Questions tagged [complex-numbers]
Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.
0
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0answers
3 views
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}}{...
1
vote
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 $a, b, c$ ...
0
votes
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 ...
4
votes
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 ...
-2
votes
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 ...
3
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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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votes
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.
2
votes
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 ...
1
vote
1answer
36 views
Proving by calculation that $\arg(2i) = \pi/2$ [on hold]
I was just wondering how you would prove that $\arg(2i)=\pi/2$
1
vote
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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vote
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 $z_1, z_2, z_3$...
1
vote
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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0answers
28 views
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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vote
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 $$z^4+6z^3+23z^2+30z+45=0$$
1
vote
2answers
62 views
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 $\Im z>0$,
$$\sup_{t\in\mathbb{R}}\left|\frac{t-i}{t-z}...
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votes
3answers
27 views
Problem of complex equations and Cauchy-Riemann
I have the following problem:
$$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$$ and later the quadratic equation, this ...
2
votes
3answers
102 views
Understanding a proof that a collection of complex numbers on one side of a line through $0$ must have a non-zero sum
I'm asked to prove that if $z_1$, $\ldots$, $z_k$ lie on one side of a straight line through $0$, then $z_1+\cdots+z_k \neq 0$.
In the proof, we let $\theta$ be the angle between the line and the ...
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0answers
15 views
Von Neumann stability analysis - almost finished exercise
I wish to analyze the stability of the FTCS scheme for the equation $u_t = iau_{xx} + cu$ where $a \in \mathbb R, c \in \mathbb C$.
I will succinctly go through what I did and where I'm stuck.
A ...
2
votes
0answers
24 views
Which complex numbers work as a (numeral) base?
When we write a real number in base k the n’th digit represents the number of k^n. I’m interested in finding a complex number w such that any complex number z can be written in base w.
I found that ...
0
votes
1answer
59 views
Is $f(z) = z^{2}$ an automorphism of $\mathbb{C}$?
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ where $f(z) = z^{2}$. I claim $f$ is an automorphism of $\mathbb{C}$ as a vector space. First $\operatorname{ker}(f) = 0$ and since $\mathbb{C}$ is ...
2
votes
3answers
54 views
Find all the $z$ such that $(1+\frac{1}{z})^{4}=1$
Question
if $S$ be the set of solution of $$\bigg(1+\frac{1}{z}\bigg)^{4}=1$$ then prove that the points are co-linear.
Attempt
$\bigg(1+\frac{1}{z}\bigg)^{4}=1$
$\implies z^4+4z^3+6z^2+4z+1=z^4$
$...
0
votes
1answer
60 views
Proof of $ \ \vert \log(z) \vert \lt \log \frac{1}{1-r} \ \ $ given $ \forall z ,\ \vert z-1 \vert \leq r \lt 1$?? [closed]
Let $\ \vert z-1 \vert \leq r \lt 1$ and let $\log(z)$ be the principal branch of the logarithmic function. How can you prove
$$ \vert \log(z) \vert \lt \log \frac{1}{1-r} \ \ ?$$
Also, how can you ...
-3
votes
0answers
38 views
Find the imaginary part of z [closed]
If $z = (a+ib)^r \times e^{x+iy}$, find the imaginary part of $z$, (note that r may not be an integer).
2
votes
0answers
54 views
Maximum value of $|Z_1-Z_2|^2 +|Z_2-Z_3|^2+|Z_3-Z_1|^2$ [duplicate]
Given three complex numbers $|Z_1|= 2 , |Z_2|= 3, |Z_3| = 4$ find the maximum value of $$|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 ...
-2
votes
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 ...
0
votes
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 ...
