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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7
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264 views

Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids using complex numbers in the proof. Existence of complex roots will ...
7
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215 views

Why does the Mandelbrot shape show up in other fractals?

In the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fractal ...
7
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72 views

Cauchy representation and branch point order

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. The main application is for dispersion ...
7
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472 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 ...
6
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113 views

Minimising a sum of roots of unity

Let $n$ be an integer, $n\ge2$. Let $m$ be a positive integer, $m\le n$, having no common factor with $n$. How can we select $m$ distinct complex $n$th roots of unity in such a way as to minimise ...
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67 views

A periodic entire function which must have a fixed point

I would like to check my work on the following problem: Suppose $f(z)$ is a non-constant periodic entire function satisfying $f(z+1)=f(z)$. Show that $f(z)$ has a fixed point. So my attempt is: ...
5
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96 views

Can you prove that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$, assuming LeClaire's approximation?

Can you prove using double series reversion that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$ (as their convergent), with initial guess for the real part $r$ to be ...
5
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140 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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111 views

Are points on the complex plane sufficient to solve every solvable equation composed of the hyperoperators, their inverses, and complex numbers?

Some background: I'm programming a maths environment. I'm computer science, so please excuse any probable ignorance and lack of precision in my question. It seems $i$ and complex numbers were ...
4
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85 views

Trigonometric Expression for $1 + \cos \alpha + \cos 2\alpha + \cdots + \cos n \alpha$ using complex numbers

This question is not a duplicate because I am asked here to use the fact that $1 + \cos \alpha + \cos 2 \alpha + \cdots + \cos n \alpha = Re (1 + z + z^{2} + \cdots + z^{n})$, where the question this ...
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35 views

The image in $\mathbb{C}$ of $\mathbb{R}^2$ under a map of counterpropagating plane waves is…?

Define $$f_n(\mathbf{r})=\frac{1}{n}\sum_{k=1}^n\exp\left(2\pi i\binom{\cos\left(2\pi k/n\right)}{\sin\left(2\pi k/n\right)}\cdot\mathbf{r} \right)$$ as the sum of $n$ counterpropagating plane waves. ...
4
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103 views

How to solve this equation in $ \mathbb{C} $?

From a small simple calculation , we get the following formulas: $ \begin{cases} e^x = \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n}}{(3n)!} \Big) + \Big( \displaystyle \sum_{n \geq 0} ...
4
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0answers
60 views

Derivatives of a Dirichlet polynomial

I am new here, so I don't know how this works exactly. If I do something wrong, please let me know. I'd like help to solve a problem I am studying: Let $A$ be finite set of positive integers and ...
4
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72 views

Symmetric Icon Fractals

I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals. There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos. I'm ...
4
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0answers
85 views

Definition of $0^z$, $z$ complex

Usually we define the function $a^b$ over the complexes using the exponential function, as $e^{b\log a}$. This function has some issues with multivalued-ness, but it still more or less satisfies ...
4
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78 views

Help computing integral of quarter contour

For a project I'm working on I need to solve the following integral. Can anyone help me tackle this challenge? The Integral: $$ \beta \int_{\pi}^{\pi/2} {i k^2 r e^{i \theta} + ikr^2 e^{2i\theta} ...
3
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15 views

Sum of Complex Numbers and Modulus Inequality

Let $z_{1}, \dots, z_{n} \in \mathbb{C}$. Then, there exists a subset $S \subset \{1,\dots,n\}$ such that: $ \left| \displaystyle\sum_{j \in S}z_{j} \right| \geq ...
3
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28 views

Is it always possible to find the roots of $P(z)=az^4+bz^3+cz^2+bz+a$, where $a,b,c \in \mathbb{R}^*$, by first dividing both sides by $z^2$?

A classic way to solve quartics in the form $P(z)=az^4+bz^3+cz^2+bz+a$, if we know that the roots lie on the unit circle, is to divide both sides by $z^2$ and then use the fact that if $$z=\cos \theta ...
3
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64 views

Is this inequality trivial?

Let $z = (z_1,z_2)\in \mathbb C^2$, $|z| \leq 1$. Then $$ \left( |z_1|^2 - |z_2|^2 \right)^2 \leq \left( |z|^2 + (z_1 \bar z_2 -\bar z_1 z_2 )^2 \right) \left( |z|^2 - (\bar z_1 z_2 - z_1 \bar ...
3
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78 views

Fundamental theorem of algebra in different functional form

Consider the polynomial function: $f(x)=c_0+c_1x+c_2 x^2+\cdots+c_{n-1}x^{n-1}+x^n$, with $x$ and $c_0,c_1,c_2,\ldots,c_{n-1}$ are complex numbers. $|f(x)|$ is continuous and there exists closed and ...
3
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67 views

Approximating $(1+\frac{1}{z})^z$ where $|z|$ is large

I know that $$\lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^x=e$$ Is there an equivalent in complex analysis for $$\lim_{|z|\rightarrow \infty}\left(1+\frac{1}{z}\right)^z=?$$
3
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43 views

Function of complex conjugate equal complex conjugate of function?

Very simply, for what type of functions $f: \mathbb{C} \rightarrow \mathbb{C}$ is the following true? $f(\bar{z})=\overline{f(z)}$ Does Schwarz reflection principle imply this is true for all ...
3
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35 views

complex numbers from real closed fields

I am very interested in first order axiomatizations of the complex numbers, but I have never actually seen one laid out. Algebraically closed fields of characteristic zero are a start, but they don't ...
3
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95 views

A question about the proof of $(z_1z_2)^a=z_1^az_2^a$

For $z_1,z_2\in \mathbb C$ if $\Im(z_1z_2)>0$ and $\Im(z_2)\ge 0$ prove that $(z_1z_2)^a=z_1^az_2^a$ , for $a$ is any real. I proved it like this: $z_1^az_2^a=\exp(a\log z_1)\exp(a\log ...
3
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63 views

Solving simultaneous equations in complex numbers

Given $z_1,z_2$ are complex numbers such that sum of their squares is a real number and $$z_1(z_1^2-3z_2^2)=2$$ and $$z_2(3z_1^2-z_2^2)=11.$$ I need to find the value of sum of squares of two complex ...
3
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0answers
135 views

Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such ...
3
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0answers
107 views

Pisier's $\epsilon$-net condition

I'm reading a book about Sidon sets and I'm stuck on the following proof. In order to facilitate the comprehension of my problems I will give the full proof and the context. Let $G$ be a compact ...
3
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0answers
53 views

What's an elegant expression for a general conic using complex numbers?

A 'nice' form of writing a line through $2$ points $z_1,z_2\in\mathbb{C}$ is $$z\overline{z_1-z_2}+z_1\overline{z_2-z}+z_2\overline{z-z_1}=0$$ Now I'm asked to give a similar expression for a general ...
3
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130 views

what is the the value of $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}$

If $\frac{a}{a+i}+\frac{b}{b+1}+\frac{c}{c+1}=1$ then what is the the value of $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}$ here I got $a=0$ and $bc=1$, when $ bc\neq 0$ but then I cant ...
3
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0answers
68 views

Complex Analysis (Complex Mapping) stuck on professor's method of simplification in math notes

I'm having an exam this university semester and need some help with my math notes. I've come accross some problems with the section "Complex Mapping." Link to Image of my Notes: Click Me (see first ...
3
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0answers
87 views

Random Sampling of vectors on the Complex Unit Sphere

This is my first post in these forums. Working in Mathematica, I would like to generate a large number (10000) of randomly sampled vectors on the complex unit sphere in n dimensions. I am not ...
3
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57 views

Is the exponential function the one this problem is hinting at?

Suppose that $f$ is holomorphic on all of $\mathbb{C}$ and that $$\lim_{n\rightarrow \infty} \left(\frac{\partial}{\partial z}\right)^nf(z)$$ exists, uniformly on compact sets, and that this limit ...
3
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57 views

Problem on Complex numbers involving a point on a Circle

Question: The Complex number $z$ is represented by the point $T$ in the Argand Diagram.Given that $$z =\frac{1}{3+it}$$ where $t$ is a variable, show that i) as $t$ varies, $T$ lies on a circle, and ...
3
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0answers
77 views

How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
3
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0answers
103 views

Can't match boundary conditions on a perturbation series solution to a non-linear ODE?

I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation: \begin{equation} \frac{d^2y}{dt^2} + y = \epsilon ...
3
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0answers
5k views

Prove the triangle inequality involving complex numbers.

Our eventual goal in this problem is to prove the triangle inequality involving complex numbers. (a) Show that for every $z ∈ C$, $|Re(z)| ≤ |z|$ and $|Im(z)| ≤ |z|$. (b) Given $z$, $w ∈ C$, show ...
3
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0answers
56 views

Complex root notation

Is there a standardized way to distinguish between real and complex roots? In other words, is there a convention about how to I formally write that I expect $\sqrt[3]1$ to be solved in $\mathbb C$, ...
3
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0answers
107 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
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0answers
436 views

If the product of two analytic functions is zero, then one must be identically zero.

I want to prove this statement: Let $f,g$ be analytic on $D(0,2)$. If $f(z)g(z) = 0$ when $z = 1/n$ for $n \in \mathbb{N}$, then either $f \equiv 0$ or $g \equiv 0$ in $D(0,2)$. My attempt: ...
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0answers
55 views

Subgroups of $P=\{z\in\mathbb{C} : z^{2n}=1 \;\mbox{for some}\; n\geq 0\}$

Investigate the subgroups of $P$ where $$P=\{z\in\mathbb{C} : z^{2n}=1 \;\mbox{for some}\; n\geq 0\}.$$ In particular, investigate the finitely generated subgroups and the infinite subgroups. ...
3
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0answers
108 views

Prove (*) by induction on k.

Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form $$\sum_{i=1}^m ...
3
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0answers
143 views

Illustrations of a line and a curve intersecting for complex field

Are there nice illustrations on the Net of say $y=a·x+b$ and $y=x^2$ intersecting where x and y are complex? I'm thinking of the amplitude of y being depicted as height above the complex plane with ...
3
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0answers
147 views

Do I understand complex differentiability correctly?

I don't use complex analysis much in my "day job", but I'm perusing the materials for personal interest. I just wanted to check with the community about my understanding of complex analytic ...
3
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0answers
180 views

On the distribution of unimodular matrices generated by the Hermite normal form

A problem I'm currently considering requires me to generate (pseudo-)random Gaussian integer matrices with Gaussian integer matrix inverses. By analogy with an algorithm I know for generating random ...
2
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0answers
17 views

get magnitude of addition of complex numbers in trigonometric form

My problem is that I have multiple complex number in trigonometric form and I want to add those and get the magnitude of the result. I am aware that the normal route would be to calculate the ...
2
votes
0answers
29 views

Limit of $\exp(z^2)$ as $|z|$ tends to infinity

Let $g(z) = \exp(z^2)$ and $L$ a ray starting at the origin. Determine those $L$ along which $g$ has a limit (finite or infinite) as $|z|$ tends to infinity and $z ∈ L$. Find the value of the limit ...
2
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0answers
45 views

What is the motivation behind the solution of this problem involving complex numbers?

The problem is Suppose for three distinct complex numbers $a, b, c$ such that $|a|=|b|=|c|>0$ all of the three numbers $a+bc, b+ac, c+ab$ are purely real. Prove that $abc=1$ By playing with ...
2
votes
0answers
26 views

The nth roots of $z_n=p_n\cdot(i)^{p_n}$, where $i=\sqrt{-1}$ and $p_n$ is the nth prime number

I want refresh some basics too in Complex Analysis. Let $p_n$ the sequence of prime numbers $2, 3, 5, 7\ldots$, thus $p_n$ is the general term of this sequence, and $i=\sqrt{-1}$ is the complex ...
2
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0answers
50 views

Functions that maps unit circle into unit circle

This and This problems discuses the characterization of Analytic functions which maps unit circle on to itself. I would like to know the characterization of functions which map unit circle into ...
2
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0answers
21 views

Complex Hadamard Matrix

Let $n$ be a positive integer. A matrix $A = [a_{ij}] \in \mathcal{M}_{n\times n}(\mathbb{C})$ is a complex Hadamard matrix if and only if $|a_{hj}| = 1$ for all $1 \leq h$, $j \leq n$ and any pair ...