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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8
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37 views

Geometric interpretation of the determinant of a complex matrix

A complex $n$-dimensional vector space $V$ can be thought of as a real $2n$-dimensional vector space equipped with a map $J:V \to V$ with $J^2 = -I$. Complex-linear maps are then linear maps $V \to V$ ...
7
votes
0answers
286 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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0answers
123 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 ...
7
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271 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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75 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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495 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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78 views

Generalization of FTA

I'm sure that this is not any hypothesis, but following came to my mind when I was reading complex analysis. Consider a function $f(z)=z^n+g(z)$, where $g(z)$ is continuous (not necessary ...
5
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89 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 ...
5
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73 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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149 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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119 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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38 views

Find how many such complex numbers exist

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=z^2+iz+1$. How many complex numbers $z$ are there such that $\text{Im}(z)>0$ and both the real and the imaginary parts of $f(z)$ are integers ...
4
votes
0answers
28 views

Complex roots of quartic polynomial

This is a question from an undergraduate course on Galois theory: Find all complex numbers which are roots of $P(T)=T^4+2T^2-\sqrt{6}T+\frac{3}{4}$ Can we use Galois theory to solves this? Or ...
4
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78 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
4
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82 views

Double contour integral in terms of real integrals

Let $\gamma$ be a curve in $\mathbb{C}$, and let $\gamma_0$ be a circle in an open connected set $A \subset \mathbb{C}$ around $z_0 \in A$. Suppose the interior of $\gamma_0$ lies in $A$. Let $z$ be ...
4
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94 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 ...
4
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143 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 ...
4
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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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109 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
62 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
88 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} ...
4
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0answers
79 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
28 views

Cosine Inequality, Geometric interpretation in the complex plane

The following identity was given as an exercise in the course notes for a complex analysis course. I am able to solve it (the proof is given below), but am unsure of the geometric interpretation of ...
3
votes
0answers
27 views

Find and classify singular points of $\cot\left(\frac{1}{z}\right)$

I need to find and classify singular points (i.e., decide whether the point is removable, a pole of order $N$, essential, or not an isolated singular point), including infinity, of ...
3
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0answers
50 views

Sum of four complex roots of unity

Let $\epsilon$ be a complex $n$ th root of unity and $i$, $j$, $k$ be positive integers so that $k^2\equiv -1\mod n$. It is relevant to see that if $i\equiv \pm j$ or $i\equiv \pm jk$, then ...
3
votes
0answers
116 views

(2016 China team selection Test) with a complex inequality

Let $z_{1},z_{2},z_{3}$ be complex numbers, such that: $z_{1}+z_{2}+z_{3}=0,|z_{i}|<1,i=1,2,3$. Find the minimum of the positive $A$ such that: ...
3
votes
0answers
72 views

Entire function approaching zero along upper half plane

Suppose $f$ is entire, i.e, $\;f: \Bbb C \to \Bbb C$ is analytic. Let $\Bbb H:= \{ z: Im(z)>0\}$ be the upper half plane. Suppose that $$\lim_{\substack{z \to \infty \\ z \in \Bbb H}} f(z)=0$$ ...
3
votes
0answers
51 views

Show that $f(z) = \ln r + i \varphi$ is differentiable in a neighborhood of $z_{0}$

I am faced with the following problem: Let $z_{0}\neq 0$ and let $f(z) = \ln r + i \varphi$, where $r = |z|$, $\varphi \in arg z$, and $\varphi$ is chosen so that $f$ is continuous in a neighborhood ...
3
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0answers
49 views

Proof that $|1 - e^{i \theta}| \geq \frac{2|\theta|}{\pi}$ for $-\pi \leq \theta \leq \pi$?

I would like to prove (geometrically if possible) the above result. Could someone help?
3
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29 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
votes
0answers
23 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 ...
3
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0answers
31 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
votes
0answers
67 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
votes
0answers
28 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 ...
3
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0answers
79 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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0answers
70 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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46 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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0answers
36 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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0answers
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
votes
0answers
67 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
113 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
56 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
131 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
72 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
99 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
57 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*} ...
3
votes
0answers
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
votes
0answers
58 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 ...