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 .