Complex Analysis Concept Question

[tylerc1991]

Homework Statement

Just to make certain that I am understanding this correctly, given a function f(z) = u(x,y) + iv(x,y), the existence of the satisfaction of the Cauchy-Riemann equations alone does not guarantee differentiability, but if those partial derivatives are continuous and the Cauchy-Riemann equations are satisfied then this is sufficient conditions for the function to be differentiable? Conversely, given a function that is known to be differentiable, the satisfaction of the Cauchy-Riemann equations must exist? So essentially the Cauchy-Riemann equations are useful in disproving the existence of differentiability of some functions, but cannot necessarily prove differentiability of others? Thank you for your help.

 

[micromass]

Yes, you are correct. A function is complex differentiable if and only if the partial derivatives are continuous and satisfy Cauchy-Riemann.