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Questions tagged [complex-numbers]

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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convergence and absolute convergence of a complex serie

Analyse convergence and absolute convergence of a complex series: $$\sum_{n=1}^{\infty}\frac{e^{in}}{n^2}$$ Can I simply use a fact that $|e^z|=|e^xe^{iy}|=e^x$, so $\sum_{n=1}^{\infty}\frac{e^{in}}{...
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2answers
50 views

Advanced Complex numbers

Let $\omega$ be a nonreal root of $z^3 = 1.$ Let $a_1,$ $a_2,$ $\dots,$ $a_n$ be real numbers such that $$\frac{1}{a_1 + \omega} + \frac{1}{a_2 + \omega} + \dots + \frac{1}{a_n + \omega} = 2 + 5i.$$...
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2answers
37 views

Solving complex roots

a) If $z = \cos x + i \sin x$, show that $z^{-1} = \cos x - i \sin x$ b) Show that $\cos (nx) = 0.5(z^n + z^{-n})$ Both of these questions are very simple and I get how to do them. It then follows ...
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1answer
40 views

Find $n \geq 4$, natural number

Find $n \geq 4$, natural number such that for every distinct complex numbers $a,b,c$ different of $0$ which satisfy $(a-b)^n + (b-c)^n + (c-a)^n =0$ implies that $a, b, c$ ...
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1answer
35 views

weird properties of complex exponentials: $e^{i 2 \pi f(x)} = 1$ [duplicate]

$$ \begin{aligned} e^{i 2\pi f(x)} &= (e^{i\ 2\pi})^{f(x)} \\ e^{i 2\pi f(x)} &= (\cos {2\pi} + i\ \sin {2\pi})^{f(x)} \\ e^{i 2\pi f(x)} &= (1 + i 0)^{f(x)} \\ e^{i 2\pi f(x)} &= 1^...
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1answer
25 views

Summation of series to n terms in trigonometry of complex numbers

The question says that: Sum the series I have solved the answer as follows: As the above picture, I don’t know what should I do after the step. The question asks to solve the problem using ...
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2answers
14 views

The cartesian form of a complex number with high power index

In the example I have : write $z^{2018}$ in its cartesian form . $z= \frac{\sqrt 2}{2}(1-i)$ What are the steps that I should follow to solve such thing? (HINT in the bottom of the page : remember ...
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1answer
37 views

If $f:[0,1]\to \mathbb{C} $ be continuous with $f(0)=0$ and $f(1)=2$, then $|f(t_0)|=1$ for some $t_0 \in [0,1]$

Question: Let $T=\{z\in \mathbb{C}:|z|=1\}$ and $f:[0,1] \to \mathbb{C}$ be continuous with $f(0)=0$, $f(1)=2$. Show that there exists at least one $t_0$ in $[0,1]$ such that $f(t_0)$ is in $T$. ...
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1answer
55 views

What is the derivative of the real part of a complex variable?

If I have the complex variable $z=x+iy$ and the function $f(z)=z$, is it possible to calculate $\frac{d\Re{f(z)}}{dz}$, or in this particular case $\frac{dx}{dz}$? It should be equal to $\frac{1}{2}$, ...
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2answers
49 views

Find the value of $\log \frac{a+b i}{a-b i}$

Find the value of $\log \frac{a+b i}{a-b i}$ The answer is 2$i\tan^{-1}{\frac{b}{a}}$ I have calculated it upto $i \tan^{-1}{\frac{2ab}{a^{2}- b^{2}}}$ But I am unable to convert it in given answer ...
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1answer
21 views

How to evaluate this partial derivative in terms of polar coordinates

How to evaluate this partial derivative in terms of polar coordinates? How to solve this question?
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0answers
14 views

Prove a given domain and a contour is close and bounded

I am having trouble proving this lemma. Given domain $\Omega \subset \mathbb{C}$ and a contour $C \subset \Omega$ (piece-wise smooth), prove the following: If $C$ is closed and bounded, then the ...
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0answers
93 views

About $\sum\limits_{r=1}^{k}\exp\left[2i\pi\sum\limits_{k=1}^{n}\frac{r^{k}}k\right]$

Context: I recently saw user @David's profile picture and description: "My icon is the graph of the exponential sum $$\sum_{n=1}^{10620}e^{2\pi if(n)}$$ for $$f(n)=\frac{n}{20}+\frac{n^2}{9}+\frac{...
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2answers
25 views

Roots of unity divisibility.

Suppose $r | n$. Then $R:= e^{2i \pi k/r}$ is an $n$-th root of unity. Thus, there exists a unique $l \in \{0, \dots, n-1\}$ such that $R = e^{2\pi i l/n}$. Does it hold that $l |n$? I tried to ...
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0answers
11 views

Multiplication of Phasors as Complex Numbers and Vectors

I am reading about phasors. Everywhere it says that phasors are complex numbers and vectors which is obvious given that every complex number is a two-dimensional vector. But there are a lot of ...
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0answers
27 views

How to prove that Re(zeta(1+i/n)) is equal to the euler constant as n -> infinity

I have noticed that $\displaystyle\lim_{n \to \infty} \Re\left[\zeta\left(1 + \frac{i}{n}\right)\right] = \gamma$ How can this be proven?
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0answers
26 views

Complex geometry, complex numbers

The locus of any point $p(z)$ on Argand plane is $\;\arg((z-5i)/(z+5i))=\pi/4$, then the total number of integral points inside the region bounded by the locus of $p(z)$ and the imaginary axis is? Any ...
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1answer
88 views

Making sense of $(-1)^\frac{1}{2}$

I just opened a thread that sparked a new question that I now want to discuss (making a new thread however to not clog the other one). Obviously looking at $(-1)^{\frac{1}{2}}$ one might have the ...
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3answers
59 views

$(a^b)^c$ and $a^{(bc)}$ for complex numbers

I've just stumbled across $$\left(e^{2\pi i}\right)^{\frac{1}{2}}=\sqrt 1=1\neq -1=e^{\pi i},$$do I have some error in my thoughts there or does $(a^b)^c=a^{(bc)}$ not hold for complex numbers?
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1answer
20 views

The principal argument of a complex number in polar form

If z=r(cos θ + isin θ), where r>0 and 0< θ < 1/2π, find in terms of r and θ the modulus and principal argument of.... a) -z I started off by: -z=-r(cos θ + isin θ) ---> = r(-cos θ - isin θ)...
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2answers
45 views

If $a \in \mathbb{C}$ and $\exists n \in \mathbb{N}$ s.t. $\{ a^n, a^{n+1} \} \in \mathbb{N}$, prove $a \in \mathbb{N}$ [on hold]

Let $a$ be a complex number. If it exists a natural number $n$ (different of $0$), such that $a^n$ and $a^{n+1}$ are integers, prove that $a$ is an integer.
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1answer
46 views

Finding $\cos(\theta)+\cos(\theta+\alpha)+\cos(\theta+2\alpha)+…+\cos(\theta+n\alpha)$ with complex variable analysis [duplicate]

We have a series as $\cos(\theta)+\cos(\theta+\alpha)+\cos(\theta+2\alpha)+...+\cos(\theta+n\alpha)=U$ How can we make use of complex variable analysis to arrive at the term below which is ...
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1answer
36 views

Proving by calculation that $\arg(2i) = \pi/2$ [on hold]

I was just wondering how you would prove that $\arg(2i)=\pi/2$
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2answers
37 views

Show that $c^2 + a^2d=abc$ for a monic quartic polynomial

I apologize in advance for asking a homework question, but I have genuinely no idea on how to approach part b). The question is as follows: Consider the polynomial equation $\rm{P}(x)=x^4 + ax^3 + ...
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4answers
80 views

If $i^2=-1$ why isn't $i$ also equal to $-\sqrt{-1}$? [duplicate]

I've read on this site several times that it is better to say $i^2=-1$ as opposed to $i=\sqrt{-1}$. If we let $i^2=-1$, why doesn't $i=-\sqrt{-1}$? This makes sense since $(-\sqrt{-1})(-\sqrt{-1})=-...
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0answers
31 views

Complex roots of a complex polynomial

In an Argand plane, $A(z_1), B(z_2), C(z_3)$ are distinct complex numbers lying on the curve $\;|z| = \sqrt 3$. If a root of $z_1z^2 + z_2z + z_3 = 0$ has a modulus equal to unity then $z_1, z_2, z_3$...
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3answers
23 views

Differentiating a complex function using the definition

I need to differentiate the complex function $f(z)=z^2+z$. I know that the definition of a derivative is $f'(z)=\frac{f(z)-f(z_0)}{z-z_0}$. In this case, $f'(z)=\frac{(z^2+z)-(z_0^2+z_0)}{z-z_0}$. ...
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0answers
6 views

Clarification of Complex Distributivity Proof

I am trying to prove complex distributivity. The proofs that I have seen proceed as follows: $z_1(z_2 + z_3) = (a + ib)[(c + id) + (e + if)]$ $= a+ib.(c+e)+i(d+f)]$ $= a.(c+e)-b.(d+f)+i[...
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3answers
39 views

Is the statement true?$|z+1|\ge|z|-1\ \forall z\in\Bbb{C}$

I have tried to prove it in the following manner- $||z|-1|=||z|-|1||\le|z-1|\ \forall z\in\Bbb{C}$ (by Triangle inequality) Now, can I write $|z-1|\le|z+1|$ in $\Bbb{C}$? If yes then the proof is ...
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1answer
46 views

If I pick $-1 = \sqrt{1}$, then why $ \sqrt{zw}= \sqrt{z}\sqrt{w} $ for only $z, w \le 0$?

This Reddit comment expatiates why the third equality (colored in red) is the one that's wrong in $\color{limegreen}{1 = \sqrt{1}} = \sqrt{(-1)(-1)} \color{red}{=} \sqrt{-1} \sqrt{-1} = i² = -1$. ...
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Evaluate fourier integral : $\int_{-\infty}^\infty \frac{e^{isx} ds}{(s-\tfrac{7i}{2})^2 + (1/2)^2}$

Evaluate $$\int_{-\infty}^\infty \frac{e^{isx} ds}{(s-\tfrac{7i}{2})^2 + (1/2)^2}$$ Here I am having trouble as using simply the fourier inverse of $\frac{2|a|}{s^2+a^2}$ which is $e^{-|ax|}$ gives ...
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0answers
24 views

Sums of norms of zeros of polynomial

For complex $z$, let $N(z)$ denote the norm of $z$, $N(z)=z \overline{z}$ where $\overline{z}$ is the complex conjugate of $z$. One day, one of my colleague ask me if I can solve the following. ...
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2answers
24 views

Finding square roots of complex number $u$

A complex number $u$ is given by $u = -1+(4\sqrt{3})i$. Find the two square roots of $u$. Now, I know we have to compare the equation with $a+bi$ but my text book doesn't square both sides in the ...
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6answers
60 views

How to raise a complex numbers to a power? [on hold]

$$((1+\sqrt{3}i)/2)^n$$ What is the cartesian form of it? We never had complex numbers at university . So i have to teach myself. My problem is the exponent.
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1answer
47 views

Help solving complex equation

$$(1 + \sqrt{3i} ) z^{4} = - 800$$ I tried doing this, but it feels wrong: Edit: I also tried multiplying with the conjugate in step 2, and then multiplying with the conjugate again to get 9 in the ...
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3answers
67 views

What is the value of $|\alpha|$?

Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha}}$ (notice the bar above $\alpha$) lie on the circles $(x-x_0)^2+(y-y_0)^2=r^2$ and $(x-x_0)^2+(y-y_0)^2=4r^2$ respectively. If $z_0=x_0+iy_0$...
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1answer
32 views

Solving $z^2+\frac{9z^2}{(3+z)^2}=-5$

Solve the equation $$z^2+\frac{9z^2}{(3+z)^2}=-5$$ PS.: The expanded form of a 4 degree polynomial is $$z^4+6z^3+23z^2+30z+45=0$$
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2answers
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An intriguing supremum… which may be inaccurate

I found the following exercise and I would like to know whether the property is true or not, and above all how to prove it: for $z$ so that $\Im z>0$, $$\sup_{t\in\mathbb{R}}\left|\frac{t-i}{t-z}...
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3answers
27 views

Problem of complex equations and Cauchy-Riemann

I have the following problem: $$u (x, y) = \sin x \sinh y.$$ I need to check that the function is harmonic in the whole plane. But I do not know where to start, I would appreciate your help. Thanks ...
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1answer
28 views

Solve $\sinh z = i$

I am trying to solve $\sinh z = i$; this has to result in $2\pi n + (1/2) \pi i$. When applying the exponential form of $$\sinh z = \frac{e ^ z - e ^ {-z}} 2$$ and later the quadratic equation, this ...
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3answers
102 views

Understanding a proof that a collection of complex numbers on one side of a line through $0$ must have a non-zero sum

I'm asked to prove that if $z_1$, $\ldots$, $z_k$ lie on one side of a straight line through $0$, then $z_1+\cdots+z_k \neq 0$. In the proof, we let $\theta$ be the angle between the line and the ...
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0answers
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Von Neumann stability analysis - almost finished exercise

I wish to analyze the stability of the FTCS scheme for the equation $u_t = iau_{xx} + cu$ where $a \in \mathbb R, c \in \mathbb C$. I will succinctly go through what I did and where I'm stuck. A ...
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0answers
24 views

Which complex numbers work as a (numeral) base?

When we write a real number in base k the n’th digit represents the number of k^n. I’m interested in finding a complex number w such that any complex number z can be written in base w. I found that ...
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1answer
59 views

Is $f(z) = z^{2}$ an automorphism of $\mathbb{C}$?

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ where $f(z) = z^{2}$. I claim $f$ is an automorphism of $\mathbb{C}$ as a vector space. First $\operatorname{ker}(f) = 0$ and since $\mathbb{C}$ is ...
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3answers
54 views

Find all the $z$ such that $(1+\frac{1}{z})^{4}=1$

Question if $S$ be the set of solution of $$\bigg(1+\frac{1}{z}\bigg)^{4}=1$$ then prove that the points are co-linear. Attempt $\bigg(1+\frac{1}{z}\bigg)^{4}=1$ $\implies z^4+4z^3+6z^2+4z+1=z^4$ $...
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1answer
60 views

Proof of $ \ \vert \log(z) \vert \lt \log \frac{1}{1-r} \ \ $ given $ \forall z ,\ \vert z-1 \vert \leq r \lt 1$?? [closed]

Let $\ \vert z-1 \vert \leq r \lt 1$ and let $\log(z)$ be the principal branch of the logarithmic function. How can you prove $$ \vert \log(z) \vert \lt \log \frac{1}{1-r} \ \ ?$$ Also, how can you ...
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0answers
38 views

Find the imaginary part of z [closed]

If $z = (a+ib)^r \times e^{x+iy}$, find the imaginary part of $z$, (note that r may not be an integer).
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0answers
54 views

Maximum value of $|Z_1-Z_2|^2 +|Z_2-Z_3|^2+|Z_3-Z_1|^2$ [duplicate]

Given three complex numbers $|Z_1|= 2 , |Z_2|= 3, |Z_3| = 4$ find the maximum value of $$|Z_1-Z_2|^2 +|Z_2-Z_3|^2+|Z_3-Z_1|^2$$ If we treat them as three vectors $a, b, c$ centred at zero the ...
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2answers
67 views

Is every complex number unique? [closed]

Is every complex number uniquely defined? Because $ i^2=-1 $ , $ i^3 = - i $ and $ e^i e^\theta=(2.71)^i\ (2.71)^\theta=\cos \theta + i \sin \theta $. Or every complex number can be written in many ...
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2answers
33 views

Equation of a line on a complex plane

Equation of a line parallel to the line $z\bar{a} + \bar{z} a + b = 0$ is $z\bar{a} + \bar{z} a + c = 0$, (where c is a real number) Equation of a line perpendicular to the line $z\bar{a} + \bar{z} a ...