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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1
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1answer
48 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 ...
0
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4answers
41 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
55 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
19 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 ...
-1
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0answers
18 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;\ ...
0
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0answers
19 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
25 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
50 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
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1answer
82 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 ...
0
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4answers
42 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 ...
1
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2answers
28 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: $$ ...
1
vote
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 ...
0
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2answers
42 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} ...
-2
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1answer
34 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 ...
1
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0answers
19 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: $$f(z)= (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y)$$ $$z = a+bi$$ I want to write $f(z)$ in terms of ...
2
votes
2answers
39 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
32 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 ...
0
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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
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1answer
28 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
votes
1answer
77 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 ...
1
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
votes
1answer
18 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
votes
4answers
37 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
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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
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1answer
30 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
24 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
15 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
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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
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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 $\frac{1}{-i8 (2\cos(3x) - 2\cos(x) - i4\sin(x))}$.
2
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1answer
43 views

Under what conditions on $f$, is $f(az)=g(a)f(z)$?

Formal Statement Given nonzero constant $a \in \mathbb{C}$, $|a|>0$ and $f:\mathbb{C} \to \mathbb{C}$, under what conditions on $f$ does the following hold? \begin{equation} f\left(a ...
3
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0answers
105 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 ...
379
votes
21answers
65k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
10
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5answers
2k views

Unexpected result from Euler's formula

I am a bit confused with a result I get from Euler's formula: $e^{2\pi i} = 1$ $\sqrt[3] { e^{2\pi i} }= \sqrt[3]{ 1 }$ $(e^{2\pi i})^{\frac{1}{3}}= 1$ $e^{\frac{2}{3} \pi i} = 1$ This last ...
4
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1answer
37 views

Graphically solving for complex roots — how to visualize?

So recently we've been doing the complex roots of quadratics, cubics and polynomials in general in school. But my question is, is there a way to see where these roots are, just like you can see where ...
1
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0answers
32 views

The singular points and residues of $\sin(\frac 1 z)$

I met a question asking all the singular points and corresponding residues of $$ \sin \frac 1 z $$ My understanding is that $$\sin \frac 1 z=\frac 1 z-\frac 1{3!z^3}+\frac 1 {5!z^5}+... $$ Thus ...
0
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3answers
63 views

Multiplication of real and complex radicals

If I have, for example, the product $\sqrt{7+\sqrt{22}}\sqrt[3]{38+i\sqrt{6}} $ Can I perform the multiplication or this cannot be done and only remains to leave the product in this form?
0
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1answer
34 views

An inequality with complex numbers.

Given $n$ complex numbers $z_1,\ldots,z_n$, is it true that $$ |z_j|\sum_{k=1}^n|z_k|\leq\sum_{k=1}^n|z_k|^2 $$ for $j\in\{1,\ldots,n\}$ ? Thank u for any help!
15
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6answers
827 views

Help me prove $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$

Please help me prove this Leibniz equation: $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$. Thanks!
0
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0answers
18 views

Loops around 0 of polynomial restricted to the unit circle [duplicate]

Given a polynomial with coefficients in C, consider the image of the polynomial restricted to the unit circle (That is plugging in only things with absolute value one). How many loops around 0 can ...
0
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0answers
39 views

Complex Matrix Representation

Lets say if $X\in C ^{m\times n}$, it does have real and imaginary parts. If I want to represent a matrix in real and imaginary form then why I write it this way where is $i$? \begin{bmatrix} X_r ...
68
votes
9answers
4k views

Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} ...
0
votes
1answer
57 views

complex numbers and locus

When the problem says that the complex number $z$ moves on the straight line $y=2x$,what "clue" do I get from that? And generally when it says that a complex number belongs/moves to a conic section ...
1
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0answers
36 views

Proving analytic function $f = 0$ under certain assumtions

I was given the following exercise: Let $f(z)$ be analytic in an open and connected set $U$ containing the point $z=0$ and assume $|f(1/n)| < \frac{1}{2^n}$ for $n \in \mathbb{N}_{> 0}$. Prove ...
0
votes
2answers
29 views

How to find the absolute value of this complex number: $\frac{-4-6i}{17+i}$

I know that, in general, $|a+bi|=\sqrt{a^2+b^2}$, however, I don't know how to make $\frac{-4-6i}{17+i}$ into the form of $a+bi$.
0
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1answer
38 views

Solving an equation involving complex numbers.

I tried solving the problem on my own. I would like to know if I have made any mistakes. If I have indeed made a mistake, I would appreciate it if someone corrects it and explains what it is. Also, I ...
0
votes
0answers
18 views

Solving systems of linear equations with complex numbers by hand

How can I solve a 3x3 system of linear equations with complex numbers by hand without making a mistake? I know that I can solve them either with Gaussian Elimination or Cramer's rule, but I find it ...
6
votes
3answers
355 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 $n \geq 1$, ...
2
votes
1answer
40 views

Is the converse of the Pythagorean Theorem false for complex inner products?

I was thinking about the converse of the Pythagorean theorem: $\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y\rVert^2 \implies x \perp y$ Does this hold if the inner product $\langle ...