Cauchy-Riemann Equations Question?

strictly speaking, if a function does not satisfy the C-R equation at a given point, then it is not differentiable at that point (because taking the derivative along a different direction in the complex plane would yield differing results). However, there are several functions that satisfy the C-R equations over some but not all points.

For example, the function f(z) = Log(z) is analytic (hence differentiable) except along its "branch cut."

In sum: as far as complex analysis goes, the terms "differentiable" and "analytic" apply to the same situations.

it particularly is particularly somewhat resembling differentiability in distinctive actual dimensions. The C-R equations handle partials, so we gained't get the full image from them. in spite of the undeniable fact that, if we additionally comprehend that the function is non-end, then all of us comprehend that it is analytic. in short: Analytic <-> (C-R holds and non-end) See the source occasion 3.6 for a function the place C-R holds at a component, however the function isn't analytic in any open community of the component. @JB: It sounds such as you may desire to have referenced wikipedia. the subject is that the wikipedia article demands u and v to be differentiable => u and v non-end => f non-end. i'm thinking that the wiki web page desires somewhat cleanup to steer away from this confusion... 2d Edit: regrettably i don't have my diagnosis texts with me on the 2d. i became utilising "not analytic at a component" to intend "not analytic in any open community of the component", it is in simple terms the belief of a singularity. I replaced it as a manner to steer away from from now on confusion.