Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Possible Duplicate:
When is a function satisfying the Cauchy-Riemann equations holomorphic?

If real the functions $u(x,y)$ and $v(x,y)$ satisfy the Cauchy-Riemann equations and have continuous partial derivatives in an open set $U$, then the function $f(z)=u(x,y)+iv(x,y)$, where $z=x+iy$, is analytic in $U$. Are there less restrictive conditions on $u$ and $v$ to ensure the analyticity of $f$? Thanks.

share|cite|improve this question

marked as duplicate by PEV, Akhil Mathew Apr 6 '11 at 4:53

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

There actually are none. One of the more beautiful theorems of Complex Analysis is that a function is holomorphic if and only if it is continuous and satisfies the Cauchy-Riemann equations.

share|cite|improve this answer
Actually, this answer might be regarded as "less restrictive conditions" since the original question required the partial derivatives to be continuous. – Robert Israel Apr 6 '11 at 4:47

Not the answer you're looking for? Browse other questions tagged or ask your own question.