I am a bit confused about differentiability in complex analysis. We showed in class that if a function $f$ is differentiable at $z_0$, then the Cauchy-Riemann equations hold at $z_0$. We also showed that if a function has continuous first partial derivatives at $z_0$, and the Cauchy-Riemann equations hold at $z_0$, then $f$ is differentiable at $z_0$. So apparently just satisfying Cauchy-Riemann at a point is not sufficient to determine differentiability.

So my question is: if one is given some function, what is a good way to see where this function is differentiable without going to the definition of the derivative?

  • $\begingroup$ You could always compute the first partials and write down the CR equations. If either one of those fails at some point, then the function is not differentiable there. $\endgroup$ – John Martin Dec 8 '12 at 5:30
  • $\begingroup$ A good way is what you described. Show that $f$ has continuous partial derivative at $z_0$ and that Cauchy-Riemann equations are satisfied at $z_0$. $\endgroup$ – P.. Dec 8 '12 at 5:44
  • $\begingroup$ But there may be points where $f$ does not have continuous partials, in which case we have to go back to the definition of the derivative, no? $\endgroup$ – nigel Dec 8 '12 at 5:56

Well, if a function $f$ is holomorphic (i.e. $\mathbb{C}$-differentiable), it is real analytic, then it has continuous partial derivatives. Therefore, checking the continuity of partial derivatives and then the CR-equations is, in some sense, a perfectly general method in order to determine whether a function is holomorphic or not.

By the way, it is absolutely true that satisfying the CR-equations is not enough: set $f(z)=e^{-z^{-4}}$ for $z\neq0$ and $f(0)=0$. Then $f$ has partial derivatives everywhere, also in the origin, and they satisfy the CR-equations everywhere, but $f$ is not $\mathbb{C}-$differentiable in $0$. Indeed, the partial derivatives are not continuous in $0$.

On a more operative side, there are many other ways to determine if a function is holomorphic or not. For example, if the partial derivatives exist (in the weak sense) and are locally integrable and satisfy the CR-equations, or if it is bounded and holomorphic outside a small enough set. Sums, products, compositions of holomorphic functions are holomorphic, and so on...

  • $\begingroup$ I an wondering whether $z^2$ or $z^3$ also works instead of $z^4$ in the exponentiation. $\endgroup$ – Groups Dec 31 '14 at 9:46

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