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

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10
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213 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 ...
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28 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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2k 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, ...
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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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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 ...
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173 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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70 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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330 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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46 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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109 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 $2^n$ ...
3
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55 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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192 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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49 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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92 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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121 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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225 views

How does quater-imaginary (and other imaginary/complex bases) work?

So I've been working on a simple base-conversion program, and having given it the ability to convert from decimal to any base $> 1$ or $< 0$, as well as the $p$-adic (bijective, I think?) bases, ...
3
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143 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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51 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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32 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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40 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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43 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?
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28 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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20 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 ...
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105 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 ...
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41 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
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48 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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88 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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47 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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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. ...
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75 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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82 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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60 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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129 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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218 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 $$ ...
2
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0answers
171 views

Definition of a complex prime

If there would be such a thing as a complex prime number, how would it be defined? A normal prime is defined as a number only dividable by one or itself.. would that be "1+1i" with complex numbers? ...
2
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913 views

Physical meaning of Fourier transform of complex signal?

I understand what is meaning of Fourier transform over function that returns only real values — it can be thought of function taking time and returning real ...
2
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0answers
125 views

Hyperbolic Universal Covering Space

I have been working with Ricci flow in the euclidean and hyperbolic space but have been having considerable trouble determining how to generate a universal covering space for complex hyperbolic ...
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76 views

Evaluating the limit $y \to 0^+$

Given $t \in \mathbb{R}$ and $z = x + iy$ and $y>0$. $\lim_{y\to0^+} \frac{1}{t - z} = \frac{1}{t-x} + \pi i \delta(t-x)$ This limit is given in the book Integral Transforms and Their ...
2
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119 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 ...
2
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296 views

Gamblers ruin difference equation

I'm not sure where my mistake is in the following. Gambler starting with k dollars and playing a $50/50$ game where he increases in wealth by one dollar or decreases by one dollar until achieving $N ...
2
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315 views

Hermitian operators and commutators

If I have three operators such that $[A,B] = C$, and I know that $A$ and $C$ are Hermitian, does it follow that $B$ is anti-hermitian? If $A$ and $B$ were Hermitian, $C$ would be anti-Hermitian, so ...
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33 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
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57 views

Is this correct: $\ln({(-1)}^{2x-1})=(2x-1)\ln(-1)$?

I would expect the answer to be positive, but it appears otherwise for some values of $x \geq 1$. Here is a simple C++ code that I have used in order to test this: ...
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0answers
9 views

Restoring an Analytic Function

Restore the analytic function $f(z)$ if its imaginary part $v(x,y) = -\frac{y}{(x-1)^2 + y^2}$ and $f(2)=2$. Now, $v_y = \frac{-[(x-1)^2 + y^2] - (-y)(2y)}{[(x-1)^2 + y^2]^2} = ...
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26 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, ...
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15 views

Alternative coordinates for the complex plane $\mathrm{Re}[e^{-is}z]=a$, $\mathrm{Re}[e^{-it}z]=b $

I am defining coordintes on $\mathbb{C}$ using a "generalized" real and imaginary part. Here $a,b \in \mathbb{R}$. \begin{eqnarray*} \mathrm{Re}[e^{-is}z]&=&a \\ ...
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59 views

Calculating Fourier magnitude spectrum for Local Binary Pattern histogram

I have the follwoing discrete Fourier transform function defined in my book (Computer Vision using Local Binary Patterns, Pietikainen et. al, 2011): $$H(n, u ) = \sum_{r=0}^{P-1}c_{nr} ...
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38 views

Determining wether a set of complex numbers is open or closed

Let $S = \{z: arg(z)= \pi/4\}$. Is S open? Is it closed? What?'s its boundary? I graph it in polar and it just seems to be a line from the origin with $\theta = \pi/4$. What is a better way to ...
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53 views

Can “polar numbers” be added in a sensible way?

Let $\mathbb{P}$ denote the set of all "polar numbers," by which I just mean pairs of real numbers $(r,\theta).$ Note in particular that $r$ is allowed to be negative. Then we can structure ...