Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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2answers
100 views

Could anyone explain this curious solution to $7^{2x}= 2^x$

I typed the following on wolfram alpha today : $7^{2x} = 2^x$ and found this as a solution besides $x=0$: $x = \dfrac{2\pi i n}{\log2 - 2\log7}$ where $log$ has a base of $e$ and $n$ is any integer. ...
5
votes
3answers
1k views

Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
5
votes
2answers
220 views

Number of Complex Roots of a Complex Polynomial

This is related to the question I asked regarding finding the complex roots of $z^3+\bar{z}=0$. It turned out that there were 5 complex roots, but because the equation was of degree 3 I was only ...
5
votes
3answers
417 views

Determine the set $\{w:w=\exp(1/z), 0<|z|<r\}$

I can't do this exercise of Conway's Book: For $r>0$ let $A=\{w:w=\exp(1/z), 0<|z|<r\}$, determine the set $A$. Any hints?
5
votes
1answer
113 views

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ .

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ ? How to find if it is convergent or not? Thanks!
5
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2answers
80 views

Is $\mathrm{GL}_n(\mathbb C)$ divisible?

A group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ Is the group $\mathrm{GL}_n(\mathbb C)$ divisible? Or more precisely, ...
5
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2answers
174 views

Complex analysis : If $z =re^{i\theta}$, then prove that $|e^{iz}| =e^{-r\sin\theta}$

Problem : If $z =re^{i\theta}$, then prove that $|e^{iz} | =e^{-r\sin\theta}$ My working : $z = re^{i\theta} = r(\cos\theta + i\sin\theta)$ $\Rightarrow iz = ir(\cos\theta +\sin\theta) $ = ...
5
votes
3answers
92 views

$w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$

I am stuck on the following problem: Let $w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$.Then the triangle in the complex plane with $w_1,w_2,-1$ as vertices ...
5
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2answers
90 views

If $\left |z-3\right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$

If $\left |z-3 \right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$. I have got to the point where I have $\left |z \right |= \sqrt{x^2+(y+1)^2} = \sqrt{(x-3)^2+y^2}$ But really ...
5
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2answers
191 views

The roots of the derivative $P'(z)$ of the polynomial $P(z)\in\mathbb C[x]$ lie in the convex hull of the set of roots of $P(z)$.

Assume $S=\{z_1,z_2,...,z_k\}, z_i\in \mathbb C$$, C(S)$ and define $$C(S):=\{z=a_1z_1+a_2z_2+...+a_kz_k | a_i\ge0 ,a_1+a_2+...+a_k=1\}$$ where $$A:=\{z\in \mathbb C:f(z)=0 ...
5
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2answers
246 views

Primitive roots of unity

I am trying to show that, If $$f\left( x\right) =a_{0}+a_{1}x+\ldots +a_{k}x^{k}$$ then $$\dfrac {1} {n}\left\{ f\left( x\right) +f\left( wx\right) +\ldots +f\left( w^{n-1}x\right) \right\} ...
5
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2answers
847 views

How to solve $z^4 +6z^2 +25 = 0$ into complex conjugate?

How to solve $z^4 +6z^2 +25 = 0$ into complex conjugate? I started with $$(z^2 + 3)^2 + 16 = 0$$ $$(z^2 + 3)^2 = - 16$$ $$z^2 + 3 = \pm 4i$$ Is this the way to start solving the equation or am ...
5
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1answer
65 views

Confused about exponents and imaginary/real answers

I am confused about some exponent behavior. $$(-2)^{7.6} = (-2)^{\frac{76}{10}} = ((-2)^{76})^{\frac{1}{10}} = ((-2)^{\frac{1}{10}})^{76}$$ Is there something wrong in this logic? When I plug the ...
5
votes
5answers
287 views

Prove that $(x^2-x^3)(x^4-x) = \sqrt{5}$, where $x= \cos(2\pi/5)+i\sin(2\pi/5)$

Prove $(x^2-x^3)(x^4-x) = \sqrt{5}$ if $x= \cos(2\pi/5)+i\sin(2\pi/5)$. I have tried it by substituting $x = \exp(2i\pi/5)$ but it is getting complicated.
5
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2answers
118 views

Expand $(4+i)(5+3i)$ and show $\pi/4=\arctan{1/4}+\arctan{3/5}$

I can't remember ever having done this before so if someone could help me out that would be great. The question is expand $(4+i)(5+3i)$ and hence show that $\pi/4=\arctan{1/4}+\arctan{3/5}$. ...
5
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4answers
157 views

Can a cubic that crosses the x axis at three points have imaginary roots?

I have a cubic polynomial, $x^3-12x+2$ and when I try to find it's roots by hand, I get two complex roots and one real one. Same, if I use Mathematica. But, when I plot the graph, it crosses the ...
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1answer
433 views

Hermitian/positive definite matrices and their analogues in complex numbers

I've heard a couple of times some people say that in a way, Hermitian matrices are to matrices as real numbers are to complex numbers. I know two examples where this is sort of true: Complex ...
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2answers
955 views

Prove that the zeros of an analytic function are finite and isolated

Let us assume that the zeros of $f = \{Z_1,\ldots,Z_n,a\}$ are infinite and converge towards $a$. The book which I am reading says that any neighborhood of $a$ will contain infinite zeros. Since $f$ ...
5
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1answer
107 views

What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?

Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.) In ...
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2answers
11k views

How to get principal argument of complex number from complex plane?

I am just starting to learn calculus and the concepts of radians. Something that is confusing me is how my textbook is getting the principal argument ($\arg z$) from the complex plane. i.e. for the ...
5
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1answer
333 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
5
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1answer
726 views

Why are imaginary numbers called imaginary numbers

Why do we call imaginary numbers "imaginary numbers"? As far as I can tell, there's nothing really imaginary about them. They exist. They're used all the time. What makes them so "imaginary"?
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1answer
72 views

Region in complex plane with $|1-z|\leq M(1-|z|)$

Let $M>0$. Describe the region in the complex plane such that $|1-z|\leq M(1-|z|)$. To start, I take $M=1$. The inequality becomes $|1-z|\leq 1-|z|$. But by triangle inequality, we have ...
5
votes
1answer
159 views

Good texts in Complex numbers?

I have asked some members on chat about good text to study complex numbers , they recommended for example , "Visual Complex Analysis" by Needham and "complex analysis" by Steins. But, I look for a ...
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1answer
241 views

Prove ${a^2+ac-c^2=b^2+bd-d^2}$ and $a > b > c > d \implies ab + cd$ is not prime

Let $a>b>c>d$ be positive integers and suppose that $${a^2+ac-c^2=b^2+bd-d^2}$$ Prove that $ab+cd$ is not prime? I don't know if this problem is true. I found that this same problem has ...
5
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2answers
313 views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
5
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1answer
434 views

Modulo complex number

I was wondering what would happen if we tried to do a modulo operand with complex numbers? For instance, what would be the answer (if any) to the next statement? $ x \mod (a + bi) $ can it be ...
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2answers
415 views

Proof of an inequality about $\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$

I've encountered an inequality pertaining to the following expression: $\frac{1}{z} + \sum_{n=1}^{\infty}\frac{2z}{z^2 - n^2}$, where $z$ is a complex number. After writing $z$ as $x + iy$ we have ...
5
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1answer
229 views

Radius of convergence of $\displaystyle\sum\limits_{n=0}^\infty2^{-n^2}z^n$

I was reading examples to find the radius of convergence for power series. The power series is defined as $\displaystyle\sum\limits_{n=0}^\infty c_n(z-z_0)^n$. And to find the radius of convergence ...
5
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1answer
158 views

Generalized addition function

I would like to have an example (or a proof that there does not exist) of a function on the complex numbers, which for lack of a better term I'll call generalized addition, such that $$x\oplus ...
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2answers
141 views

Complex Number Roots

When I am solving to find the root of a complex number what exactly am I finding? Does it relate somehow to the complex plane? What would be it's geometrical representation if it has one?
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1answer
86 views

a question on transcendental entire function

$f$ is a transecendental entire funtion I need to show $\{w : f^{-1}(w) \text{ is infinite}\}$ is dense in $\mathbb{C}$ I have no idea how to prove it, please help.
5
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2answers
101 views

Complex integral prove

$f(z)$ is analytic in the unit circle, and $u=Re(f), v=Im(f)$. Please prove that if $u(0)=v(0)$, then $\int_0^{2\pi}(u(re^{i\theta}))^2d\theta=\int_0^{2\pi}(v(re^{i\theta}))^2d\theta$ for every ...
5
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1answer
205 views

Why do Quaternions and octonions exist?

Ok so I have known about imaginary numbers for quite some time now. I also understand why we want them to exist (to have a solution for $x^2=-1$). I also remember reading that the complex numbers are ...
5
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1answer
159 views

Preimage of discs under a complex polynomial

Let $a_0, \ldots, a_n \in \mathbb{C}$, with $a_n \neq 0$. Consider set $$U_R = \{~z \in \mathbb{C} ~:~ |a_nz^n + \dots + a_1z + a_0| < R~\}$$ for each $R > 0$. How do I prove that $U_R$ is ...
5
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2answers
186 views

Lerch-$\small \zeta(\varphi,0,-n)$ of integer *n* purely real and imaginary ($\small \zeta_\varphi (-n)^2 $ is real) for $\small n \ge 2$?

Are the Lerch-$\zeta(\varphi,0,-n) $ of integer n (for shortness I use the notation of my earlier question $\small \zeta_\varphi(-n)$) periodically purely real and imaginary: $\zeta_\varphi (-n)^2 $ ...
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1answer
196 views

For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero

Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. ...
5
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2answers
137 views

Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
5
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1answer
158 views

Do there exist complex algebraic $α,β$ such that $α^β=π$ or $α^β=e$?

Given the algebraic operations and complex exponentiation $(a+bi)^{c+di}$ and logarithm, is it possible to derive $\pi$ and $e$? If one is derivable then so should be the other, as $e^\pi = ...
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1answer
173 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 ...
5
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1answer
163 views

Show $|a|+|b|+|c|+|a+b+c| \geq |a+b|+|b+c|+|c+a|$ for complex $a$, $b$, $c$

How to prove for any complex numbers $a$, $b$, $c$, the inequality $$|a|+|b|+|c|+|a+b+c| \geq |a+b|+|b+c|+|c+a|$$ is correct?
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1answer
1k views

Simple Complex Number Problem: 1 = -1 [duplicate]

Possible Duplicate: -1 is not 1, so where is the mistake? I'm trying to understand the exact point of failure in the following reasoning: $1 = \sqrt{1} = \sqrt{(-1)(-1)} = ...
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1answer
56 views

Simple-looking bound on root of unity

I am trying to prove some bound and stuck with the following: If $|n|\leq 3N/4$, then $\left|e^{2\pi in/N}-1\right|\geq\dfrac{n}{N}$ ($n,N$ are integers) How can I prove it?
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1answer
466 views

Zero sum of roots of unity decomposition

It's known that sum of all $n$'th roots of some $z \in \mathbb C$ with $|z| = 1$ is zero (if $n \geqslant 2$). Is it true that any zero sum of roots of unity can be decomposed in this way? That is if ...
5
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1answer
243 views

How to compute the sum of every $k$-th binomial coefficient?

My teacher was discussing binomial expansions of $(1 + x)^n$ and he gave as an interesting example with $x = i$ whereby you could obtain the sum of all the odd coefficients ($C_n^1+ C_n^3+ C_n^5 ...$) ...
5
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1answer
426 views

Minimizing l-infinity norm of complex vector

I have an $n$-dimensional complex vector space, and I want to minimize the $L_\infty$ norm of a point that is constrained to an $m$-dimensional affine subspace. That is, Given $\mathbf{z} \in ...
5
votes
4answers
519 views

Why do all circles passing through $a$ and $1/\bar{a}$ meet $|z|=1$ are right angles?

In the complex plane, I write the equation for a circle centered at $z$ by $|z-x|=r$, so $(z-x)(\bar{z}-\bar{x})=r^2$. I suppose that both $a$ and $1/\bar{a}$ lie on this circle, so I get the equation ...
5
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1answer
44 views

Detailed Visual Introduction to Complex Numbers with Problems and Solutions

I'm aware of http://tutorial.math.lamar.edu/Extras/ComplexPrimer/ComplexNumbers.aspx. It's very detailed and helpful but I'm looking for something with more pictures. It also doesn't have enough ...
5
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0answers
69 views

Complex Numbers vs. Matrix

I have a line starting at the origin, and i extend it to a point $(a,b)$ in the plane. This thing can be called a vector and be represented as $(a,b), [a\text{ }b]^T$ (column vector) or by ...
5
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0answers
265 views

Natural extension of the divisor function $\sigma$ to the complex integers $\mathbb{Z}[i]$

The divisor function $\sigma$ of a positive integer $n$ is defined as the sum of the positive divisors of $n$. It turns out that this definition is equivalent to ...