query: Let $f$ be an entire function. And assume $f(z)$ is real on $Im(z)=0$ and $Im(z)=\pi$ lines. Show $f$ is $2\pi i$ periodic.
positive_passages.docid: 152701
positive_passages.text: A question on periodicity of complex functions
negative_passages.docid: 33489
negative_passages.text: |f| is periodic implies f is periodic
negative_passages.docid: 39921
negative_passages.text: show that f is constant on I
negative_passages.docid: 122356
negative_passages.text: Complex Analysis, Entire functions
negative_passages.docid: 7755
negative_passages.text: Question on entire function
negative_passages.docid: 31319
negative_passages.text: An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.
negative_passages.docid: 64650
negative_passages.text: If $a\operatorname{Re}(f(z))+b\operatorname{Im}(f(z))=c$ then f is constant
negative_passages.docid: 58647
negative_passages.text: Show that $f$ is integrable and $\int_{I} f=0$
negative_passages.docid: 26687
negative_passages.text: Show that $f$ maps the unit circle to itself
negative_passages.docid: 82488
negative_passages.text: Periodic solution of differential equation $y′=f(y)$
negative_passages.docid: 84103
negative_passages.text: Let $f(z)$ be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem
negative_passages.docid: 65548
negative_passages.text: Prove if f(z) is entire function s.t $|f(z)|\leq M|z|$ then $f(z)$ Is polynomial of degree $1$
negative_passages.docid: 147270
negative_passages.text: Entire function. Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$
negative_passages.docid: 54499
negative_passages.text: How do I show that all continuous periodic functions are bounded and uniform continuous?
negative_passages.docid: 82229
negative_passages.text: Show if complex function $f(z)$ contains all real coefficients then $f(\bar z)$ = $\bar {f(z)}$
negative_passages.docid: 14759
negative_passages.text: Prove that $ f(z) = 0 $ for all $ z \in \overline{D}(0,1) $.
negative_passages.docid: 27772
negative_passages.text: Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.