query: Prove $\ \frac{z-1}{z+1} $ is imaginary no' iff $\ |z| = 1 $
positive_passages.docid: 147986
positive_passages.text: Prove: if $|z|=1$ than $\frac{z+1}{z-1}$ is an imaginary number
negative_passages.docid: 25967
negative_passages.text: Prove $\csc(x)=\sum_{k=-\infty}^{\infty}\frac{(-1)^k}{x+k\pi}$
negative_passages.docid: 111976
negative_passages.text: Prove that a product of two complex numbers has zero imaginary part
negative_passages.docid: 61782
negative_passages.text: How to prove that $f(x) = \frac{-2x+1}{(2x-1)^2-1}$ is one-to-one on $(0,1)$?
negative_passages.docid: 90343
negative_passages.text: Proving the inequality $|e^z-1|\leq e^{|z|}-1$
negative_passages.docid: 89728
negative_passages.text: Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
negative_passages.docid: 64096
negative_passages.text: Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.
negative_passages.docid: 52800
negative_passages.text: Show that $z^5-1=(z-1)(z^2+2z\cos{\pi \over 5}+1)(z^2-2z\cos{2\pi \over 5}+1)$
negative_passages.docid: 138559
negative_passages.text: How can I show that $\frac{1}{1+|z|^2}$ isn't analytic at $z=1$?
negative_passages.docid: 156766
negative_passages.text: Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
negative_passages.docid: 87941
negative_passages.text: Prove by mathematical induction that $\forall n \in \mathbb{N} : \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k} $
negative_passages.docid: 73892
negative_passages.text: Prove $Σ_{cyc}\frac{1}{\left(a+1\right)^2}+\frac{1}{a+b+c+1}\ge 1$
negative_passages.docid: 64941
negative_passages.text: Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?
negative_passages.docid: 158524
negative_passages.text: Complex numbers $z$ such that $|z|= 1$
negative_passages.docid: 5526
negative_passages.text: Prove that $ℤ[i]^*= \{1,-1,i,-i\}$
negative_passages.docid: 79657
negative_passages.text: Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$
negative_passages.docid: 77076
negative_passages.text: Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$