l'Hopital rule on $\mathbb{C}$

Question

Is the L'Hopital's rule true when using limits on $\mathbb{C}$ (the complex field)?

I don't know if it is only valid using the real numbers $\mathbb{R}$... is that the case?

Answer 1

The answer is both trivial and difficult. If you translate "differentiable" to "holomorphic", then much is true in $\mathbb{C}$. Indeed, holomorphic functions are power series, and many limits of the form $[\frac{0}{0}]$ are solved by expansions in power series; The l'Hopital is essentially a first-order expansion. If a limit $$\lim_{z \to a} \frac{f(z)}{g(z)} = \left[\frac{0}{0}\right],$$ you may expand both $f$ and $g$ around $z=a$ and hope to solve the limit.

The case $[?/\infty]$ is harder, since there is no $\infty$ in $\mathbb{C}$ that playes the same role as $\pm\infty$ in $\mathbb{R}$. You may take the modulus of a function, but this can break its analyticity.

Answer 2

It is valid in $\mathbb C$ as well. You need the compex version of differentiability (aka. analytic/holomorphic function), though.