Is L’Hopitals rule applicable to complex functions?

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:

limz0log(1+z)z?=limz01(1+z)=1

Now i’m wondering if this is correct? Also if we take |z|<1, is it then correct?

Thanks,

Answer

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 limz0, then it's automatic that only small values of z are in play. Saying "take |z|<1" is redundant.

Generally, you have a point aC and some neighborhood of a in which f,g are holomorphic. If f(a)=g(a)=0, then
limzaf(z)za=f(a),limzag(z)za=g(a)
hence
limzaf(z)g(z)=limzaf(z)/(za)g(z)/(za)=f(a)g(a)
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.

Attribution
Source : Link , Question Author : user112167 , Answer Author : Post No Bulls

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