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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?

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2 Answers 2

up vote 5 down vote accepted

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 $[0/0]$ are solved by expansions in power series; De l'Hospital 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 $g$ arounf $z=a$ 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.

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

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