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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13
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0answers
5k views

What's more common? Re / Im or Fraktur-R / Fraktur-I for real / imaginary part?

Title says it all. What's more common? Is there one to prefere (maybe due to some norm)? This: $\operatorname{\mathfrak{R}} z, \operatorname{\mathfrak{I}} z$ or that: $\operatorname{Re}z, ...
7
votes
0answers
59 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
votes
0answers
413 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
votes
0answers
98 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 ...
5
votes
0answers
80 views

Writing circles as $|z-a| = \lambda |z-b|$ for the same $a,b$

My problem is in the context of the complex plane. I want to know if given two disjoint, not concentric circles $C_1,C_2\subset \mathbb{C}$, can you find $a,b\in \mathbb{C}$ such that $$C_1=\{z\in ...
5
votes
0answers
127 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
votes
0answers
109 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 ...
5
votes
0answers
186 views

Primality using $\Gamma(x)$

Wilson's theorem states $n \in \mathbb N$ is prime iff$(n-1)! \equiv -1\pmod n$. $\Gamma$-function extends the usual factorial to complex numbers. What are the complex numbers such that $\Gamma(z)+1 ...
4
votes
0answers
72 views

Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
4
votes
0answers
33 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
votes
0answers
83 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
votes
0answers
34 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
votes
0answers
60 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
votes
0answers
54 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: ...
4
votes
0answers
83 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
votes
0answers
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} ...
4
votes
0answers
131 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 ...
4
votes
0answers
85 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 ...
3
votes
0answers
37 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
votes
0answers
109 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
votes
0answers
102 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
votes
0answers
48 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
votes
0answers
119 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
votes
0answers
63 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
votes
0answers
60 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
votes
0answers
55 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
votes
0answers
49 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
votes
0answers
84 views

Factorial of Complex Values

Since the gamma function is an analytic continuation of the factorial function, we can find the factorial of complex values. How does one go about doing so? I've looked far and wide on the internet ...
3
votes
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
votes
0answers
86 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
votes
0answers
51 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
votes
0answers
103 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
0answers
335 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: ...
3
votes
0answers
53 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
votes
0answers
102 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
votes
0answers
139 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
votes
0answers
168 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
votes
0answers
32 views

Moving limit inside a contour integral

I'm trying to compute this integral as part of a larger problem I'm working on. I'm trying to solve the integral $\int_0^\infty \frac{\sin(x)}{x}dx$ and to do it I'm using the method where you ...
2
votes
0answers
27 views

Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
2
votes
0answers
45 views

Are there famous complex constants?

Are there any famous constants (like $\pi$ and $e$) that are complex? More specifically, to rule out trivial complex numbers, are there any famous constants of the form $a+bi$ with $a \neq 0 \wedge b ...
2
votes
0answers
35 views

Question based on geomerical properties of complex number.

Let $z$ and $w$ be two complex numbers such that $\left| z\right| \leq 1$, $\left| w\right| \leq 1$ and $\left| z+iw\right|= \left| z- i\bar{w} \right|=2$, then z equals (A) $1$ or $i$ (B) $i$ or ...
2
votes
0answers
46 views

A (analytical) Geometric Way to solve this complex number problem?

Problem Let $a_0,a_1,a_2$ be three complex numbers that lie on the circle $C$ with center $K(2,0)$ and radius $r=1$. Let $v \in \mathbb C$ such that $$v^3 + a_2v^2 + a_1v + a_0 = 0$$ ...
2
votes
0answers
41 views

compute the complex-valued integral for the branch cut

Let $C$ be the circle of radius $2$ centered at origin. Let $f(z)$ be the branch cut of the function $z^{2−i}$ on the domain $−π < θ < π$. Compute the integral $$ \int_C f(z) dz$$ My attempt: ...
2
votes
0answers
21 views

Solve Intergal Equation of form g.u1=Int(K.u2) for u1 and u2

I'm trying to find a solution to a differential equation of an unusual form: $$g(x) u_1(x)=\int_a^b K(x,y) u_2(y) dy$$ where $g(x)$ and $K(x,y)$ are known and $u_1(x)$ and $u_2(x)$ are complex ...
2
votes
0answers
84 views

Can an ordered field contain complex numbers?

I read a question about ordering of complex numbers, and saw an answer showing that there cannot exist an ordering of the complex numbers because regardless of how $i$ is placed in that order, it ...
2
votes
0answers
27 views

Determine the set of points $M$ of affix of $z\in\mathbb{C}$

Determine the set of points $M$ of affix of $z\in\mathbb{C}$ such that there exists at least one real $t$ satisfying $z^2=t(t-i)$ My attempt: We look for the form $z=x+yi$ and we want there is a ...
2
votes
0answers
21 views

How to express vectors with more than 2 components in complex coordinates

It is straightforward to extend the notion of a 2D vector in the Cartesian $x-y$ plane to 3D $(x,y,z)$ or to any dimension. Sometimes it is useful to express vectors in the complex plane, where the ...
2
votes
0answers
44 views

Laurent Series Expansion of $e^{z+\frac{1}{z}}$ about $z_0=0$

As the question title states, I'm tasked with finding the Laurent series expansion for $e^{z+\frac{1}{z}}$ about $z_0=0$. My approach is as follows. $$ \begin{align*} ...
2
votes
0answers
19 views

Finding the set {$z\in \mathbb{C}|2z+2i-3\in [0,\infty)$}

This is how I tried this but not really sure: {$z\in \mathbb{C}|2z+2i-3\in [0,\infty)$}$\implies${$z\in \mathbb{C}|2z+2i\in [3,\infty)$}$\implies${$z\in \mathbb{C}|z+i\in [3/2,\infty)$}. Now if ...
2
votes
0answers
80 views

If the sum of absolute values of complex numbers is at least $1$, then some subset of these numbers has absolute value at least $C$

There is a challenging problem in a book of mine on complex analysis, and I seriously do not even know where to start. I'm more than sure I don't properly understand the problem. Prove that there ...