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Let $U\subseteq \mathbb{C}$ be a region and let $h,g$ meromorphic functions in $U$. Suppose that there exist a set $W\subseteq U$ with a acumulation point in $U$ with $h(w)=g(w)$ for all $w\in W$.

If $h$ has an essential singularity in $U$. Can I say that $h=g$?

Remark: The case that $h$ has no essential singularities was solved in the comments of the post:'Identity theorem' for Meromorphic functions.

Therefore, I think that the important question is what happens if $W$ has the limit point in an essential singularity of $h$ in $U$.

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  • meromorphic on $U$ functions don't have any essential singularities on $U$, so maybe you mean an essential singularity on $\partial U$ ? – reuns May 10 '16 at 04:15
  • and a counter-example is $g(z) = 0, h(z)= \sin(1/z)$ on $Re(z) > 0$, which are equal whenever $\sin(1/z)= 0$ i.e. $1/z = k \pi$, $z = \frac{1}{k \pi}$, so $W = {\frac{1}{k\pi} \ \mid \ k \in \mathbb{N}^*}$ works. of course if $W$ contains a non-empty open (or even a curve), then clearly $g= h$ on this open and hence on $U$. (and replacing open by $C^1$ curve also works) – reuns May 10 '16 at 04:19
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    @user1952009 If the essential singularity is not an accumulation point, then we can say $f = g$? – Diego Fonseca May 10 '16 at 15:49
  • @DiegoFonseca : ?? – reuns May 11 '16 at 04:00
  • I think that everything is answered, see the linked discussion for the sufficient conditions for saying that $f=g$ (on $U$) – reuns May 11 '16 at 04:07

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