If we have a region R is $f(z)$ analytic in the region R if and only if it satisfy the Cauchy-Riemann equations for every point in R. If not what are the other conditions it must satisfy? Do we have to check that $f'(z)$ is continuous or is that given by the Cauchy-Riemann equations?
[Math] Are the Cauchy-Riemann equations a necessary and sufficient condition for a function to be analytic
analyticitycomplex-analysis
Best Answer
The standard result is that for functions $f$ of class $C^{1}$, $f$ is complex analytics iff $u,v$ satisfy the CR equations. There is however, a more powerful result: The Looman-Menchoff theorem allows us to replace $C^1$ above with $C^0=C$ (=the class of continuous functions).