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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1answer
4 views

Product of distants from 1 to the corner of n gonal inscribe inside a unit circle.

Let $P_j=\cos \frac{2\pi}{n}+i\cos \frac{2\pi}{n}$ for $n\in \mathbb{N}.$ Show that $$ \prod_{i=1}^{n-1}|P_j-P_0|=n.$$
0
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2answers
49 views

Showing a complex sum $\sum_{n\geq 1} a_n/n^s$ converges for $\Re(s)>1$

The conditions on $a_n$ is that it is real and sums $a_1+....+a_n$ are bounded independently of $n$. I am given a hint that I should instead consider the sum $$\sum_{n\geq ...
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0answers
33 views

If $I=\int_{0}^R \frac {1}{1+x^{2n}}dx$, why$ \int_{0}^R \frac {1}{1+(x e^{i\frac{\pi}{n}})^{2n}}d(xe^{i\frac {\pi}{n}})=e^{i\frac {\pi}{n} }I$?

Just read a proof in the textbook Basic Complex Analysis, and there is one point that I do not understand: Let $I=\int_{0}^R \frac {1}{1+x^{2n}}dx$, where $x$ is real. We have the following: $$ ...
1
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3answers
37 views

Are all solutions to the ODE $ay''(t) + by'(t) + cy(t) = 0$ of the form $y(t)= \alpha e^{(\beta + i\gamma)t}$?

Let $a$ $b$ and $c$ be complex numbers. Consider the complex solution of the ODE $$ay''(t) + by'(t) + cy(t) = 0.$$ If there exist solutions to this, are they necessarily of the form $$y(t)= \alpha ...
-1
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1answer
26 views

Complex variable, multiplication of numbers

Question: Let a and b be complex numbers with $a \neq 0.$ Describe the set of points $az + b $ as $z$ varies over the first quadrant, $\{z = x+iy: x>0 \,and \,y>0\}$ Solution: Let $a = ...
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0answers
38 views

Can integration relate real and complex numbers?

We all know that $$\int\frac{dx}{1+x^2}=\tan^{-1}x+\mathbb C$$ Let's evaluate this a bit differently, $$\int\frac{dx}{1+x^2}=\int\frac{dx}{(1+ix)(1-ix)}$$ ...
0
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1answer
14 views

Arg function algebra

I have a question regarding the Arg function. Now I know, that $\text{Log}(z)=\log(|z|)+i\text{Arg}(z)$ for $z\in\mathbb{C}$. So it holds, that $\Im(\text{Log(z)})=\text{Arg}(z)$. My problem arises ...
6
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3answers
377 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

$\DeclareMathOperator{\Arg}{Arg}$ Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, ...
0
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1answer
21 views

How can I simplify this equation to not use complex numbers at all?

If I have the equation: $$ x^{a + ib} + y^{a+ib} = r^{a+ib} $$ It can be simplified to: $$ x^a e^{i b \ln x} + x^a e^{i b \ln y} = r^a e^{i b \ln r} $$ and then to: $$ \left(x^a\cos (b\ln ...
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1answer
150 views
+50

Multiplication of unitary matrices to make symmetric off-diagonal elements zero

Context Starting with a unitary matrix $U$ of size $m \times m$, I have read of a way to obtain a diagonal matrix by sequentially multiplying $U$ from the right by unitary matrices $V$ of a certain ...
1
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2answers
235 views

Logical explanation of Euler's formula

This question is a about (if not proving) at least guessing the Euler's formula. I don't want the proof using the infinite sums. We can guess by logic that for example that the equation ...
1
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2answers
16 views

Plot the numbers z in the complex plane that fufill $|z−2i|≤1$ and $\text{Im}(z){\ge}2$

I have recently started learning more about complex numbers and stumbled upon this problem: Plot the numbers $z$ in the complex plane that fulfil $|z-2i| ≤ 1$ and $\text{Im} (z) ≥ 2$ I know that ...
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0answers
45 views

A curious trigonometric equality

Let's consider the following expression: $(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} +  \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$ The left ...
2
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0answers
35 views

Polynomial division for identifying an expression in terms of complex numbers.

This question is blatantly copied from here, for the sake of learning more I specify it a bit more: $$f(z)= (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y)$$ $$z = x+yi$$ I want to write $f(z)$ in terms of ...
1
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2answers
28 views

Find all real numbers m for which equation $z^3+(3+i)z^2-3z-(m+i)=0$ has ..

Problem : Find all real numbers m for which equation $z^3+(3+i)z^2-3z-(m+i)=0$ has atleast a real root. No idea of approach : I am not getting idea how to approach a cubic polynomial in complex ...
0
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0answers
19 views

Let $A\in M_{n \times n}(\mathbb{C})$, $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a C.P. for $g(A)$.

Let $A\in M_{n \times n}(\mathbb{C})$. Its characteristic polynomial $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a characteristic polynomial for $g(A)$. I believe ...
1
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3answers
32 views

Matrix and scalar multiplication

Say we have the following variables: A, a matrix that is nxn in size containing complex numbers B, a matrix that is also nxn in size containing complex numbers x, a scalar If you multiply, does it ...
0
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0answers
16 views

Linear stability of an ODE $\frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right)$

This is a part of exercise: Consider the following equation: $$ \frac{dN}{dt}=-\mu N(t)+\mu N(t-T)\left(1+q\left(1-\left[\frac{N(t-T)}{K}[\right]^z\right)\right) $$ where all involved constants are ...
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0answers
17 views

Neighbourhoods with proper multiplication

Assume we have two closed subsets $F$ and $G$ of $\mathbb{C}^*$ which are proper for the multiplication, i.e. $$KF^{-1}\cap G$$ is a compact of $\mathbb{C}^*$ when $K$ is a compact of $\mathbb{C}^*$. ...
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1answer
30 views

number $-14+ \sqrt{15}+ \sqrt{12} $ which is located between two consecutive numbers? [on hold]

number of $-14+ \sqrt{15}+ \sqrt{12} $ which is located between two consecutive numbers? An interesting way , please
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1answer
49 views

$2^z$ behavior when changing real and imaginary components of $z$

I'm reading The Music of the Primes by du Sautoy and I've come across a section that I'm having difficulty understanding: Euler fed imaginary numbers into the function $2^x$. To his surprise, out ...
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4answers
72 views

Is this recurrence relation $g_{n+1}=ig_n-g_{n-1}$ is a trivial?

Let $g_1=i$ and $g_2=-1$, where $i=\sqrt{-1}$, and $$g_{n+1}=ig_n-g_{n-1}$$ For $n=1,2,3,4, ...$ then $g_n:={i, -1, -2i, 3, 5i, -8, -13i, 21, ...}$ respectively. Is this recurrence relation is ...
0
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1answer
57 views

how to evaluate this definite integral $\int_0^\infty\frac{\sin^2(x)}{x^2}dx$? [duplicate]

For $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx$. I considered using residue theorem. But since the function inside is holomorphic except for a removable singularity at the origin. So whatever contour I ...
0
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1answer
24 views

How many complexes modulo a prime $p$ are of multiplicative order $p^2 - 1$?

If $i = \sqrt{(-1)} \bmod p$, $p$ prime, does not exist, then we can form numbers of the form $a+b i \bmod p$ with multiplicative order $p^2 - 1$. How often do these numbers occur modulo $p$? In ...
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0answers
19 views

Stereographic projections.

1) Stereographicly project $\arg z =\frac\pi4$ 2)Using Inverse Stereographic projection map the 30th parallel south. My solutions: 1) That's a semi-cricle defined by $X^2+Y^2+Z^2=1;\ X=Y;\ ...
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0answers
26 views

Use Galois theory to find all complex roots of $T^4-2T^2-\sqrt{6}T+\frac{3}{4}$

I am currently studying Galois theory and a question that often comes up is "find all complex numbers which are roots of the polynomial $T^4+aT^2+bT+c$" where the coefficients are of the form ...
2
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1answer
27 views

If $a =\cos (\frac{2 \pi}{7})+i \sin (\frac{2 \pi}{7}) $ then construct a quadratic equation.

If $a =\cos (\frac{2 \pi}{7})+i \sin (\frac{2 \pi}{7}) $, then find a quadratic equation whose roots are $\alpha = a + a^2 + a^4$ and $ \beta = a^3 + a^5 + a^7$ . Using the fact that sum of $7th$ ...
2
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3answers
54 views

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$?

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? In other words, how quickly can we determine if a natural, $n$ exists where $n^2 \equiv -1 \bmod p$? NOTE This $n$ is ...
9
votes
1answer
87 views

Prove the n-th power of a matrix is the null matrix

Let $A,B$ squared matrixes with complex elements, $dim(A)=dim(B)=n, AB=BA, \det(B)\ne0$, having the following property: $|\det(A+zB)|=1, \forall z \in \mathbb{C}, |z|=1$. Prove ...
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4answers
43 views

How to understand $|1+z|<1$ where $z\in\mathbb{C}$ geometrically?

I am trying to understand how to plot $|1+z|<1$ where $z\in\mathbb{C}$, is it a circle centred at real axis $\text{Re}(z)=-1$ with radius $1$? My question is how can we show it algebraically? I ...
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2answers
33 views

Geometrical Description of $ \arg\left(\frac{z+1+i}{z-1-i} \right) = \pm \frac{\pi}{2} $

The question is in an Argand Diagram, $P$ is a point represented by the complex number. Give a geometrical description of the locus of $P$ as $z$ satisfies the equation: $$ ...
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1answer
77 views

Difference between Euler form and polar / trig form of a complex number

After some readings, I have found out that the difference between the polar / trigonometric form and the Euler form of a complex number consists on the fact that in the first case is expressed the ...
0
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1answer
436 views

complex numbers find greatest value of z

I've to sketch the complex number $z$ such that it satisfy both the inequality $|(z-2i)|\le2$ and $ 0\le \arg(z+2)\le 45\deg $ I was able to sketch and shade the region that satisfies both ...
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2answers
45 views

Complex logarithms when computing real-valued integral

My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral \begin{equation} \int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx \end{equation} ...
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1answer
36 views

Challanging problems on [Grade-12]Complex Number [on hold]

recently we are introduced to interesting world of complex number but except for 3-5 problems in the my books,all the problems are just plug-and chug,expression manipulation,etc.. which bores me out ...
2
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2answers
41 views

Prove (∀z∈ℂ\{1,-1} : |z|=1)(∃x∈ℝ) where z=(x+i)/(x-i)

I am having trouble proving the next problem: Prove that (∀$z$∈ℂ \ {1,-1} : |$z$|=1)(∃x∈ℝ) where $z=\frac{x+i}{x-i}$ What have I done: I observed complex number $z$ as dots on a circle with radius ...
0
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3answers
33 views

Let $z \in C^*$ such that $|z^3+\frac{1}{z^3}|\leq 2$ Prove that $|z+\frac{1}{z}|\leq 2$

Problem : Let $z \in C^*$ such that $|z^3+\frac{1}{z^3}|\leq 2$ Prove that $|z+\frac{1}{z}|\leq 2$ My approach : Since : $(a^3+b^3)=(a+b)^3-3ab(a+b)$ $\Rightarrow ...
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0answers
48 views

I need to compute $gcd(6+7i, -1+3i)$

I need to compute the $gcd(6+7i, -1+3i)$. I tried to calculate the function $\phi = (6 + 7i)(6-7i) = 85$ decomposed into prime factors is $85 = 5*17$, and $\phi = (-1+3i)(-1-3i) = 10$ decomposed into ...
1
vote
1answer
29 views

Let a,b,c be distinct non zero complex numbers with $|a|=|b|=|c|$ If each of …

Problem : Let a,b,c be distinct non zero complex numbers with $|a|=|b|=|c|$ If each of the equations $az^2+bz+c=0$ and $bz^2+cz+a=0$ has a root having modulus 1, then prove that : ...
2
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1answer
79 views

Find the min and max distance from origin of the curve $\vert z+\frac{1}{z}\vert=a$

$z$ is a complex number, by the way. I've tried a lot of things and always end up with a huge algebraic mess and I've wondered if anyone of you has any idea on how to approach this problem. One of ...
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vote
2answers
64 views

Solve $z^6+7z^3-8=0$

I want to find the solutions $z^6+7z^3-8=0$ but I don't know where to start because of the high degree of the equation. This is an exercise that involves complex numbers, so I have to transform the ...
0
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1answer
20 views

Let a be a positive real number and let $M_a=\{z \in C^* : |z+\frac{1}{z}|=a\}$ Find the minimum… [duplicate]

Problem : Let a be a positive real number and let $M_a=\{z \in C^* : |z+\frac{1}{z}|=a\}$ Find the minimum and maximum value of $|z|$ when $z \in M_a$ My approach : $|z+\frac{1}{z}|=a$ Squaring ...
2
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4answers
38 views

Simplifying sum of powers of conjugate pairs

The result of summing a conjugate pair of numbers each raised to the power $n$: $$ (a + bi)^n + (a - bi)^n $$ Produces a real number where $a + bi$ is a complex number. Given the result is real, is ...
2
votes
2answers
1k views

Finding the least value for points in a loci

The question goes like this: On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing the complex numbers $w$ satisfying ...
0
votes
1answer
35 views

Prove $f(z_0)I(\gamma;z_0)=\frac {g'(z_0)}{2\pi i}\int_{\gamma} \frac {f(z)}{g(z)-g(z_0)}dz. $

Let $f(z)$ and $g(z)$ be analytic in a region A and let $g'(z) \neq 0$ for all $z \in A$. Let g(z) be one to one and let $\gamma$ be a closed curve in A. Show that $$ f(z_0)I(\gamma;z_0)=\frac ...
0
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3answers
18 views

isolating x with two variables and negative exponents

I have: $$ 4^y = x^{-2} $$ Can someone hint to me what I need to do to isolate $x$? I'm not sure what to do.
1
vote
1answer
25 views

Higher degree polynomial with complex roots

I'm working on the following problem: $$ r^4 - 3r^2 -4r = 0 $$ I factor out one $r$ and leaving me $ r(r^3 - 3r -4) = 0 $. One real root is $r=0$, and I'm unable to find the other ones. I tried ...
0
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0answers
16 views

calculating $\int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2}$ using Residual Theorem [duplicate]

Could anyone help me provide a way to calculate $$ \int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2} $$ using the Residue theorem in complex analysis? Many thanks
2
votes
2answers
166 views

Complex projective line homeomorphic to $2$-sphere

Define an equivalence relation $\sim$ on $X={\bf C}^2\setminus \{(0,0)\}$ by $(x_1,y_1)\sim(x_2,y_2)$ if and only if there exists $t \in C\setminus\{0\}$ such that $(x_1,y_1)=(tx_2,ty_2)$ show that ...
0
votes
2answers
36 views

Express $\sin^3x$ in terms of cosines of multiples of $x$

I am studying complex numbers, and I have been trying to figure that out. Just not getting it. I keep getting $\frac{1}{-i8 (2\cos(3x) - 2\cos(x) - i4\sin(x))}$.