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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Derivative of fourier series in exponential form

I encountered the following passage in a paper, in witch $K$ is a general function of $\theta$ and $\phi$ expanded in Fourier series: $K=\sum_{lm} K_{lm}\exp[i(l\theta-m\phi)]$ $\implies \frac{\...
Lucas Amaral's user avatar
-2 votes
2 answers
101 views

An integral Wolfram Alpha cannot take: $\displaystyle\int_{-1}^{+1}(-1)^{i\pi e^{-x\pi}}dx$

So I was bored yet again, and decided to evaluate some integrals. After a while, I came up with this:$$\int_{-1}^{+1}(-1)^{i\pi e^{-x\pi}}dx$$which is based off of $\color{red}{\text{this}}$ integral ...
CrSb0001's user avatar
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How Often do Complex Numbers Follow Fermat's Little Theorem? How can you predict them? [closed]

Essentially, for a gaussian integer $x$, raising it so $y=x^p$ (mod prime p), how many $x$ given dimension $d$ and prime $p$ satisfy $y≡x$ (mod p)? Is there any way to predict these? Any upper or ...
Luke Bright's user avatar
3 votes
1 answer
55 views

Prove that there is no holomorphic map $s: \mathbb{P}^n \to \mathbb{C}^{n+1} \setminus \{0\}$ with $p \circ s = \operatorname{id}$.

Let $p:\mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{P}^n$ be the natural projection. Prove that there is no holomorphic map $s: \mathbb{P}^n \to \mathbb{C}^{n+1} \setminus \{0\}$ with $p \circ s = \...
Nathaniel Johonson's user avatar
-1 votes
0 answers
71 views

How do you say "ALL the $n$th roots"?

$\sqrt[2]{1}$ is strictly only $1$, despite the equation $x^{2}=1$ having two solutions: $1$ and $-1$. Same with cube roots; $\sqrt[3]{1}$ is strictly just $1$, despite $\left(-\frac{i\sqrt[2]{3}+1}{2}...
The_Animator's user avatar
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30 views

Fast addition of logarithmic values

Given two values $\log(a)$ and $\log(b)$ of complex values $a$ and $b$. Is there a numerically fast way to compute $\log(a + b)$ (on a CPU)? I'm aware that, $\log(a + b) = \log(a) + \log(1 + \exp(\log(...
Jiro's user avatar
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6 votes
1 answer
96 views

Compute $f(z)$ and show it's well defined.

I'm given the function $f$ in integral form as follows $$f(z)=\int_{\gamma} \frac{dw}{w-z},$$ where $\gamma(t)=t, \, 0<t<1$ and $z \notin [0,1]$. I'm asked to compute this integral and show that ...
Tropax's user avatar
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1 answer
57 views

Geometry with complex numbers. Equilateral triangle.

I have to do such a question: Let ABC be a triangle, whose vertices A, B, C correspond to the complex numbers α, β, γ (the origin is not necessarily at one of the vertices), respectively. Let $ ω = e^...
WOWnas's user avatar
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$Z=\ln(i)/i$ , find $Z$.

It feels like a trivial question, but i don't know if my answer is correct. My attempt : $$Z = \frac{\ln(i)}{i}$$ $$iZ= \ln(i)$$ $$e^{iZ} = i$$ $$\cos Z + i\sin Z = i$$ $$\cos Z = i(1-\sin Z)$$ ...
Avish Bhatia's user avatar
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0 answers
25 views

When does convergence of the difference of the moduli of two complex sequences imply the convergence of the difference of the two complex sequences

Let $a_i$ and $b_i$ be convergent sequences of complex numbers such that (i) for all $i$, $|a_i|, |b_i| \leq 1$, (ii) $a_i \neq e^{i\theta}b_i$, (iii) the real sequence $c_i = |a_i| - |b_i|$ converges ...
trillianhaze's user avatar
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51 views

is $\frac{(-1)^kk!}{(s+a)^k}$ ODD?

Is the following $$\frac{(-1)^kk!}{(s+a)^{k+1}}$$ an odd function? $f(-s) = -f(s)$? It looks like the function does $\mathbb R\to \mathbb R$, but not $i\mathbb R\to i\mathbb R$ (pure imaginary), so ...
David Lee's user avatar
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prime factorization and complex number fields [closed]

if we define complex number fields in the following manner: ( a + b( (-1)^(1/n) ) ) it would appear as if n = 2 is the only value for which unique prime factorization holds, atleast in reference to a^...
Mark McCane's user avatar
-2 votes
1 answer
89 views

Find all complex numbers $𝑧$ for which $𝑧^6+5𝑧^3+6=0$

Find all complex numbers $𝑧$ for which $𝑧^6+5𝑧^3+6=0$ this is a question we had in class but I'm somehow confused/lost in the explanation of how to solve it. I'd very kindly appreciate it if you ...
kokos123's user avatar
-1 votes
1 answer
139 views

How to evaluate: $\int_0^1(-1)^{\ln(x)}dx$

So I was looking through the homepage of Youtube to see if there were any integrals that I might want to evaluate when I came across this video by Maths$505$ which showed how to evaluate$$\int_0^1(-1)^...
CrSb0001's user avatar
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2 answers
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Translation/Rotation properties for equations in the complex plane

Prove that complex numbers ($z_1,z_2,z_3$) that satisfy the relation below form an equilateral triangle in the complex plane. $$z_1^2 + z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1$$ This answer first shows that ...
Starlight's user avatar
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How to simplify the complex expression of the Fourier series for a square wave?

I'm trying to find the expression of the fourier series (using complex numbers) for the following odd squarewave function with period T: $V(t) = V_0$ for $0<t<T/2$ $V(t) = -V_0$ for $T/2<t&...
Matthew_R's user avatar
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How do we prove this nested radical solution?

I’m submitting this question because my previous one was perhaps too verbose and did not get the point very quickly. A well known $\sqrt{2}$ nested radical has the following solutions as found in the ...
Cye Waldman's user avatar
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1 vote
2 answers
107 views

Why can we plug complex numbers into maclaurin series?

When finding a Maclaurin series for a function f(x). We evaluate f '(0), f ''(0), etc, to find our coefficients for each term. I have done this for the standard functions, $f (\mathbb{R}):->\...
DrVendetta's user avatar
2 votes
1 answer
65 views

n complex numbers inside a disk with center $A$ and radius 1

Inside the disk with center $A(2,0)$ and a radius of $1$, a set of $n \geq 1$ points is considered, each having the respective affixes $z_1, z_2, \ldots, z_n$. Show that $ \left| \sum_{i=1}^{n} z_i \...
math.enthusiast9's user avatar
-1 votes
0 answers
34 views

Solved a complex equation, but got 2 answers, which is the right one?

The problem and my half solution $z^4+2(i+1)z^2+i=0$ So i did: $x^2+2(i+1)x+i=0 \rightarrow x_{1,2}=\frac{-2i-2\pm\sqrt{8i-4i}}{2}=?$ $\sqrt{4i} => \sqrt{4}\cdot(cos\frac{\pi/2+2k\pi}{2}+i\cdot sin\...
Lolis4TheWin's user avatar
4 votes
1 answer
89 views

Inequality with complex numbers with the same modulus

I got stuck at the following problem. Prove that for every three distinct complex numbers $a$, $b$, $c$ with $|a| = |b| = |c| > 0$, the following inequality holds: $ \sum_{\text{cyclic}} |(a+b)(b-c)...
math.enthusiast9's user avatar
4 votes
1 answer
196 views

Inequality for a complex polynomial

Let $p(z), q(z)$, and $r(z)$ be polynomials with complex coefficients in the complex plane. Suppose that $|p(z)|+|q(z)| \leq|r(z)|$ for every $z$. Show that there exist two complex numbers $a, b$ such ...
Snowball's user avatar
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2 votes
1 answer
33 views

if $|z-3-2i| \leq 2 , \text{ find the minimum of } |2z-6+5i|$

if $|z-3-2i| \leq 2 , \text{ find the minimum of } |2z-6+5i|$ My attempt:- Let $z=x+iy$ so we have $(x-3)^2+(y-2)^2 \leq4$ let $|2z-6+5i | \text{ be }\phi $ so ${\phi}^2 =4(x-3)^2+(2y+5)^2$ I ...
math and physics forever's user avatar
-1 votes
1 answer
89 views

find the region where $\frac{i}{z-z^7}$ is analytic

I have tried it by putting $z=x+iy$ Then reduced the whole into $u(x,y)+i\cdot v(x,y)$ form. However, the expression I am getting is quite complicated and would be hard to put put in the CR eqn and ...
tushar umbarkar's user avatar
-3 votes
0 answers
36 views

Determine the points where $\frac{z}{z-1}$ is analytic.

I have tried it by puting $z =x+iy$ then reduced the whole fn to $u(x,y)+iv(x,y)$ form. However, the expression I am getting is quite complicated and would be hard to put put in the CR eqn and Laplace ...
tushar umbarkar's user avatar
-2 votes
1 answer
95 views

Is any value of $(\sqrt i)^{\ln(i)}$ real?

Is any value of $(\sqrt i)^{\ln(i)}$ real? Inspired by similar expressions, I was playing around a bit on julia prompt and typed $(\sqrt i)^{\ln(i)}$ ...
FirstName LastName's user avatar
0 votes
0 answers
16 views

A Nested Radical Arising from a Nonlinear Recurrence

I have been looking into analytic continuation of regular recurrence relations into the negative, real, and complex domains ($\mathbb{N} \mapsto \mathbb{Z}, \mathbb{R},\mathbb{C}$). In doing so, we’ve ...
Cye Waldman's user avatar
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0 answers
40 views

Solving for the derivative at the point $z = 0$ for the function $f(z) = z^2 - \bar{z}^2$

I'm trying to solve for the derivative of $f(z) = z^2 - \bar{z}^2$ using the limit definition (I'm positive that it can be done with power rule, but given my track record recently, I'm not going to ...
ekorel's user avatar
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0 votes
1 answer
48 views

How to solve complex equation with i in the bx term?

First of all, sorry for the English, i have never asked question about math in English before, so i don't know if my problem is understandable. So, basically i don't know anything about this stuff, i ...
Lolis4TheWin's user avatar
1 vote
0 answers
64 views

Removing "i" from the triangle inequality?

In my complex analysis class, I have to prove that a limit exists for a function and I think I can use the Triangle Inequality in my proof, but I don't know if it's possible. My question: knowing that ...
ekorel's user avatar
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2 votes
1 answer
72 views

Must every Cauchy-Riemann condition be fulfilled simultaneously?

Working through problems in my complex analysis book, and I have to determine where the derivative exists for a function. I know that the derivative can exist only along a certain curve, however I don'...
ekorel's user avatar
  • 57
1 vote
1 answer
47 views

Finding the number of distinct common roots of unity

Consider equations: $${x}^{p} = 1\\ {x}^{q} = 1$$ Then the number of common roots is equal to the $\gcd(p,q).$ But, how can I prove this statement?
Motivix's user avatar
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4 votes
0 answers
88 views

Confinement result for unit complex numbers

Inspired by this problem, and some computer simulations, I almost convinced myself of the following result. However, I am coming short on a proof. Result: Let $n\geq 3$, and $2n+1$ complex numbers $z_{...
Nathan Portland's user avatar
-4 votes
0 answers
75 views

Can someone teach a class 10th student calculus or advanced mathematics? Like starting from what are functions to the end? [closed]

I like math, even though the level of math I do is rather easy but I like learning new things in math and envy people who solve complex equations on their copy or board or whatever they are using ...
memeguy's user avatar
  • 59
1 vote
2 answers
83 views

If $z$ is a non-real complex number , find the minimum value of $\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5}$

If $z$ is a non-real complex number , find the minimum value of $$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5}$$ where $\operatorname{Im}(z)$ is the imaginary part of a given complex number ...
math and physics forever's user avatar
0 votes
1 answer
63 views

Is there a way of describing why multiplying complex numbers adds their angles intuitively?

Everywhere I'd looked for an explanation of this angle-adding phenomenon, it seemed to have been in one of two forms: Either something roughly like this: $$\left(\cos\left(a\right)+i\sin\left(a\right)\...
NaiDoeShacks's user avatar
0 votes
0 answers
24 views

Critical point of complex function

An exercise I'm doing is asking me to show $\partial_z f=\partial_{\bar{z}}f=0$ is equivalent to $\partial_x f=\partial_{y}f=0$ using these two definitions: $$\partial_{z}z^{n}=nz^{n-1}$$ and $$\...
Jordy1113's user avatar
-2 votes
0 answers
41 views

Roots problem in linear equations by using complex numbers

Let there be an equation $$x = \sqrt{144}$$ This can be written as $$x = \sqrt{(-12)^2}$$ We know that, $$\sqrt{k^2} = \sqrt{k} \times \sqrt{k}$$ $$x = \sqrt{-12} \times \sqrt{-12}$$ $$x= \sqrt{12} \...
Aditya Jha's user avatar
-2 votes
0 answers
30 views

i^2 is not equal to -1? [duplicate]

as $i = \sqrt{-1}$ so $i^2 = -1$ however i seem to have thought of a counter argument $$1 = \sqrt1 = \sqrt{(-1)(-1)} = i^2 = -1$$ but 1 can't be equal to -1 I must be going wrong in the $\sqrt{(-1)(-1)...
Saif's user avatar
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2 votes
1 answer
31 views

Does pairwise phase incoherence satisfy the triangle inequality?

Let $S$ be the unit circle in the complex plane, $$ S = \{z \in \mathbb{C} : |z| = 1\}. $$ For values $z_1^{(1)},z_1^{(2)},z_2^{(1)},z_2^{(2)},\ldots,z_k^{(1)},z_k^{(2)} \in S$, letting \begin{align*}...
Julian Newman's user avatar
1 vote
2 answers
62 views

If a complex number, $z$, satisfies the equation $z+ \sqrt2 |z+1| +i = 0, \text{ find } |z|$

If $z+ \sqrt2 |z+1| +i = 0, \text{ find } |z|$ My attempt: As the RHS $= 0$, the sum of the real parts and imaginary parts are both $0$. As the amplitude of a complex number is always real, $z + i = ...
math and physics forever's user avatar
3 votes
1 answer
79 views

Why must be $i^2=-1$? [duplicate]

These days I am explaining radicals to my 15-year-old students, and I wanted to add that square roots or roots with even index and negative radical can be solved with complex numbers. I know that n ...
Sebastiano's user avatar
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1 vote
1 answer
76 views

Specific doubts about complex numbers

On the one hand, in the problem of the next link, Proof that $e^{i\bar{z}}=\overline{e^{iz}}$ if and only if $z=k\pi\in Z$ we are having that $$\overline{e^{iz}} = e^{i\bar{z}} \Longleftrightarrow (\...
Blue Tomato's user avatar
-1 votes
0 answers
36 views

Complex matrix is similar to a real one [closed]

I have a complex (square ) matrix with all real eigenvalues, I've seen a similar question but it I'm not entirely sure how it affects that we have real eigenvalues.
Emilia's user avatar
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-3 votes
0 answers
63 views

Are there things which we neither own completely (+) nor owe completely (-) to others?

I think this might allow knowing about things which complex numbers represent in reality. It seems to me that 0 degree can be seen as +, and 180 degree as - sign; angle as sign. Then numbers can be ...
Sensebe's user avatar
  • 901
-1 votes
0 answers
92 views

Properties of solutions of equation $z^n+(-z-1)^n+1=0$

Let $n\ge 2$ be a positive integer and a complex number $z$ satisfies the following equation: $$ z^n+(-z-1)^n+1=0 $$ Prove that $\Re(z)=-\dfrac{1}{2}$ or $|z|=1$ or $|1+z|=1$. I used Wolfram ...
John_zyj's user avatar
0 votes
0 answers
43 views

can epsilon in a epsilon-delta proof rely on the independent variable if it cancels out at the end?

I’m taking a class in Complex Analysis and trying to prove the same result as found in this question. In the accepted answer, a delta is chosen such that it’s the minimum of two values - one of those ...
ekorel's user avatar
  • 57
2 votes
1 answer
71 views

Sufficient condition for two holomorphic functions to have the same zeros.

Let $\mathbb{D}$ be the open unit disc in $\mathbb{C}$ and $f,g$ two holomorphic functions on $\mathbb{D}$ with the following properties: \begin{align*} f(z) \overline{f(\overline{z})} = g(z) \...
anonym's user avatar
  • 73
3 votes
1 answer
112 views

$a,b,c$ be three complex numbers satisfying $|a|=|a-b|=3\sqrt3,|a+b+c|=21$,$a^2+b^2+c^2-ab-bc-ca=0$. Find $|b|^2+|c|^2$

Let $a,b,c$ be three complex numbers satisfying $$|a|=|a-b|=3\sqrt3\\|a+b+c|=21\\a^2+b^2+c^2-ab-bc-ca=0$$ Find $|b|^2+|c|^2$. I started by squaring the first relation to get $|b|^2=a\bar{b}+b\bar{a}.$...
Maths's user avatar
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0 answers
21 views

Does $\lim_{z\to 0}\;z\times \sin(\frac{1}{z})$ exist for $z \in \mathbb{C}$? [duplicate]

So I stumbled across the question where i was asked whether $\lim_{z \to 0}\;z\times \sin(\frac{1}{z})$ exists for $z \in \mathbb{C}$ . I know that if $z \in \mathbb{R}$ ,then the limit exists , but ...
Raghav Madan's user avatar

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