Something interesting in Complex Analysis is the Wirtinger derivatives:

They are often simply defined as such, but one would be curious how to derive them, at least heuristically.

## How to derive Wirtinger derivatives

It turns out we can derive them as such. Any complex function can be viewed as a function by considering . Since , we can also view as .

Then by the Chain Rule (for multivariable calculus), we have .

Similarly, we get .

Then, solving the simultaneous equations we get the Wirtinger derivatives.

. Thus, .

Similarly, we can get that .

Using Wirtinger derivatives, we can express the Cauchy-Riemann equations in a succinct manner: A function satisfies the Cauchy-Riemann equations iff .