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

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9
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4k 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
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0answers
50 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 ...
6
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0answers
69 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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94 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
96 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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0answers
337 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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0answers
179 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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71 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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75 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
83 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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356 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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0answers
28 views

Convergence of a complex function

I need to proof if the following function is bounded and convergent. $f(n)=\left(\frac{10+in}{n^{2}+2in}\right)^{n}$ Status: This should be correct. Can anybody confirm this? I tried it with ...
3
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0answers
22 views

Find a value for a number to the power of a complex number

Find a value for $2^{-4i}$? I have no idea what to do or how to find the value. My thoughts are that I should use logarithm. Can someone please show me how to solve this?
3
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25 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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65 views

If $|z +\frac{1}{z}|=a$ find extreme values ​​of $|z|$

Let $a> 0$. Knowing that $z$ is complex number with $|z +\frac{1}{z}|=a$ find extreme values ​​of $|z|$. My partial solution: $$|z +\frac{1}{z}|=a \iff (z +\frac{1}{z})(\overline{z} ...
3
votes
0answers
49 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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0answers
36 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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63 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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63 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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0answers
43 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
244 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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0answers
52 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. ...
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98 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
132 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
153 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
25 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
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0answers
14 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 D. Sometimes it is useful to express vectors in the complex plane, where the 2D vector has ...
2
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0answers
29 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
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0answers
16 views

Progression of complex numbers

In the complex plane, consider $z_1,z_2,z_3$ as distinct complex numbers lying on the curve $|z| = 3$. Suppose that a root of $f(z) = z_1z^2+ z_2z+ z_3$ satisfies $|z| = 1$. I want to prove that ...
2
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0answers
17 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
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0answers
64 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 ...
2
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0answers
59 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
votes
0answers
51 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
votes
0answers
49 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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0answers
71 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
votes
0answers
68 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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0answers
67 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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0answers
75 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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0answers
43 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
55 views

For which $m$ is this sum of roots of unity $0$?

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed ...
2
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0answers
62 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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0answers
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 $H^i(X)$.
2
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41 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
47 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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0answers
62 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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0answers
34 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 ...
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0answers
23 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
117 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
votes
0answers
58 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
votes
0answers
105 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 ...