Tagged Questions
0
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0answers
19 views
Complex Coordinate Foot Perpendicular
We suppose $A$ and $M$ two points their complex coordinates respectively are $a=1$ and $z(\theta)$ as $z(\theta)=\dfrac{1}{2}(1+e^{i\theta})^2$ and $\theta \in (-\pi;\pi)$.
What are the complex ...
1
vote
1answer
32 views
Proving $\arg(a)\equiv \alpha\Longleftrightarrow z_1z_2 \in \mathbb{R}$
Suppose the complex equation $iz^2+(2-i)az-(1+i)a^2=0$ as $a\in \mathbb{C}^{*}$.
$z_1$ and $z_2$ are the solution of this equation and we have also $z_1*z_2 = a^2(i-1)$.
How can I prove that ...
-1
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1answer
46 views
Set of points $M(z)$
Suppose $z \in \mathbb{C}$ and $a=-1+i$ and $\forall z \in \mathbb{C}\setminus{a}$ $\quad f_a(z)=\dfrac{az}{z-a} $ we suppose in a plan $(P)$:
$(D)=\{{M(z) \in (P), ...
0
votes
2answers
45 views
Set of points that fulfill a formula
Suppose $u=\frac{z(1+i)-i}{z+1}$ as $z\in \mathbb{C} \setminus\{-1\}$
What is the set of points $M(z)$ for which $u$ is a real number?
What is the set of points $M(z)$ for which $u$ is pure ...
2
votes
2answers
66 views
The Roots of Unity and the diagonals of the n-gon inscribed in the unit circle
I want to prove that the sum of the squares of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $2n$. So what I've done is I considered the $n$th roots of unity and said ...
0
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3answers
89 views
Interpretation of Complex Angles [duplicate]
Possible Duplicate:
Do “imaginary” and “complex” angles exist?
Do Situations ever arise where angles can be complex?
If they are already in use what geometrical or other interpretation do ...
7
votes
2answers
275 views
Why isn't $\log(-1)=i\pi$?
Reading http://people.math.gatech.edu/~cain/winter99/ch3.pdf, $\log(z)$ is defined as $=\ln|z|+i\arg(z)$. Looking on the Wessel plane, isn't $\arg(-1)=\pi$ (more generally $\pi \pm 2 \pi n$)? And ...
1
vote
1answer
96 views
Complex Numbers geometry question
on each edge of a quadrilateral ABCD you build a square such that the points H, G, F, E are the centers of these squares (the intersection of the diagonals).
I need to use complex numbers to prove ...
4
votes
2answers
127 views
Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?
I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
2
votes
1answer
201 views
Is there a geometric projection for every complex function
I was wondering about the best way to visualize complex functions. As they're $$ R^2 \rightarrow R^2$$ i think best way are complex plane image/grid transforms like they used in the Dimensions movie ...
1
vote
3answers
132 views
Geometrically interpret the set $\{z \in \Bbb C :$ $|z+a|+|z-a| \leq 2b,$ $b\in\Bbb R^+,$ $|a|<b\}.$
Geometrically interpret and determine the following set of complex numbers:
$$\{z \in \Bbb C : |z+a|+|z-a| \leq 2b , b\in\Bbb R^+,|a|<b\}.$$
I understand it means the sum of distance between ...
1
vote
1answer
707 views
Cross product in complex vector spaces
When inner product is defined in complex vector space, conjugation is performed on one of the vectors. What about is the cross product of two complex 3D vectors?
I suppose that one possible ...
0
votes
1answer
71 views
A complex function
I just need some help with the penultimate question of my coursework:
Let $w=f(z)=\coth(z/2)$. Show that $w=f(z)=h(C) = (C+1)/(C-1)$ where $C=g(z)=e^z$. Find the image of the given points, ...
3
votes
2answers
141 views
Complex sets of the form $|f(z)| = c$
I'm looking for information on sets of the form $|f(z)| = c$, possibly with additional restrictions (e.g. $|z| < d$). I can probably also assume that $f$ is as nice as you'd like.
I'm interested ...
11
votes
4answers
738 views
Equation of the complex locus: $|z-1|=2|z +1|$
This question requires finding the Cartesian equation for the locus:
$|z-1| = 2|z+1|$
that is, where the modulus of $z -1$ is twice the modulus of $z+1$
I've solved this problem algebraically ...
8
votes
2answers
249 views
If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear
If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear.
I really can't seem to get anywhere on this problem, ...
