# what is the identity principle for meromorphic functions

Let $$F(z)$$ and $$F(z,\theta)$$ be complex function with $$z \in \mathbb{C^+} \equiv\{z \in \mathbb{C}: Im(z)>0\}$$ $$,\theta \in \mathbb{C}$$ . Suppose that for $$\theta \in \mathbb{R}$$ we have $$F(z)=F(z,\theta)$$.

Now suppose that for $$\Omega^+\equiv\Omega \cap \mathbb{C^+}\neq \varnothing,F(z,\theta),\Omega^-\equiv\Omega \cap \mathbb{C^-}\neq \varnothing$$ can be meromorphically continued in $$z$$ from $$\Omega^+$$ to $$\Omega^-_\epsilon \subset \overline{ \Omega^-}$$ ,where $$\Omega \subset \mathbb{C}$$.

To proof that there exist a meromorphic continuation form $$\Omega^+$$ into $$\Omega^-_\epsilon \subset \overline{ \Omega^-},$$ textbook says that the identity principle for meromorphic functions says that there exists a function, meromorphic on $$\Omega$$ which equals to $$F(z)$$ on $$\Omega^+$$ and this function provide the continuation.

My question is what is the identity principle for meromorphic functions?

• I don't understand your question because of the horrible naming, the identity principle is that if a meromorphic function vanishes somewhere (on a small disk) then it vanishes everywhere. Then the only difficulty is to distinguish between the clockwide and the counterclockwise analytic continuation of $\log z$, the two are different because it has a branch point at $0$. Commented Dec 30, 2019 at 20:05
• the identity principle for meromorphic functions it is cited here books.google.lu/… Commented Dec 30, 2019 at 20:13
• The identity principle more generally says that a function determined on a space with an accumulation point has a unique continuation Commented Dec 30, 2019 at 20:20
• Could you give me a reference where it is proved? Commented Dec 30, 2019 at 20:23
• I presume any book on complex analysis? Are you studying this for a class? Commented Dec 30, 2019 at 21:04

The Identity Theorem for meromorphic Functions says that in region $$\Omega$$ the following statements about the meromorphic Functions $$f,g$$ are equivalent :
1. $$f=g$$
2. The set $$\{w \in \Omega \setminus (P(f) \cup P(g)) : f (w) = g(w)\}$$ has a cluster point in $$\Omega \setminus (P(f) \cup P(g))$$
3. There is a point $$c \in \Omega \setminus (P(f) \cup P(g))$$ such $$f^{(n)}(c)=g^{(n)}(c)$$ for each non-negative integer $$n$$
where $$P(.)$$ is the pole set.