query: If $f : D(0,1) \rightarrow \mathbb{C}$ is a function, $f^2$ is holomorphic, and $f^3$ is holomorphic, then prove that $f$ is holomorphic.
positive_passages.docid: 19399
positive_passages.text: $f^2$ and $f^3$ are holomorphic implies $f$ is holomorphic.
negative_passages.docid: 65901
negative_passages.text: Solution verification to complex analysis problem, prove $f$ is a polynomial.
negative_passages.docid: 131999
negative_passages.text: Can a holomorphic function satisfy $f(1/n)=1/(n+1)$?
negative_passages.docid: 23585
negative_passages.text: Show that a holomorphic function satisfying $|f(z)|\leq c|z|^n$ must be $f(z)=a\cdot z^n$ for some a.
negative_passages.docid: 92376
negative_passages.text: Let $f:A \rightarrow B$ and C ⊂A. Define f[C]={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
negative_passages.docid: 132907
negative_passages.text: The inverse of a bijective holomorphic function is also holomorphic
negative_passages.docid: 43136
negative_passages.text: Branch of the n-th root of a holomorphic function is holomorphic
negative_passages.docid: 23663
negative_passages.text: Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.
negative_passages.docid: 74622
negative_passages.text: Composition of holomorphic functions is holomorphic
negative_passages.docid: 111635
negative_passages.text: Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$.
negative_passages.docid: 123467
negative_passages.text: Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B $ onto $\mathbb{C} \setminus \{0\}$.
negative_passages.docid: 95632
negative_passages.text: Where is this function holomorphic?
negative_passages.docid: 141115
negative_passages.text: $f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$
negative_passages.docid: 114488
negative_passages.text: If $|f|+|g|$ is constant on $D,$ prove that holomorphic functions $f,~g$ are constant on $D$.
negative_passages.docid: 130679
negative_passages.text: Difference between Analytic and Holomorphic function
negative_passages.docid: 98118
negative_passages.text: Prove that a holomorphic function injective in an annulus is injective in the whole ball
negative_passages.docid: 87723
negative_passages.text: Show a holomorphic function on unit disc with with $|f'(z) - f'(0)| < |f'(0)|$ is injective