query: Determine all $z \in \mathbb{C} $, so that:$ | \frac{z-a}{1-\bar{a} z} | = 1,$ where $a \in \mathbb{C}$, $|a| < 1$.
positive_passages.docid: 26501
positive_passages.text: Proof using complex numbers
negative_passages.docid: 39492
negative_passages.text: Finding $z\in \mathbb C$ with $\sin z = 2$?
negative_passages.docid: 17677
negative_passages.text: Prove or show a counter example for: $\forall a, b, c \in \mathbb Z$, if $ab\mid c$ then $a\mid c$ and $b\mid c$
negative_passages.docid: 128256
negative_passages.text: How to prove $H_1(\mathbb{C}/\Lambda, \mathbb{Z})$ to the lattice $\Lambda$ ,$\gamma \mapsto \int_\gamma dz$ is well defined?
negative_passages.docid: 103757
negative_passages.text: how can I prove that $\frac{\arctan x}{x }< 1$?
negative_passages.docid: 127198
negative_passages.text: What is $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$?
negative_passages.docid: 69219
negative_passages.text: Prove there is a point $z\in[a,b]$ at which $f(z)=\frac{f(x_1)+f(x_2)+\cdots+f(x_k)}{k}$
negative_passages.docid: 55673
negative_passages.text: Alternate proof that $\exists\ x \in \Bbb{Z}$ such that $ \gcd (a+bx,c) = 1$?
negative_passages.docid: 96265
negative_passages.text: Why does $x= \frac{1}{2}(z+\bar{z}) = \frac{1}{2}(z+\frac{r^{2}}{z})$ on the circle?
negative_passages.docid: 45219
negative_passages.text: Proving that $C$ is a subset of $f^{-1}[f(C)]$
negative_passages.docid: 35242
negative_passages.text: If $a$, $b$, $c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$, and $\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are all integers, then $|a|=|b|=|c|$
negative_passages.docid: 74977
negative_passages.text: When does $az + b\bar{z} + c = 0$ represent a line?
negative_passages.docid: 158605
negative_passages.text: Finding The Laurent Series for $\sin(\frac{z}{1+z})$ at $z = -1.$
negative_passages.docid: 61466
negative_passages.text: what is $\int_{|z|=3}\frac{1}{(z-2)^n(z-4)^m}dz$ for $m,n\in\mathbb{N}$?
negative_passages.docid: 15604
negative_passages.text: Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$
negative_passages.docid: 36
negative_passages.text: Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$
negative_passages.docid: 33052
negative_passages.text: How to prove $\bar{m}$ is a zero divisor in $\mathbb{Z}_n$ if and only if $m,n$ are not coprime