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

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18
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349 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
8
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96 views

Complex Exponential False “Proof” That All Integers Are $0$

The following false "proof" is attributed to Thomas Clausen in 1827, and was stated on page 79 of Nahin's An Imaginary Tale. $e^{i2\pi n}=1$ for all integers $n$. So \begin{align*} ee^{i2\pi ...
7
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45 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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3k 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, ...
5
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48 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
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82 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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82 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
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297 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 ...
5
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176 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
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61 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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74 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
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55 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
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139 views

What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
4
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341 views

What are applications of Lagrange's identity?

I recently proved for homework the following identity on $\mathbb{C}$: if $a_1, \ldots , a_n, b_1, \ldots, b_n\in\mathbb{C}$, then $$ \left|\sum_{i=1}^na_ib_i\right|^2 = ...
3
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56 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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60 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
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206 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
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51 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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96 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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131 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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146 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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22 views

On the criterion of convergence of infinite products of complex numbers

I have troubles in understanding the proof of the criterion which states that an infinite product exists iff the series of the complex logarithms of the terms of the product converges. In particular, ...
2
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33 views

Proof of Cauchy-Riemann equations using differentials as quotients?

In my analysis 2 book the proof goes like this: If a complex function $f = P(x,y) + iQ(x,y)$ is differentiable at a point $z$, then $$ \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} ...
2
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38 views

Irrational Roots of Complex Numbers

Given a complex number $e^{ix}$, the nth root can be computed using Euler's formula: $$e^{i(x + 2k\pi)/n} = \cos((x+2k\pi)/n)+i \sin((x+2k \pi)/n).$$ If $n$ is an irrational number, can $k$ vary ...
2
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56 views

A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
2
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0answers
50 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 ...
2
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60 views

How the total order property of $\mathbb{R}$ is related to not being algebraicaly closed?

The field of real numbers $\mathbb{R}$ is total-ordered and not algebraicaly closed, the field of complex numbers $\mathbb{C}$ is not ordered but is algebraicaly closed. Intuitively how these two ...
2
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68 views

Imaginary number in relativistic speeds

I am layman in field of mathematics but when I was reading about theory of special relativity I have come across speed limit of light and the book said that no one can cross that limit because ...
2
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20 views

Cyclic sum inequality involving five numbers with modulus one and zero sum

When working on this MSE question, I was led to conjecture the following : If $z_1,z_2,z_3,z_4,z_5$ are five complex numbers with modulus $1$, such that $z_1+z_2+z_3+z_4+z_5=0$, then $$ ...
2
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38 views

Complex integral and parametrization of a circle

I am trying to compute the following integral of $$\int \frac{1}{z^3+3} dz$$ over a circle of radius $2$, centerd at $(2,0)$. Thus I am trying to compute the residue and have found that the function ...
2
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0answers
57 views

Extension to complex numbers

Is there an extension to the complex numbers in which $zz^* = i$ has a solution? (The star denotes conjugation.) EDIT: I'm mathematically ignorant, but I'm guessing such an extension can't be a ...
2
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35 views

Algebraic Variety compact cohomology and singular cohomology.

Given an algebraic variety $X$ defined over the complex numbers and its compact cohomology $H^i_c(X)$, under what conditions it is possible to compute its singular cohomology $H^i(X)$.
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37 views

Real-valued Irreducible Representations of Lie Groups

I'm interested in the real-valued irreducible representations of a number of Lie groups. For concreteness I'll use the group $M(2)$ of Euclidean motions, which can be parameterized as follows: $$ g(t, ...
2
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0answers
42 views

Find a holomorphic function that satisfies this equation

Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this ...
2
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56 views

What properties do we lose when moving from the rational numbers to the real numbers?

When we pass from the real numbers to the complex numbers, we lose total ordering. But what do we lose when we move from the rational numbers to the real numbers?
2
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30 views

Accurate computation of arcsec near branch points

The direct numerical implementations of the usual definitions of the complex $\mathrm{arcsec}(z)=\arccos(1/z)$ and similar for $\mathrm{arccsc}(z), \mathrm{arcsech}(z), $ etc are not accurate near ...
2
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22 views

Give an appropriate region on which the function is analytic.

$F(z)=\log(z^4+4i)$ Here is what I did, Let $z=re^{i\theta} \Rightarrow z^4 =r^4e^{i4\theta}=-4i \Rightarrow r^4=4, \theta=\frac{\frac{-\pi}{2}+2\pi k}{4}$ for $k=0,1,2,3$. Now $F$ is analytic on ...
2
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0answers
112 views

Find all solutions for a complex equation: $(1+i)z^2 - (6+i)z + 9+7i=0$

There is this math assignment that we've been given to find all the answers for some diffrent math problems. The problem is: $(1+i)z^2 - (6+i)z + 9+7i=0$, find all the solutions and answer in ...
2
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0answers
42 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 $\mathbb C$, ...
2
votes
0answers
53 views

Complex numbers question (finding the Im)

So I've been wrestling with this for about an hour and am still not sure how to solve it. The task is to find the Im part of the complex number (it has to be as short as possible, or else it'd be ...
2
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0answers
96 views

Evaluating integral with branch of log

I'm having trouble understanding what this question even means really. "Let $\gamma$ be the semi-circle from $2i$ to $-2i$ that passes through $2$ in the positive direction. Find $\int_\gamma ...
2
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48 views

Taking the non-relativistic limit of a Lagrangian

This question has to do with question 1 on this sheet. I'm trying to self-learn QFT, please be patient with me! I have problem to get the exact form of the Lagrangian in the non-relativistic limit as ...
2
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165 views

Swap real and imaginary part of a complex number

Say I have some complex number $z=a+ib$ with $a,b\in \mathbb{R}$. Now I want to "swap" its real and imaginary parts, i.e. I want to get $\tilde{z}=b+ia$. While I figure that the appropriate mapping ...
2
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0answers
168 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. ...
2
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0answers
87 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 ...
2
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0answers
84 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 ...
2
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0answers
63 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
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131 views

Is there a finite number of solutions to $\mathrm{Re}(a^n)+\mathrm{Im}(a^n)=b^n$, where $a$ is a Gaussian integer and $b \in \Bbb Z$?

Let $$E_n=\{(x,y,b) \in \mathbb{Z}^*\times \mathbb{Z}^*\times \mathbb{Z}^* ~|~ \gcd(x,y,b)=1 ~ \mathrm{and}~\mathrm{Re}((x+iy)^n)+\mathrm{Im}((x+iy)^n)=b^n \}$$ For $n \geq 3$, is $E_n$ finite or not ...
2
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265 views

Probability distribution of the product of two independent complex gaussian random variables

I have to calculate the pdf of $Z = X*Y$, where $X \in \mathcal{C}(\mu_x,\Sigma_x)$ and $Y \in \mathcal{C}(\mu_y,\Sigma_y)$, where $\mathcal{C}$ is a complex distribution. It can be assumed that ...
2
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0answers
86 views

convergence of a sum with zeroes of zeta function

Can it be proved that the sum of this series is smaller than $x$? $$ \sum_{\zeta(a+ib)=0}u_{a,b}(x)\lt x, $$ for all $x$, with $$ ...