0
votes
0answers
28 views

Complex Analysis Questions Compilation

All of my questions are in relation to Gamelin's Complex Analysis. How was the parametric form of the line from the North Pole on the unit sphere through a point P come to be? It is $$ N + t (P-N) ...
3
votes
1answer
74 views

Show that a complex expression is smaller than one

This is the question I'm stumbling with: When $|\alpha| < 1$ and $|\beta| < 1$, show that: $$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$$ The chapter that contains ...
1
vote
1answer
41 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
votes
1answer
52 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
58 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
126 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
votes
3answers
161 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
568 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
115 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 ...
6
votes
2answers
227 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
225 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
4answers
169 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
1k 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
90 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
152 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
2k 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
302 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, ...