23 at 15:01