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

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Sketching complex numbers on an Argand diagram help

for {${z\in \mathbb C : Im(z)>0}$}, we simply sketch the upper half of the Real axis, right? Then, if we have $z=a+ib$, and we sketch that, and we have $w=iz=-b+ai$ which means $w+1=(1-b) + ai$ ...
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29 views

Notion of complex optima

Consider the function: $$y = \frac{1}{3}x^3 + x$$ Suppose we wanted to determine its local optima, but instead of looking at local optima with domain $R$ we instead consider domain $C$ and range ...
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35 views

Prove that if a complex number raised to nth power equals one, then product of the solutions to this euqations is one

I've thought of this "proof", but it seems lame to me (I'm not sure it is even a proof), I probably should have used other properties of complex numbers to write one. So, $z^n=1 \implies ...
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3answers
34 views

Solving $z^3=-1+i$

First, is there a better way than using x+iy and solving the system? I tried letting $z=e^{i3\theta}$ and using the cosine and i*sine way but I don't see how that can equal -1 and i at the same time, ...
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4answers
48 views

argument of the complex number $1-\cos x-i\sin x$

How can I find the argument of $1-\cos(x)-i\sin(x)$? Can I use the exponential form of a complex number? If so, how do I continue to find the the answer?
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27 views

Complex matrix calculations

Sorry about the vague subject but I really found some difficulties in calculating complex matrices. Assume $Z$ is a square Hermitian non-singular complex matrix, then we denote $$F= \left[ ...
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1answer
52 views

$\cosh(iz) -\cosh(z)=0$

$\cosh(iz) -\cosh(z)=0$ Apparently $iz=z$ and $z=0$ is a solution. How do I proceed next? Do I need to convert $\cosh$ into $\exp$ form? I tried that I get $e$ to complex and real power and I don't ...
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arguments in complex numbers

I need to show that if $\arg\left(\frac{z_1+z_2}{z_1-z_2}\right)=\frac{\pi}{2}$ then $|z_1|=|z_2|$. How should I work this out? I know that $\arg\left(\frac{z_1+z_2}{z_1-z_2}\right) = ...
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30 views

Complex number forming equilateral triangle

Suppose we have 3 complex numbers , such that $$|z_1|=|z_2|=|z_3|=1$$ and they form equilateral triangle then will condition $$z_1.z_2.z_3=1$$ always be true? I know cube roots of unity , that is ...
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Suppose $x = 3 - 2i$ and $y = 4 + i$. Find both square roots of y. Then indicate which one is the principle square root.

Suppose $x = 3 - 2i$ and $y = 4 + i$. Find both square roots of $y$. Then indicate which one is the principle square root. Use the polar form of complex numbers to accomplish this task. I'm not ...
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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 ...
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Expressing complex numbers in standard form problem

Express the following complex number in the standard form $x + iy$ $ie^{\frac{i\pi}{2} +3} $ I have made an attempt and got the answer $\cos(\frac{\pi}{2} +3) +i\sin(\frac{\pi}{2} +3)$. Is this an ...
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1answer
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Recursive sequence with Complex numbers, missing conclusion.

I am solving the following task: Let $a_1 = \sqrt{2}\sqrt{3}*i, a_{n+1} =\frac{i* a_n}{n+1}$ What can you say about the convergence of $a_n$? I already found out a lot. What i concluded so far, is: ...
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39 views

Values of the complex power $1^\sqrt{2}$

I have to show that the values of the complex power $1^\sqrt{2}$ all lie on the unit circle, i.e. that $|1^\sqrt{2}|=1$. $1^\sqrt{2} = e ^ {\sqrt{2} \ln{1}}$ by definition, and $\ln{1} = 2k \pi i$ ...
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3answers
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imaginary algebraic inequality equation

This problem was actually given to me as a typo. I decided to work it despite it being a typo and it presented a couple of questions regarding applying imaginary results to an inequality equation. ...
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1answer
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Complex numbers finding real values of x and y

Im having trouble where to go next? Solving the real values of x and y $2+x+jy=(x-jy)(5+j6)$ Here's my working $2+x+jy=(x-jy)(5+j6)$ $2+x+jy= 5x+6y+j6x-j5y$ Real numbers $2+x=5x+6y$ Imaginary ...
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Homework: Restriction on |z+1|+|z-1|=2, z∈C

After simplifying the relation $|z+1| + |z-1| = 2, z∈\mathbb{C}$ to $\Im(z)=0$, I plotted the original relation on my TI-nspire calculator and WolframAlpha. As expected it simplifies to $\Im(z)=0$ ...
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1answer
47 views

Is it possible to have Logarithm with base 1 or 0?

I am wondering is there any definition that allows logarithm to have base 0 or 1 in real or complex fields (considering Euclidean space)?? Out-coming question is if you can define a logarithm with ...
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How to evaluate $\lim_{z \to i} \frac{z^2+i}{z^4-1}$?

This is a pretty basic question but I've been stuck on it for a while now. $$\lim_{z \to i} \frac{z^2+i}{z^4-1}$$ My attempt: $z^4-1=(z^2+i)(z^2-i)-2$ and then dividing the numerator and ...
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Convert $\frac{1+ \sqrt{3i} }{1- \sqrt{3i} }$ to polar form

How do I convert $\frac{1+ \sqrt{3i} }{1- \sqrt{3i} }$ to polar form? I came across it in this question but I don't know much about complex numbers and have no idea how to figure out $\theta$.
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Complex number calculation

I'm supposed to show $ z^{10} $, when z = $ \frac{1+ \sqrt{3i} }{1- \sqrt{3i} } $ I can work it out to $ \frac{(1+\sqrt{3}\sqrt{i})^{10}}{(1-\sqrt{3}\sqrt{i})^{10}} $ However this is inconclusive ...
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3answers
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How to prove $\lim_{x\to \infty}\Im\left(xi^{1/x}\right)=\frac{\pi}{2}$?

I have yet to study complex analysis, but I sperimentally found $$\lim_{x\to \infty}\Im\left(xi^{1/x}\right)=\frac{\pi}{2}.$$ W|A agrees with me too, and while I know that's not so significant, ...
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55 views

Limes of $a_n = i^n$

Out of couriosity and for my understanding i want to ask: When i have the sequence $a_n = i^n$ While i is the imaginary number, i will of course have four accumulation points: $-1,1,-i,i$. So the ...
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2answers
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Problem involving complex conjugate

I have the equation : $3z-\bar{z}=2-3i$ First I write this as : $3(x+yi)-(x-yi) = 2-3i$ $= 3x-3yi-x+yi = 2-3i$ $= 2x+4yi = 2-3i$ Now the following must be true : $2x = 2\wedge 4y = -3$ So $x = ...
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1answer
18 views

Complex root of equation

Suppose we have $$z^2 + kz + m=0$$ where $k,m$ are real and $z$ is complex such that two distinct roots of this equation lie on $Re(z)=1$ so what will be range of m? Since root of this this equation ...
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1answer
24 views

Rotating the complex number

Suppose we have square circumscribing circle $$|z-1|=\sqrt{2}$$ and one of its vertices is $$2+\sqrt{3}i$$ so what will be other vertices ? I simple tried rotating half diagonal of square in ...
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Linear Transformation of Complex Numbers

I have recently started taking a Linear Algebra course and we have been given a question beyond what we have studied so far. I was hoping I could find some guidance here. We are told to find out ...
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Division of complex numbers when to use what sign

I have two examples of dividing complex numbers, but both do the sign differently. The first is: $$\frac{a+bi}{c-di} \cdot \frac{c+di}{c-di}$$ the other is: ...
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23 views

Show that a non-constant entire function has a dense image.

Let $f$ be a nonconstant entire function and $U$ be an open set in the plane. Show that there is a $z_0$ such that $f\left(z_0\right)\in U$. This question is an exercise for the Maximum Modulus ...
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Finding $\tan \pi/8$ from $\sqrt{1+i}$.

I want to prove that $\tan \pi/8 = \sqrt{2} - 1$ using $\sqrt{1+i}$ in some way. Write: $\sqrt{1+i} = a+bi$, and let's find $a$ and $b$. We have: $$1+i = a^2+b^2 + 2abi,$$ so $a^2+b^2 = 1$ and $ab = ...
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1answer
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Another way to prove that if n$^{th}$-degree polynomial $r(z)$ is zero at $n+1$ points in the plane, $r(z)\equiv 0$?

The original problem is as follows Let $p$ and $q$ be polynomials of degree $n$. If $p(z)=q(z)$ at $n+1$ distinct points of the plane, the $p(z)=q(z)$ for all $z\in \mathbb{C}$. I attempted ...
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Determine the number of zeros for $4z^3-12z^2+2z+10$ in $\frac{1}{2}<|z-1|<2$.

I'm faced with the problem in the title Determine the number of zeros for $4z^3-12z^2+2z+10$ in the annulus $\frac{1}{2}<|z-1|<2$. Clearly this requires a nifty application of Rouche's ...
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278 views

Prove $|\cos^2(z)| + |\sin^2(z)| > 1$ for complex numbers $z$ with nonzero imaginary part

Prove $$|\cos^2(z)| + |\sin^2(z)| > 1$$ for $\operatorname{Im}(z) \ne 0$ I know from using the triangle inequality, $|x+y| \leq |x| + |y|$, that $|\cos^2(z)| + |\sin^2(z)| \geq 1$ but I don't ...
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1answer
39 views

Inequality which involves complex numbers and absolute values

How can I solve the following inequality: $|\frac{(1+(1-\theta)z)}{1-\theta z}| \leq 1$ ? $z$ is a complex number. I have to find the values of $\theta$ for which the inequality is satisfied.
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proving $zw+\frac{16}{zw}=-8\cos(\theta+\beta)$

$z,w$ are complex numbers (not zero) that are in the first quarter of Gauss's plane such that: $$z+\frac{4}{z}=4\sin\theta$$ $$w+\frac{4}{w}=4\sin\beta$$ Need to prove that: ...
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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 ...
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1answer
56 views

Convergence of $a_n = (1+i)^n+(1-i)^n$

I am asking a question related to Is there a formula for $(1+i)^n+(1-i)^n$? I am looking on the exact same term, just as a sequence, so i want to find out: Is $a_n = (1+i)^n+(1-i)^n$ convergent or ...
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Multiplying complex numbers

I'm doing some AC circuits problems and it involves complex numbers. In the textbook, it was given that the multiplication of complex numbers, in polar form is: $$z_1 \cdot z_2 = r_1 r_2 \angle ...
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If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$?

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$? I'm asked to give a proof or a counterexample. I'm a bit confused on the notation of $\mathbb R(a)$, what does this ...
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1answer
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Upper bound for $\left|(a+\pi)^k e^{i \pi x }- (a-\pi)^k e^{-i \pi x }\right|$.

I want to find an upper bound for $$\left|(a+\pi)^k e^{i \pi x }- (a-\pi)^k e^{-i \pi x }\right|\leq ?$$ where $a,k\in\mathbb{N}, x\in \mathbb{R}$. thanks a lot
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2answers
26 views

Convergence of sequences in $\mathbb{C}$

I am currently getting into the field of complex numbers, with the imaginary unit $i^2 = -1$ and stuff. At the moment i am looking onto a few sequences in $\mathbb{C}$, regarding convergence. I have ...
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1answer
39 views

Elementary question about imaginary numbers.

I'm a physics student and I seem to be having trouble accepting a glaring inconsistent with regards to $i^2$. From all the math sources I see, $i^2$ is defined as -1. While, on the other hand, ...
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Balance of forces in a mechanics problem

I tried to solve a particular problem of mechanics and found some difficulties in the vector analysis part that I can't get rid of. It's probably some stupid mistake I made, but I can't see it now, ...
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Complex polar co-ordinates

We know that rectangular co-ordinates $(x, y)$ can be written as a complex number $re^{i\theta}$ where $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1} \big(\frac{y}{x}\big)$ and $r,\theta \in ...
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3answers
61 views

Prove using De Moivre's formula,that $\sum\limits_{k=0}^{n}\sin(kx)=\frac{1}{2}\cot(x/2)-\frac{\cos(nx+(x/2))}{2\sin(x/2)}$

I've been asked to prove that: $$ \sum\limits_{k=0}^{n}\sin(kx)=\frac{1}{2}\cot(x/2)-\frac{\cos(nx+(x/2))}{2\sin(x/2)} $$ When $0<x<2\pi$. I know there are many similar posts on this site, but ...
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38 views

Roots of a polynomial $f(x)$ in $\mathbb{C}[x]$

"Find all the roots of the polynomial $f(x)=x^2+(3i-2)x-2(1+i)$. Why does the answer not violate the $Conjugate \space Roots \space Theorem \space (CJRT)$" I tried using the quadratic formula and ...
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1answer
46 views

Proving geometry using complex numbers?

Consider the following figure: Let $A(z_1),B(z_2),C(z_3),E\equiv P(z),O(\mathtt{0})$Q(-z), I need to prove BQ=AC, I can prove it anyways but using complex numbers. Anyways what I tried is as ...
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3answers
26 views

Number of Complex satisfying a given condition

Suppose we have a complex number $z$ such that $|z|=1$ and $$|\frac{z}{z'} + \frac{z'}{z}|=1$$ where $z'$ is conjugate . How many complex number satisfy this? So I simplified second condition as ...
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1answer
42 views

Gemetric representation of complex numbers

Find the geometric representation of; |z-2| - |z+2| < 2 things i know; | z - 2 | is the distance between a point z and the point (2,0) in the complex plane. It suppose to represent a hyperbola, ...
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2answers
33 views

Complex roots forming a equilateral triangle

Suppose we have relation $$z^2 + az + b=0 $$ where $a$ and $b$ are real and roots of this equation $z_1$ and $z_2$ form equilateral triangle with origin then what could be relation between $a$ and ...