Ok. I am used to see written everywhere the following Cauchy-Rieamman equations for complex functions:
Now, today I saw this other notation
and I'd like to understand how it maps to the first. It must be pretty simple, I guess?
Yes it is, but only after one understands what's going on here: you have a complex variable function , but we can put , where and is the representation of a complex number in the complex plane.
Thus we can write , with the usual division of the function in its real and imaginary functions
Now, finally, applying the rule for partial derivatives of multivariable functions we get:
Now just compare real and imaginary parts in both sides.