So I understand how to show that there is no solutions, but my lecturer said before that, I must show that $\displaystyle f(z) = \cot (z)$ for all $\displaystyle z \in \Omega$Suppose that a function $\displaystyle f$ that is analytic in some arbitrary region Ω in the complex plane containing the interval [1,1.2]. Assume $\displaystyle f(x) = \cot (x)$ for all $\displaystyle x \in [1,1.2]$. Show that $\displaystyle f(z) = -i$ has no solutions in Ω.

I'm not sure how to explain it. There are some theorems that say if $\displaystyle f(z) = ...$ (for z on some interval) has an accumulation point a the origin then $\displaystyle f(z) = ...$ for all z in the complex plane, but this equation has no accumulation point at the origin.

So how do I go about showing f(z) = cot(z) for all z, instead of just the small interval given?