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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1answer
61 views

Has anyone ever explored $(\sin{x})^x$ , $(\cos{x})^x$, etc?

I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis. It occurred to me that complex analysis ...
0
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0answers
33 views

Is the square root of -1 really “i” [duplicate]

I know that the imaginary unit i is a number with the following property: i^2 = -1 But I often see people turn that into this ...
1
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1answer
67 views

Proving that a complex number lies on the imaginary axis.

Given that there are two complex numbers - $z, w$ - such that $w\overline{w} = 1$ and $z = \frac{1+w}{1-w}$, how do I deduce that $z$ lies on the imaginary axis?
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1answer
22 views

Doubt in raising a power to a complex number

What's the value of $$i^{i^{i^{...}}}$$? I tried to take log on both sides. But I could realise that I can't go with that.
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1answer
18 views

Let $z,w\in \mathbb{C}$ where $|w|<1$ (modulus). What is the set of all $z\in \mathbb{C}$ that satisfies $|z-w|\leq|1-\bar{w}z|$.

Let $z,w\in \mathbb{C}$ where $|w|<1$ (modulus). What is the set of all $z\in \mathbb{C}$ that satisfies $|z-w|\leq|1-\bar{w}z|$. I've tried a few things with no luck. I wrote $z,w$ are complex ...
1
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1answer
33 views

Is it necessary for the Imaginary-axis to be perpendicular to the Real-axis?

The Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...
1
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1answer
35 views

An alternative method to find $\sum_{k=1}^{2n-1} | \beta ^k - 1|$

Let $\beta \in \mathbb{C}$ such that $\beta ^n = 1$ but $ \beta ^k \neq 1$, $\forall k=1,2,\cdots, n-1$. Find the value of $$ \sum_{k=1}^{2n-1} | \beta ^k - 1|.$$ I came across such a question ...
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2answers
24 views

calculating complex numbers - help needed [on hold]

Probably it is simple, but I am blind right now and I do not see how to solve this task: $e^{i \frac{2\pi}{3}}+e^{i\frac{4\pi}{3}}+1$
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1answer
33 views

Find the number of elements of a complex subset

How many elements does the set $\{z\in \mathbb C:z^{60}=-1,z^k\not=-1\text{ for } 0<k<60\}$ have ? $z^{60}=-1=\cos(2k\pi+\pi)+i\sin(2k\pi+\pi)$. Then , ...
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1answer
21 views

Expression of reflection isometry in the complex plane

Using the fact that an anti-displacement in the plan has the form $$f(z) = a \overline{z} + b$$ I have done some computation to find the reflection about the line passing through two points $P$ and ...
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3answers
81 views

Find all complex numbers satisfying $x^4+x^2+1=0$ [on hold]

Find all solutions that fit: $$x^4+x^2+1=0$$ I did it couple of days ago, now I can't remember.
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1answer
18 views

Solve complex equation graphically

I have this problem that is split in 2, A and B, and Im struggling with B in particular, but I also dont know if I have done A correctly, which I suppose is necessary.. A) "Let $z$ be the complex ...
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0answers
32 views

Fourier series for a Sinusoid in a conventional way?

So my TA in class introduced this amazing way of finding fourier series coefficients for a sin wave, by writing $ sin( \omega t ) = (e^{i\omega t}-e^{-i\omega t}) / 2i $ ----(1) Hence getting the ...
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2answers
22 views

Solve Complex Equation with several terms

I have a complex number $z = 3 + 3i$ And I want to find all solutions of $z^{10} + 2z^{5} + 2 = 0$ I'm kinda lost. I recognise the fact that I can substitute $u = z^{5}$ and rewrite the equation as ...
2
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4answers
109 views

Is $i^i$ mathematically valid? [duplicate]

WARNING: SLIGHT NSFW http://www.smbc-comics.com/index.php?db=comics&id=2934#comic Uhh...guys, mathematically speaking, how accurate is this comic. From what I remember in High School $$a^b= ...
0
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1answer
37 views

Exponential Complex Number

I need assistance in solving the following: http://i.stack.imgur.com/EcGLD.jpg I am not very sure on how to remove the exponential to convert it into complex numbers and get the arguments in the ...
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0answers
28 views

Is there a name for complex numbers over affinely extended reals?

Is there a name for the set of complex numbers over affinely extended real line, that is $\mathbb{C}\cup \{-\infty\}\cup\{+\infty\}$? I think this set is the most commonly used in analysis ...
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3answers
37 views

$|z|=1$ should represent semi-circle or circle?

Suppose we have complex number $z=x+iy$ and we are given locus $|z|=1$ which should be $\sqrt{x^2+y^2} =1$ this should be a semi-circle above x axis , it's when we square our equation we get a circle ...
2
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1answer
39 views

Find the min and max distance from origin of the curve $\vert z+\frac{1}{z}\vert=a$

$z$ is a complex number, by the way. I've tried a lot of things and always end up with a huge algebraic mess and I've wondered if anyone of you has any idea on how to approach this problem. One of ...
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2answers
38 views

multiple sets of complex roots of a number?

I am not sure if this question was asked before but I couldn't find the right keywords to choose for searching. So today I discovered a weird problem: If we take this equation: $$x^2=1=e^{(0i)}$$ ...
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2answers
48 views

find the coefficient

If $n$ is an odd natural number, and $\sin(n\theta) = \Sigma_{r=0}^{n} b_r \sin^r\theta$, then find $b_r$ in terms of $n$. I have tried this using trigonometric expansion but unable to find solution ...
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1answer
35 views

Find the mistake in calculation [duplicate]

$(-1)^3 = (-1)^{6/2} = ((-1)^6)^{1/2} = 1^{1/2} = 1$ So it comes $(-1)^3 = 1$ can anybody explain where exactly the mistake in calculation?
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2answers
25 views

the locus of $z$ in the complex plane

Describe the locus of $z$ in the complex plane if $z$ satisfies: $$ arg(z)=arg(z+3+i)\quad (mod\ 2\pi) $$ Indeed Let $O$ be the origin and $B=-3-i$. \begin{align*} arg(z)&=arg(z+3+i)\quad ...
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3answers
60 views

Finding the modulus and argument of a complex number

I would need help with this question: $$Z = \frac{(1+j2)^2(4-j3)^3 }{ (3+j4)^4 (2-j3)}$$ My starting point for this question is to expand the complex numbers first then continue doing but after ...
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0answers
20 views

Show that if $u_n (z_0) \rightarrow 0$ for some $z_0 \in D$, then $u_n \rightarrow 0$ uniformly on compact subsets of $D$.

Let $D \subseteq \Bbb C$ be a connected open subset and let {$u_n$} be a sequence of harmonic functions $u_n: D \rightarrow (0,\infty)$. Show that if $u_n (z_0) \rightarrow 0$ for some $z_0 \in D$, ...
2
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2answers
46 views

What is the interval for possible values of the argument of a complex number?

It looks like there are different intervals in which the argument of a complex number can be. Some say it goes from $-\pi$ to $+\pi$ others say it goes from $0$ to $2\pi$. For the most part, both ...
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0answers
17 views

If $f:B_N\rightarrow \mathbb{D}$ and $z_n\in B_N$ with $\{f(z_n)\}$ thin, is $\{f(\phi(z_n))\}$ thin for any autmorphism $\phi$ of $B_N$?

Let $B_N$ denote the open unit ball in $\mathbb{C}_N$. A sequence $\{z_n\}$ of distinct points in $\mathbb{D}$ is called thin if $\lim_{k\rightarrow \infty}\displaystyle\prod_{j: j\not =k}^\infty ...
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3answers
41 views

Reducing to upper triangular form

I've just had some difficulty with this transforming this matrix into upper triangular form: $$ \pmatrix{ i& 2i& -1\\1 & 1& i\\ 2-i& 1& i } $$ I've tried almost everything. ...
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2answers
37 views

Express the complex number in rectangular form $a + ib$

$12e^{2-\pi*i/3}$ express this in rectangular form $a + i\cdot b$ Not sure how to solve when fractions are involved Example $2.6\cdot e^{3+i} = 2.6\cdot e^3\cdot e^i$ ?
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2answers
63 views

2+2=square root of 16. What's the appropriate answer? [closed]

4? Positive and negative 4? I just got into an argument with a buddy about this. He argues if it's not an i, it's not included as a imaginary number, but only the real positive number.
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3answers
275 views

Finding complex number defined by 3 equations

Let $z$ be a complex number satisfying $$\DeclareMathOperator{\Re}{Re}\Re[z^4]=1/2$$ $$z\bar{z}+2|z|-3=0$$ $$\arg z \leq \frac{\pi}{4}.$$
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1answer
39 views

Compute the integral $\int_\mathbb{R^{n}}\lvert x\rvert ^{2m}\exp(- \lvert x\rvert^2/2) \mathrm{dx} $ [closed]

Please help me to compute the integral $\int_\mathbb{R^{n}}\lvert x\rvert^{2m} \exp(- \lvert x\rvert^2/2) \mathrm{dx} $, where $ m = 1,2,...$ using some complex analysis results. Thanks in advance
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2answers
56 views

Powers of complex numbers property

I would like to prove the following statement. Let $\lambda_1,\dots,\lambda_s \in \mathbb{C}$ be such that $|\lambda_1| = \dots = |\lambda_s|=1$. Then $\forall \varepsilon \gt 0$ there exist ...
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1answer
26 views

How to show that this works (complex numbers)

So if I have a set of complex numbers: $A= \{z\in\mathbb{C} |\ \text{Re}\,(z) > 0, |z|<1\}$ So I have a problem showing this: For any $z\in A$ exists $w\in A$ such,that this works: ...
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1answer
17 views

Exercise: Prove some properties about some numbers of a set.

Let $W=\left\{w = \dfrac{1+\dfrac{iy}{2}}{1-\dfrac{iy}{2}} \ \colon \ y \in \mathbb{R}\right\}$. Then show that: $w\in W \Longrightarrow |w|=1$ $-1\not\in W$ . What I've done so far: I took the ...
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1answer
42 views

Quadratic field extensions and complex conjugation

If you consider any quadratic extension $K$ of $\mathbb{Q}$, it has to be fixed by complex conjugation, because from $[K : \mathbb{Q}] = 2$ we know $K | \mathbb{Q}$ has to be a normal extension and as ...
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1answer
21 views

Is there a notion of finite fields containing complex scalars? [closed]

Does a finite field $\mathbf F = \{1,i,...,+,*\}$ exist? Is there such an F containing only imaginary scalars?
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1answer
52 views

argument of $\dfrac{i}{\bar{z}^{2}}$

let $z \in \mathbb{C}$ and $arg z\equiv \dfrac{\pi}{6} (\textrm{mod}\ 2\pi)$ then calculate: $$arg \dfrac{i}{\bar{z}^{2}} $$ indeed, $$ \begin{align*} arg \dfrac{i}{\bar{z}^{2}}&=arg i - ...
0
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1answer
30 views

there exists a constant $M$ such that $\mid f^{(k)}(0) \mid \leq k^4 M^k$. Show that $f$ can be extended to be analytic on $\Bbb C$. [closed]

(a) Suppose that $f$ is analytic on the open unit disc and that there exists a constant $M$ such that $\mid f^{(k)}(0) \mid \leq k^4 M^k$ for all $k \geq 0$. Show that $f$ can be extended to be ...
3
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0answers
33 views

complex numbers from real closed fields

I am very interested in first order axiomatizations of the complex numbers, but I have never actually seen one laid out. Algebraically closed fields of characteristic zero are a start, but they don't ...
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0answers
29 views

Solutions to the equation $\langle z,u_1\rangle \langle z,u_2\rangle =\langle u_1,u_2\rangle$.

Suppose $u_1$ and $u_2$ are elements of $\mathbb{C}^N$ of norm $1$, and that $\langle u_1,u_2\rangle\not =0$. If $z$ is in $\mathbb{B}_N$ (the unit ball of $\mathbb{C}^N$), how many solutions does ...
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1answer
48 views

Solve complex equation $5|z|^3+2+3 (\bar z) ^6=0$

I'm stuck in trying to solve this complex equation $$ 5|z|^3+2+3 (\bar z)^6=0$$ where $\bar z$ is the complex conjugate. Here's my reasoning: using $z= \rho e^{i \theta}$ I would write $$ ...
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1answer
69 views

Will this punch a hole in the field of complex number? [closed]

According to this, complex number is algebraically closed, i.e. every polynomial has complex root. What if we allow other type of equations? I ask this question because equations seemingly can extent ...
1
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1answer
40 views

inequality on the unit disk

$n$ points $z_1,z_2,\cdots,z_n$ in the unit open disk are given. Prove or disprove that there exists $z$ in the unit circle such that $\prod_{i=1}^n |z-z_i|^i \ge 1$. I think it can be solved by ...
7
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1answer
175 views

Complex Numbers $\stackrel{?}{=} \mathbb{R}^ 2$

Suppose we have a vector field over real numbers $\mathbb R^2$. In additon to vector field proporties define inner product $(x,y) = x_1\cdot y_1 + x_2\cdot y_2$, where $x_1,x_2,y_1,y_2$ are real ...
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3answers
41 views

Question about complex numbers [duplicate]

Proof that if $z = 1$, then $|z-w| = |1- \overline{w}z|$, $\forall w \in$ $\mathbb{C}$ My attempt below: $(z-w)\overline{(z-w)} = |z-w|^2$ $(z-w)\overline{(z-w)} = (z-w)(\overline{z}-\overline{w}) = ...
3
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0answers
91 views

A question about the proof of $(z_1z_2)^a=z_1^az_2^a$

For $z_1,z_2\in \mathbb C$ if $\Im(z_1z_2)>0$ and $\Im(z_2)\ge 0$ prove that $(z_1z_2)^a=z_1^az_2^a$ , for $a$ is any real. I proved it like this: $z_1^az_2^a=\exp(a\log z_1)\exp(a\log ...
1
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1answer
23 views

Show that there exists a constant C depending on $U$ and $K$ such that $|f(z)| \leq C(\int_{U} |f|^2)^{1/2}$.

Let $f$ be analytic in an open set $U \subseteq \Bbb C$ and let $K \subseteq U$ be compact. Show that there exists a constant C depending on $U$ and $K$ such that $|f(z)| \leq C(\int_{U} |f|^2)^{1/2}$ ...
1
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0answers
46 views

Practical application of Gauss-Lucas theorem

Let $z_1,z_2,z_3 \in \mathbb C$ pairwise distinct be the affix of points $A, B$ and $C$. Let $P(x)=(x-z_1)(x-z_2)(x-z_3)$. Let $z_4$ and $z_5$ be the roots of $P'$ (with the possibilty that ...
0
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1answer
28 views

What region of the complex plane does $\left|{z-1+i}\right|+\left|{z+1-i}\right|=6$ fill?

What region of the complex plane does $\left|{z-1+i}\right|+\left|{z+1-i}\right|=6$ fill? I'm having a tough time figuring what region this fills up. Maybe its easy, but for some reason I cant think ...