Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.
5
votes
1answer
56 views
Why is the bailout value of the Mandelbrot set 2?
For the past few days I've been studying the Mandelbrot set, and many say that if the iterations of a point stay within a magnitude of 2, the point converges. A very natural question of "why is the ...
1
vote
1answer
29 views
Factoring any single-variable polynomial in $\mathbb C$
The fundamental theorem of algebra says
$$
\forall p(x):\mathbb C \to \mathbb C,\ p(x) = a\prod_{n=0}^m\big(b_nx+c_n\big)
$$
where $p(x)$ is a single-variable polynomial, and $\{a;m\}\cup\{\forall ...
0
votes
2answers
42 views
If $w= \frac{3z+i}{i-z}$ show that $\Re(z) \ge0$ implies $\Im(z)\le0$
I have exam after two days and I want answer of the question please help me as fast as you could , thanks everyone .
If $w= \frac{3z+i}{i-z}$ show that $\Re(z) \ge0$ implies $\Im(z)\le0$?
6
votes
1answer
81 views
Why does $\sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )+{1\over2}-{1\over3} = \gamma$?
How could one prove that
$$x = \sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )$$
is such that $x+{1\over2}-{1\over3} = \gamma$ ?
I am having problems just calculating ...
2
votes
1answer
27 views
Inequality holding for complex numbers in the unit disk
In Nehari's book Conformal Mapping he gives it as an exercise to prove that for $a,b\in \mathbb{C}$, $|a|, |b| <1$ we have $$\frac{|a|-|b|}{1-|ab|} \leq \left|\frac{a-b}{1-\overline{a}b}\right| ...
0
votes
2answers
34 views
Find $\sum(\alpha_i \alpha_j)^5$
If $\alpha_1,\alpha_2,...,\alpha_{100}$ are the 100th roots of unity then find
$\sum(\alpha_i \alpha_j)^5$
0
votes
2answers
12 views
Finding measure angle
If $P$ and $Q$ are points on the unit circle corresponding to complex numbers $\frac{1}{2}+\frac{\sqrt{3}}{2}i$ and $- \frac{1}{2}+\frac{\sqrt{3}}{2}i$ resp. What is the measure of angle POQ?
Let ...
0
votes
2answers
70 views
Could we define multiplication of “complex numbers” in this way?
If we define multiplication of complex numbers as follows:
$$z_1 \cdot z_2=(x_1x_2+y_1y_2, x_1y_2+x_2y_1)$$ then it can be shown that it induces a group structure $(G, \cdot)$, because it has inverse ...
1
vote
1answer
57 views
Proving that: $-\frac{2 \;i\log(i^2)}{2} = \pi$
I'm trying to prove that: $$-\frac{2 \;i\log(i^2)}{2} = \pi$$
This is what I've tried:
$$-\frac{2 \;i\log(i^2)}{2} = -i \log(i^2) = -i (i \pi)\implies$$
$$-x\;(x y)=-x\;y\;x\implies$$
$$-i\;(i \pi) ...
1
vote
0answers
20 views
Numerical Methods for eigen values of $A \in \mathbb{C}^{n \times n} $
I've been writing a linear algebra library in c# for a while as an intellectual exercise and its gotten vastly more sophisticated that I originally thought it would and when I started adding methods ...
2
votes
0answers
79 views
Number of solutions for $x$ of such form that $\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$
Consider
$$\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$$
where $y$ is an integer.
In relation to solutions for $x$; How could one prove that:
$(1)$: There are $y$ solutions for $x$, in total.
...
1
vote
1answer
21 views
BesselI and BesselK with complex arguments
I would like to check with you the following issues.
During my work, I ended up with following Bessel Functions.
BesselI(0, i*x) and BesselK(0, i*x). (Modified bessel functions of first and second ...
0
votes
1answer
44 views
Does $i = -\frac{(2\;W({\pi\over2}))}{\pi}$
Let $x = -\frac{(2\;W({\pi\over2}))}{\pi}$, where $W$ denotes the Lambert W-function.
As
$${\log(i^2)\over i} = \pi$$
and $${\log(x^2)\over x}=\pi$$
Does $x = i$?
1
vote
2answers
28 views
Expressing Complex Number in terms of its conjugate [duplicate]
Given a complex number $z$ , is it possible to express its conjugate $\bar z$ in terms of $z$ using only operations of addition , subtraction , multiplication , division and exponentiation on $z$ as a ...
2
votes
0answers
28 views
Set of all odd complex polynomials - complex vector space
Is the set of all odd complex polynomials a complex vector space?
I'm given the following definition of a vector space:
A vector space $V$ over the field $\mathbb F$ is a set $V$ of vectors, a field ...
0
votes
4answers
86 views
Complex numbers properties
I know that $z \overline z$ is $|z|^2$, but what about $z^5{\overline {z}}^5$ and $z^{25}{\overline {z}}^{25}$ or $z^{100}{\overline {z}}^{50}$?
0
votes
4answers
70 views
Find all $z \in \Bbb{C}$ such that $z^3 = \overline{(-5z^2)}$
Find all $z \in \Bbb{C}$ such that $z^3 = \overline{(-5z^2)}$
How to solve this?
I have to apply de formula of Moivre?
2
votes
1answer
90 views
Complex numbers true or false
Are there any complex numbers "z" that satisfy this equation? $$z=-\bar z?$$
2
votes
2answers
68 views
As inspired by a previous question, why is this derivation invalid?
So I worked along the lines of the following:
$$
\left( \cos \left( \theta \right) + i \sin \left( \theta \right) \right)^{\alpha} = \left( e^{i \theta} \right)^{\alpha} = e^{i (\theta \alpha)} = ...
6
votes
4answers
147 views
If $\theta\in\mathbb{Q}$, is it true that $(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$?
Is the following true if $\theta\in\mathbb{Q}$?
$$(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$$
Is it true if $\alpha\in\mathbb{R}$? In each case, prove or give a ...
1
vote
1answer
39 views
finding all complex roots of equation
let $z = 1 +i$
Find all complex solutions such that $z^2 + \bar z^2 = 0$.
My working out:
$z^2 = -\bar z^2 = -(1-i)^2 = 2i$
so $z^2 = 2i$
hence $r^2 = 2 \implies r = \sqrt 2$
mod: $2\theta = ...
1
vote
2answers
54 views
Do any issues arise if we try to raise an element of $\mathbb{R}^+$ to an element of $\mathbb{C}$?
If $a$ and $b$ are non-zero natural numbers, the definition of $a^b$ is clear. Now it seems to me that there are (at least) two distinct ways of generalizing to larger number systems.
Firstly, given ...
2
votes
5answers
80 views
Complex number questions.
I have an exam on this kind of stuff on Monday and was wondering if anyone could help me with these questions and can you tell me what this type of question is so i can go away and revise it for ...
0
votes
1answer
26 views
Proof using Möbius transformation
Let D be the open unit disc, and $f:D\to D$ an analitic function.
How can I prove that $|f'(0)|\le1$?
5
votes
4answers
88 views
complex Analysis - Absolute values
A. It is possible to find distinct $z_1,z_2,z_3\in \mathbb C$ such that $|z_1-z_2|=|z_1-z_3|=|z_2-z_3|$.
Answer: True
B. It is possible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that ...
1
vote
1answer
44 views
If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Is $f(x+iy) = \exp(x+iy)$?
$(1)$ If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Then, is $f(x+iy) = \exp(x+iy)$ for every $x$ and every $y$?
$(2)$ If $f$ is an entire function such that ...
1
vote
1answer
50 views
Trouble with complex numbers
Is my following calculation true?
$e^{a+ib}e^{\overline{a+ib}}=e^{a+ib}e^{a-ib}=e^{2a}$? for a,b real numbers
or in general, what is $\overline{{z}^{w}}$ if $z,w$ are complex numbers?
4
votes
5answers
65 views
find the complex number that satisfies the following conditions
Find all values of $z \in \Bbb C$ such that: $z + \bar{z} = 18$ and $z.\bar{z} = 84$.
I don't know how to get that values, someone can help me to solve this?
0
votes
0answers
23 views
Problem of finding General value
The general value $e^i$ is given by___
$$e^i=e^{\cos(2n\pi+\pi/2)+i\sin(2n\pi+\pi/2)}=e^{e^{2n\pi+\pi/2}}, \quad \forall n\in I$$
Is it right? But here, I need answer $e^{-(2n\pi+\pi/2)}, \forall ...
0
votes
0answers
20 views
(Theoretical question) How to evaluate $|f(z_n)|^n$ if $|f(z_n)|<1$ but $\lim z_n=z_0$ with $|z_0|=1$
I often have a problem as follows: Let $f$ be a holomorphic function and $(z_n)$ be a sequence of complex numbers such that $|f(z_n)|<1$ for every $n$. So how can we evaluate $|f(z_n)|^n$ when $n$ ...
1
vote
2answers
50 views
Calculating $\sum^{10}_{k=1}\left(\sin\frac{2k\pi}{11}+i\cos\frac{2k\pi}{11}\right)$
Find the value of $$\sum^{10}_{k=1}\left (\sin\left (\frac{2k\pi}{11} \right )+i\cos\left (\frac{2k\pi}{11}\right ) \right)$$
My approach:
Since $\cos\theta + i\sin\theta = e^{i\theta}$, we can ...
3
votes
2answers
62 views
Finding two eigenvalues which add to $1$
$\textbf{Question}$: For $0<t<\pi$, the matrix
$$
\left( \begin{array}{cc}
\cos t & -\sin t \\
\sin t & \cos t \\
\end{array}
\right)
$$
has distinct complex eigenvalues $\lambda_1$ ...
9
votes
2answers
177 views
Wild automorphisms of the complex numbers
I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could ...
3
votes
1answer
45 views
complex exponential equation
I am trying to solve the following exponential equation: $z^{1+i} = 4$ where the argument of $z$ is between $-\pi$ and $\pi$. Here is what I have gotten so far: If $z = a + bi$ then the magnitude of ...
1
vote
4answers
76 views
Square roots of complex numbers [duplicate]
I know that the square root of a number x, expressed as $\displaystyle\sqrt{x}$, is the number y such that $y^2$ equals x. But is there any simple way to calculate this with complex numbers? How?
0
votes
3answers
42 views
complex equation to be solved
I need to find all solutions to the complex equation $e^{1/z} = \sqrt{e}$
Then I need to show that all these solutions are on the circle $|z-1|=1$
Using the fact that $e^{2\pi i}=1$, I solved the ...
0
votes
2answers
48 views
Remainder of a complex function
Dividing $f(z)$ by $z-i$, the remainder is $1-i$ and by dividing $z+i$ the remainder is $1+i$, then what is the remainder when $f(z)$ is divided by $z^2+1$?
I just started solution using division ...
1
vote
0answers
26 views
Representation of Complexification of Lie Algebra
Is the following obvious? I think it is, but wanted to make sure before an exam tomorrow!
"There is a bijection between the complex representations of a real Lie algebra and the complex ...
0
votes
2answers
65 views
Finding all the solutions to a complex equation
I am asked to find all the solutions to $z^{42}=-1$. I go a head and square root both sides to produce $z^{21}= i$. Then I can write $z^{21}= r^{21} (\cos(21\ θ) + i\sin(21\ θ)) = 0+i. $ Hence ...
0
votes
2answers
23 views
Sides of the Right angled Triangle in Complex notation.
If $z=a+ib$ is a complex number, then $z, iz, z+iz$ represents sides of the right angled triangle.
I got this result through Cartesian form, i,e. $(a,b),(-b,a) and (a-b,a+b)$ are the vertices of the ...
1
vote
5answers
66 views
How does $Ae^{4ix}+Be^{-4ix}=A\cos(4x)+B\sin(4x)$?
$e^{ix}=\cos(x)+i\sin(x)$
$Ae^{4ix}=A(\cos(4x)+i\sin(4x))$
$Be^{-4ix}=B(-\cos(4x)-i\sin(4x))$
What am I doing wrong?
I am trying to find the complimentary function of $\frac{d^2y}{dx^2} ...
0
votes
5answers
74 views
How do you solve $w^4=16(1-w)^4$?
Giving you answer in Cartesian form.
$\dfrac{w^4}{(1-w)^4}=16$ Are you supposed to let $w=x+yi$?
$w^4=x^4+4x^3yi-6x^2y^2-4xiy^3+y^4=16$
I then know that you get routes 2,-2,2i,-2i But I ...
2
votes
0answers
40 views
First time dealing with limits with complex numbers in it.
I am solving the following problem.
Investigate the behavior (convergence of divergence) of $\Sigma a_n$ if
$$a_n = \frac{1}{1+z^n}, \quad \text{ for } z \in \Bbb C.$$
First of all, I am ...
3
votes
3answers
59 views
Simplifying a quotient of complex numbers
Given the equation I am supposed to simplify :
$$\frac{(7 - 4i)}{(5 + 3i)}$$
I conclude that I should first multiply both the numerator and denominator by $(5 - 3i)$ (note : or by $7 + 4i$ but ...
1
vote
2answers
37 views
Show that $\sin6\alpha\equiv \sin2\alpha(16\cos^4\alpha-16\cos^2\alpha+3)$
$$\sin6\alpha\equiv \sin2\alpha(16\cos^4\alpha-16\cos^2\alpha+3)$$
Can you help me with De Moivre's theorem and how I would go about tackling this question.
I understand that De Moivre's theorem ...
2
votes
2answers
64 views
Showing $\left(\frac{z+i}{z-i}\right)^n = -1$ implies $z$ is real
I have shown the following identity: $$ \frac{1+e^{i \theta}}{1-e^{i \theta}} = \frac{1}{2}\cot\left(\frac{\theta}{2}\right) $$ And I now need to use this to show that the the equation: $$ ...
0
votes
1answer
29 views
Complex numbers - separate real/imaginary parts
$$K(\omega) = \frac{1}{1 + j\omega RC}$$
Uhm...How do I separate the real part from the imaginary part here? :U And how can I find the argument? I mean, if the document I got this formula from is ...
1
vote
1answer
36 views
Radius of convergence of power series (complex)
I don't know if my reasoning is right on this exercise:
If the power series $\sum a_n z^n$ has radius of convergence $R$, which is the radius of convergence of the series $\sum a_n^2 z^n$ and $\sum ...
2
votes
0answers
44 views
Overdetermined system - showing that there are no roots that satisfy the set of equations
We consider an overdetermined set of equations, consisting of two equations for one complex variable $x$. I want to show that there are no roots for $x$ in the complex unit disc but without the ...
2
votes
1answer
40 views
How to show that $|z| \geq 0$ and $|z|=0$ if and only if $z=0$?
Teaching myself Linear Algebra and got stuck on the following question for complex numbers:
Show that $|z| \geq 0$ and $|z|=0$ if and only if $z=0.$
Now, the question itself seems pretty obvious where ...
