I'm just doubtful whether my proof(s) for showing functions are complex differentiable suffice as valid proofs.
e.g. let's take the classic example $f(z)=\bar{z}$.
You inevitably know from studying complex analysis that this function isn't differentiable for any $z \in \mathbb{C}$ and I can prove such using limits.
However, is there anything wrong with instead using the Cauchy Riemann equations and just saying
$f(z)=x-iy \Rightarrow \frac{\partial u}{\partial x}=1, \frac{\partial v}{\partial y}=-1$
Therefore $f(z)$ isn't complex differentiable for any $z \in \mathbb{C}$ as $\frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y}$?
Just in general for many functions, when it comes to showing they're not complex differentiable, I'm able to show this using limits but it's a whole lot easier just using the Cauchy Riemann equations. Am I allowed to do this for a valid proof?
Best Answer
Yes, the cauchy reimann equations are a necessary condition for complex differentiability.
So if they do not hold, then it is not complex differentiable. but it is good to know how to use the limits in case you are asked to specifically not use the equations.