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Before I ask the question, I know for sure this following question makes sense:

Prove that a necessary condition for $ f(z) $ to be analytic is that the Cauchy-Riemann equations are satisfied.

This question makes sense because if the Cauchy-Riemann equations are satisfied, $f(z)$ need not be analytic. Hence, why it's called a necessary condition.

But in order for $ f(z) $ to be analytic , there are 2 sufficient conditions, the Cauchy-Riemann equations must be satisfied and the partial derivatives must be continuous.

Now for the main question I want to ask:

Prove that a sufficient condition for $ f(z) $ to be analytic is that the Cauchy-Riemann equations are satisfied.

I'm asking this because I had it on my exam yesterday, now how could this make sense, the Cauchy-Riemann equations being satisfied should not be enough, the question did not mention the continuity of the partial derivatives, thus I treated this question as if it was asking "Prove that a sufficient condition for the Cauchy-Riemann equations to be satisfied is that $f(z)$ must be analytic".

So am I right and should I argue with my professor about that?

My question is not about when a function is holomorphic, my question is if the following makes sense as a proof question or not:

Prove that a sufficient condition for f(z) to be analytic is that the Cauchy-Riemann equations are satisfied.

khaled014z
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  • That's not my question. – khaled014z Jan 06 '19 at 23:18
  • They discuss why the Cauchy Riemann equations are sufficient for a continuous function to be analytic. Hence, the question makes sense. – SmileyCraft Jan 06 '19 at 23:23
  • Provided that the function and the partial derivatives are continuous, which are not mentioned in the question? – khaled014z Jan 06 '19 at 23:27
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    I may have misunderstood your question then. Does this answer your question? "The function given by $f(z)=e^{-z^{-4}}$ for $z\neq0$, $f(0)=0$ satisfies the Cauchy–Riemann equations everywhere but is not analytic (or even continuous) at $z=0$." https://en.wikipedia.org/wiki/Looman%E2%80%93Menchoff_theorem – SmileyCraft Jan 06 '19 at 23:31
  • Yes that is precisely what I meant, the C.R condition is satisfied, the other condition for analyticity is not mentioned in the question, that's the problem. – khaled014z Jan 07 '19 at 00:04

1 Answers1

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There is a diffrence between differentiability at a point and differentiability/analyticity in a neighborhood of a point. For example $f(x+iy)=\sqrt {|xy|}$ satisfies C-R equations at $0$ but $f$ is not differentiable at $0$. However, in an open set analyticity is equivalent to validity of C-R equations. Continuity of partial derivatives need not be assumed. In fact C-R equations imply that $f$ has continuous partial derivatives of all orders.

Kavi Rama Murthy
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  • So that means that the question should still mention that the function is differentiable everywhere, correct? So that the cauchy riemann equations when satisfied, imply the analyticity of $ f(z) $ 100 % (even at zero) – khaled014z Jan 06 '19 at 23:49
  • @khaled014z You are right. For differentiability in an open set the functions must have first partial derivatives at all points and these derivatives must satisfy C-R equations. – Kavi Rama Murthy Jan 06 '19 at 23:56