# Tagged Questions

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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### How find all complex numbers such that: $|\,1 - z\,| < k\ (1 - |\,z\,|\, )$?

Let $k > 1$ be a real number. How may one find all complex numbers such that: $|\,1 - z\,| < k\ (1 - |\,z\,|\, )$? ..................................................................................
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### Complex numbers $x$, $y$, and $z$ are collinear, show that there exist $a,b,c$ for which $ax+by+cz=0$ where $a+b+c=0$

Suppose $x$, $y$, and $z$ are collinear (complex numbers). Prove that there exist real constants $a$, $b$, $c$, not all zero, such that $ax+by+cz=0$ where $a+b+c=0$. This is how I did it: First, ...
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### Evaluate the given limit in $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$

Let , $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$ denotes the semicircle traversed clockwise. Show that $$\lim_{r\to 0}\int_{C_r}\frac{e^{iz}}{z(z^2+1)}\,dz=-\pi i$$ I can not use the Jordan's ...
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### Questions on whether imaginary number is larger than $0$

The imaginary number implies $i=\sqrt{-1}$ But this this not say whether $i$ is larger or smaller or equal to $0$ So i wonder if $i$ can be larger or smaller than $0$ and if so how do we see it
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### Why is $\sqrt{xy}=\sqrt{x}\sqrt{y}$ also true when $x=-1$, making it $i$?

A : If: $$\sqrt{xy}=\sqrt{x}\sqrt{y}$$ only when $x,y>0$, B : Then why can I do this: $$\sqrt{-4}=\sqrt{4\times-1}=\sqrt{4}\sqrt{-1}=2i$$ which violates A since $y<0$ C : But why can I not ...
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### Show that $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ represents a circle

Suppose $|a|<1$ and $r\in (0,1)$. Show that the set of complex number $z$ satisfying $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ is a circle in complex plane. Find the centre and radius of this circle....
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### Find all the complex numbers that satisfy this quotient.

A certain problem that I have been working on involves the equation $$1 = \frac{1}{1-n}$$ One can see that the only real-number solution is $n=0$. As far as the original problem goes, that is ...
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### Solving $z^z=z$ in Complex Numbers

I wanted to find all complex numbers $z\neq0$ such that $z^z=z$. I observed that $z=\pm1$ satisfies the equation. But I had problems when tried to find all the possible solutions since $z^z$ may take ...
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### Complex inequality question

I am trying to understand why the following holds: \begin{align*} \Re((1-\imath)(A+B)) \geq \Re((1-\imath)A) - \sqrt{2}|B|, \end{align*} where, \begin{align*} A:= \sum_{x=1}^{[\sqrt{k}]} e\left(\frac{...
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### Why is $\sqrt{-x}*\sqrt{-x}=-x?$

Q1 - Why is $\sqrt{-x}*\sqrt{-x}=-x?$ Q2 - I was thinking it would be: $\sqrt{-x}*\sqrt{-x}=\sqrt{-x*-x}=\sqrt{x^2}$ but apparently not (why not?) Q3 - What are the formal algebra rules to use? Can ...
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### Evaluating a complex integral using resiue theorem

Evaluate the integral $$\int_{|z+1|=2} \frac{z^2}{4-z^2}dz$$ Solution : So $|z+1|=2$ is the circle of radius 2 centered at -1. Now inside this circle $\frac{z^2}{4-z^2}$ is analytic except for a ...
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### Cardinality of a set of complex numbers

The question is basically to find the number of elements in the set $\{z \in \mathbb{C} : z^{60} = -1 , z^k \neq -1, 0<k<60 \}$. As is quite obvious with the kind of question,I am a self-...
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### Proving that two or three segments are concurrent using complex numbers or vectors.

For example if we have a triangle and we want to prove that the medians all intersect at a point, using complex numbers (or vectors); how do we do that? (This is not my main question) My problem is ...
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### Polynomial with exactly one complex root

Is it possible that a polynomial of degree $n$ with real coefficients has exactly one complex root? I saw https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem but wondered if this can happen ...
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### Inverse image of $[-2,2]$ under cosine.

I solved the following problem: Let $g(z) = \cos z$. Find $g^{-1}[-2,2]$. but my solution was kind of long. I was wondering if there was a faster way to do this problem. Here's my solution: ...
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### Why is $-i^3 = i$?

Why is the value of $-i^3$ equal to $i$? After experimenting, I got this result - $-i^3=-i^2\cdot -i=1 \cdot -i=-i$ What is the error in my proof? EDIT Here is the original proof - -i^3=\left(\...