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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2answers
46 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 e^{(...
1
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0answers
43 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, $$\int\frac{dx}{1+x^2}=\int\frac{dx}{(1+ix)(1-ix)}$$ $$=\frac{1}{2}\int\frac{(1+ix+1-ix)dx}{(1+ix)(...
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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 ...
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1answer
25 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 x)+y^a\...
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0answers
48 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 ...
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2answers
20 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
19 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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3answers
35 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 ...
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0answers
22 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 ...
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1answer
31 views

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

number of $-14+ \sqrt{15}+ \sqrt{12} $ which is located between two consecutive numbers? An interesting way , please
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0answers
27 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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2answers
30 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 ...
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4answers
83 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 ...
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0answers
33 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 $\pm\...
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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 $X^2+Y^2+Z^2=1;\ X=Y;\ X,Y>0$...
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1answer
26 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 ...
2
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1answer
28 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
57 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 ...
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4answers
44 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 ...
9
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1answer
101 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 $A^n=0_n$. ...
0
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2answers
49 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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2answers
58 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 ...
2
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0answers
37 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 $z$...
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0answers
56 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 ...
0
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3answers
34 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 z^3+\frac{1}{z^3}=(z+\frac{1}{...
1
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1answer
31 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 : $|a-b|=|b-c|=|c-a|$...
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1answer
22 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 ...
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4answers
39 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 ...
1
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2answers
38 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: $$ \arg\left(\frac{z+...
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3answers
19 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.
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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
1
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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 ...
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1answer
40 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 {g'(...
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1answer
44 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 z\...
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2answers
37 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 $\frac{1}{-i8 (2\cos(3x) - 2\cos(x) - i4\sin(x))}$.
4
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1answer
49 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 ...
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0answers
33 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 ...
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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 ...
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1answer
35 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!
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2answers
67 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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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 ...
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3answers
66 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?
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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 ...
1
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0answers
38 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
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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 $a+bi$.
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1answer
41 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
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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 ...
2
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1answer
49 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 \cdot,\...
3
votes
1answer
67 views

About prefactor in book's Gamma function identity

In "Mathematical Methods for Physicists" (Arfken & Weber, 7th ed.), exercise 13.1.16 says the following, Prove that $$|\Gamma (\alpha+i\beta)|=|\Gamma(\alpha )|\prod_{n=0}^{\infty}\left[1+\...
2
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1answer
48 views

Simplifications of nth roots of complex numers.

Is it easy to find that $( a + ib)^n$ is equal to a certain complex number, say $p+iq$, by just using Newton's binomial theorem. But how to find in general that $\sqrt[n]{p+iq} = a+ib$, where $a, b, p,...