I have a question about something I’m wondering about. I’ve read somewhere that
L’Hopitals rule can also be applied to complex functions, when they are analytic.
So if have for instance:
Now i’m wondering if this is correct? Also if we take |z|<1, is it then correct?
L'Hopital's rule is a local statement: it concerns the behavior of functions near a particular point. The global issues (multivaluedness, branch cuts) are irrelevant. For example, if you consider limz→0, then it's automatic that only small values of z are in play. Saying "take |z|<1" is redundant.
Generally, you have a point a∈C and some neighborhood of a in which f,g are holomorphic. If f(a)=g(a)=0, then
Note that the above is a simple special case of the L'Hopital's rule, because we have (1). It's basically just the definition of derivative.