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
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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{\...
-2
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2
answers
101
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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 ...
-3
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53
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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 ...
3
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1
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55
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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 = \...
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71
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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}...
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30
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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(...
6
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1
answer
96
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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 ...
1
vote
1
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57
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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^...
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2
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57
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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 : $$Z = \frac{\ln(i)}{i}$$ $$iZ= \ln(i)$$ $$e^{iZ} = i$$ $$\cos Z + i\sin Z = i$$ $$\cos Z = i(1-\sin Z)$$ ...
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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 ...
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51
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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 ...
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1
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65
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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^...
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1
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89
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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 ...
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1
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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$$\int_0^1(-1)^...
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2
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53
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Translation/Rotation properties for equations in the complex plane
Prove that complex numbers ($z_1,z_2,z_3$) that satisfy the relation below form an equilateral triangle in the complex plane.
$$z_1^2 + z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1$$
This answer first shows that ...
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23
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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&...
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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 ...
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2
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107
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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}):->\...
2
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1
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65
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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 \...
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34
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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\...
4
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1
answer
89
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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)...
4
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1
answer
196
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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 ...
2
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1
answer
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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 ...
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1
answer
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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 ...
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0
answers
36
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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 ...
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1
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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)}$
...
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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 ...
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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 ...
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1
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48
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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 ...
1
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0
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64
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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 ...
2
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1
answer
72
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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'...
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1
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47
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Finding the number of distinct common roots of unity
Consider equations:
$${x}^{p} = 1\\
{x}^{q} = 1$$
Then the number of common roots is equal to the $\gcd(p,q).$
But, how can I prove this statement?
4
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0
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Confinement result for unit complex numbers
Inspired by this problem, and some computer simulations, I almost convinced myself of the following result. However, I am coming short on a proof.
Result: Let $n\geq 3$, and $2n+1$ complex numbers $z_{...
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Can someone teach a class 10th student calculus or advanced mathematics? Like starting from what are functions to the end? [closed]
I like math, even though the level of math I do is rather easy but I like learning new things in math and envy people who solve complex equations on their copy or board or whatever they are using ...
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If $z$ is a non-real complex number , find the minimum value of $\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5}$
If $z$ is a non-real complex number , find the minimum value of $$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5}$$
where $\operatorname{Im}(z)$ is the imaginary part of a given complex number ...
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1
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63
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Is there a way of describing why multiplying complex numbers adds their angles intuitively?
Everywhere I'd looked for an explanation of this angle-adding phenomenon, it seemed to have been in one of two forms:
Either something roughly like this:
$$\left(\cos\left(a\right)+i\sin\left(a\right)\...
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0
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24
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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
$$\...
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0
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41
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Roots problem in linear equations by using complex numbers
Let there be an equation
$$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} \...
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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)...
2
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1
answer
31
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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*}...
1
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2
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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
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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 ...
1
vote
1
answer
76
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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
$$\overline{e^{iz}} = e^{i\bar{z}} \Longleftrightarrow (\...
-1
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0
answers
36
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Complex matrix is similar to a real one [closed]
I have a complex (square ) matrix with all real eigenvalues, I've seen a similar question but it I'm not entirely sure how it affects that we have real eigenvalues.
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0
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63
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Are there things which we neither own completely (+) nor owe completely (-) to others?
I think this might allow knowing about things which complex numbers represent in reality.
It seems to me that 0 degree can be seen as +, and 180 degree as - sign; angle as sign. Then numbers can be ...
-1
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0
answers
92
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Properties of solutions of equation $z^n+(-z-1)^n+1=0$
Let $n\ge 2$ be a positive integer and a complex number $z$ satisfies the following equation:
$$
z^n+(-z-1)^n+1=0
$$
Prove that $\Re(z)=-\dfrac{1}{2}$ or $|z|=1$ or $|1+z|=1$.
I used Wolfram ...
0
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0
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43
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can epsilon in a epsilon-delta proof rely on the independent variable if it cancels out at the end?
I’m taking a class in Complex Analysis and trying to prove the same result as found in this question.
In the accepted answer, a delta is chosen such that it’s the minimum of two values - one of those ...
2
votes
1
answer
71
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Sufficient condition for two holomorphic functions to have the same zeros.
Let $\mathbb{D}$ be the open unit disc in $\mathbb{C}$ and $f,g$ two holomorphic functions on $\mathbb{D}$ with the following properties:
\begin{align*}
f(z) \overline{f(\overline{z})} = g(z) \...
3
votes
1
answer
112
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$a,b,c$ be three complex numbers satisfying $|a|=|a-b|=3\sqrt3,|a+b+c|=21$,$a^2+b^2+c^2-ab-bc-ca=0$. Find $|b|^2+|c|^2$
Let $a,b,c$ be three complex numbers satisfying $$|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 $|b|^2=a\bar{b}+b\bar{a}.$...
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0
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21
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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 ...