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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?

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    $\begingroup$ 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 $0$. $\endgroup$
    – reuns
    Commented Dec 30, 2019 at 20:05
  • $\begingroup$ the identity principle for meromorphic functions it is cited here books.google.lu/… $\endgroup$ Commented Dec 30, 2019 at 20:13
  • $\begingroup$ The identity principle more generally says that a function determined on a space with an accumulation point has a unique continuation $\endgroup$ Commented Dec 30, 2019 at 20:20
  • $\begingroup$ Could you give me a reference where it is proved? $\endgroup$ Commented Dec 30, 2019 at 20:23
  • $\begingroup$ I presume any book on complex analysis? Are you studying this for a class? $\endgroup$ Commented Dec 30, 2019 at 21:04

1 Answer 1

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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.

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