Questions tagged [complex-numbers]
Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.
18,847
questions
0
votes
0
answers
11
views
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{\...
-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 ...
- 1,651
-3
votes
0
answers
53
views
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 ...
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 = \...
-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}...
- 146
0
votes
0
answers
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(...
- 577
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 ...
- 99
1
vote
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^...
- 11
0
votes
2
answers
57
views
$Z=\ln(i)/i$ , find $Z$.
It feels like a trivial question, but i don't know if my answer is correct.
My attempt : $$Z = \frac{\ln(i)}{i}$$ $$iZ= \ln(i)$$ $$e^{iZ} = i$$ $$\cos Z + i\sin Z = i$$ $$\cos Z = i(1-\sin Z)$$ ...
- 13
-2
votes
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 ...
- 191
0
votes
0
answers
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 ...
- 103
-3
votes
1
answer
65
views
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^...
-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 ...
- 7
-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$$\int_0^1(-1)^...
- 1,651
0
votes
2
answers
53
views
Translation/Rotation properties for equations in the complex plane
Prove that complex numbers ($z_1,z_2,z_3$) that satisfy the relation below form an equilateral triangle in the complex plane.
$$z_1^2 + z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1$$
This answer first shows that ...
- 1,105
0
votes
0
answers
23
views
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&...
0
votes
0
answers
33
views
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 ...
- 6,781
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}):->\...
- 21
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 \...
- 169
-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\...
- 17
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)...
- 169
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 ...
- 1,052
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 ...
-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 ...
-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 ...
-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)}$
...
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 ...
- 6,781
0
votes
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 ...
- 57
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 ...
- 17
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 ...
- 57
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'...
- 57
1
vote
1
answer
47
views
Finding the number of distinct common roots of unity
Consider equations:
$${x}^{p} = 1\\
{x}^{q} = 1$$
Then the number of common roots is equal to the $\gcd(p,q).$
But, how can I prove this statement?
- 461
4
votes
0
answers
88
views
Confinement result for unit complex numbers
Inspired by this problem, and some computer simulations, I almost convinced myself of the following result. However, I am coming short on a proof.
Result: Let $n\geq 3$, and $2n+1$ complex numbers $z_{...
- 2,203
-4
votes
0
answers
75
views
Can someone teach a class 10th student calculus or advanced mathematics? Like starting from what are functions to the end? [closed]
I like math, even though the level of math I do is rather easy but I like learning new things in math and envy people who solve complex equations on their copy or board or whatever they are using ...
- 59
1
vote
2
answers
83
views
If $z$ is a non-real complex number , find the minimum value of $\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5}$
If $z$ is a non-real complex number , find the minimum value of $$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5}$$
where $\operatorname{Im}(z)$ is the imaginary part of a given complex number ...
0
votes
1
answer
63
views
Is there a way of describing why multiplying complex numbers adds their angles intuitively?
Everywhere I'd looked for an explanation of this angle-adding phenomenon, it seemed to have been in one of two forms:
Either something roughly like this:
$$\left(\cos\left(a\right)+i\sin\left(a\right)\...
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
$$\...
- 31
-2
votes
0
answers
41
views
Roots problem in linear equations by using complex numbers
Let there be an equation
$$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} \...
-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)...
- 1
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*}...
- 350
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 = ...
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 ...
- 6,719
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
$$\overline{e^{iz}} = e^{i\bar{z}} \Longleftrightarrow (\...
- 489
-1
votes
0
answers
36
views
Complex matrix is similar to a real one [closed]
I have a complex (square ) matrix with all real eigenvalues, I've seen a similar question but it I'm not entirely sure how it affects that we have real eigenvalues.
- 1
-3
votes
0
answers
63
views
Are there things which we neither own completely (+) nor owe completely (-) to others?
I think this might allow knowing about things which complex numbers represent in reality.
It seems to me that 0 degree can be seen as +, and 180 degree as - sign; angle as sign. Then numbers can be ...
- 901
-1
votes
0
answers
92
views
Properties of solutions of equation $z^n+(-z-1)^n+1=0$
Let $n\ge 2$ be a positive integer and a complex number $z$ satisfies the following equation:
$$
z^n+(-z-1)^n+1=0
$$
Prove that $\Re(z)=-\dfrac{1}{2}$ or $|z|=1$ or $|1+z|=1$.
I used Wolfram ...
- 1
0
votes
0
answers
43
views
can epsilon in a epsilon-delta proof rely on the independent variable if it cancels out at the end?
I’m taking a class in Complex Analysis and trying to prove the same result as found in this question.
In the accepted answer, a delta is chosen such that it’s the minimum of two values - one of those ...
- 57
2
votes
1
answer
71
views
Sufficient condition for two holomorphic functions to have the same zeros.
Let $\mathbb{D}$ be the open unit disc in $\mathbb{C}$ and $f,g$ two holomorphic functions on $\mathbb{D}$ with the following properties:
\begin{align*}
f(z) \overline{f(\overline{z})} = g(z) \...
- 73
3
votes
1
answer
112
views
$a,b,c$ be three complex numbers satisfying $|a|=|a-b|=3\sqrt3,|a+b+c|=21$,$a^2+b^2+c^2-ab-bc-ca=0$. Find $|b|^2+|c|^2$
Let $a,b,c$ be three complex numbers satisfying $$|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 $|b|^2=a\bar{b}+b\bar{a}.$...
- 499
0
votes
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 ...
- 23