Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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679 views

Complex numbers - finding minimum value

For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2-3-4i|=5$ , find the minimum value of $|z_1-z_2|$ Can we go like this : Let $z_1 = x +iy$ therefore $|z_1| = \sqrt{x_1^2+y_1^2}$ and ...
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1answer
92 views

Compact sets of the complex plane with countable boundary

Suppose $E$ is a compact set of the complex plane and the boundary of $E$ is a countable set. How does one prove that $E$ is equal to its boundary?
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3answers
355 views

Distance between point and line in the complex plane

Let $a,b$ be fixed complex numbers and let $L$ be the line $$L=\{a+bt:t\in\Bbb R\}.$$ Let $w\in\Bbb C\setminus L$. Let's calculate $$d(w,L)=\inf\{|w-z|:z\in L\}=\inf_{t\in\Bbb R}|w-(a+bt)|.$$ The ...
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1answer
87 views

Demonstrating the coefficients of the power series of $\frac{1}{1-z-z^2}$ satisfies a recurrence relation.

I have the power series $$\frac{1}{1-z-z^2} = \sum_{n=0}^{\infty} c_nz^n$$ and I'd like to show that the coefficients of this power series satisfy $c_n=c_{n-1}+c_{n-2}$. I thought the most obvious way ...
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2answers
102 views

is $1^z=1$ for all complex values of $z$?

i would like to see if $1^z=1$ is valid for all complex variable $z$,first of all you can rewrite above equation as $1^{a+b*i}=e^0$ here i think that instead of $+$ sign, we may take take ...
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0answers
35 views

check validity of following manipulation

in my algebra book,there is written following well known identity $e^{2*\pi*i}=1$ generally we can use also this identity $e^{k*\pi*i}=(-1)^k$ and if instead of $k$,we put $2$ we get ...
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1answer
168 views

A finite sum of trigonometric functions

By taking real and imaginary parts in a suitable exponential equation, prove that $$\begin{align*} \frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases} 1&\text{if } k ...
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0answers
34 views

Compare one real number to one complex number. [duplicate]

I understand that complex numbers can be neither ordered nor compared by 'size', but if mapped one for one by a transformation, then they can be. Latter point aside, can I say that $2-xi < 2 < ...
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5answers
199 views

If $z$ is a complex number of unit modulus and argument theta

If $z$ is a complex number such that $|z|=1$ and $\text{arg} z=\theta$, then what is $$\text{arg}\frac{1 + z}{1+ \overline{z}}?$$
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3answers
239 views

Complex numbers $z$ such that $|z|= 1$

There are infinitely many complex numbers $z$ such that $|z|= 1$. Can anybody just explain this to me so I can get the picture.
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1answer
225 views

How can I calculate the bode magnitude and frequency as well as their plots?

I've been trying to figure this problem out for a while now. I've been given a transfer function $$H(s) = \frac{s(s+100)}{(s+2)(s+20)}.$$ I'm supposed to calculate the bode magnitude and frequency for ...
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3answers
86 views

Adding real and imaginary parts

When trying to add $x$ to $x^{*}$ is it allowed to say that it would be equal to $2|x|$ i.e. so that $$x+x^{*}=2|x| $$ If this isn't the case is there any way to add them or should they be left as ...
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1answer
90 views

3 complex-variable equation

Moderator Note: This is a current contest question on Brilliant.org. $x,y,z$ are complex numbers satisfying $$ \begin{align} x+y+z & =1\\ x^2+y^2+z^2 & =2\\ x^3+y^3+z^3 & =3 ...
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0answers
61 views

Show the steps from step 1 to step 2 (2D Potential - Physics + Maths)(Really Urgent)

I have been trying to show $$ 2 \operatorname{Im}\left( \sum_{n=1}^{\infty} \dfrac{z^n}{n} \right) = \tan^{-1} \left( \dfrac{\sin(\pi x / a)}{\sinh(\pi y / a)} \right) $$ where $$ \large{z = e^{i ...
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1answer
52 views

How to find the phase of a complex-valued function not in trigonometric form?

I have the following function: $$(1 + (jw/w_i)^2 - 2j\alpha_i(w/w_i))\over(1 + (jw/w_i)^2 + 2j\alpha_i(w/w_i))$$ I know that the magnitude is 1 since this is a ratio of complex conjugates, but how ...
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1answer
50 views

Determinant formula and invertibility.

I am working on a problem where I need to find the determinant of $$ \begin{bmatrix} b & a & & \\ & b & a \\ & & & \ddots \\ & & & & ...
4
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1answer
842 views

Rudin Theorem 1.35 - Cauchy Schwarz Inequality

Any motivation for the sum that Rudin considers in his proof of the Cauchy-Schwarz Inequality? Theorem 1.35 If $a_1,...,a_n$ and $b_1, ..., b_n$ are complex numbers, then ...
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2answers
162 views

Finding the number of different ordered quadruples $(a,b,c,d)$ of complex numbers

Find the number of different ordered quadruples $(a,b,c,d)$ of complex numbers such that: $$a^2=1$$ $$b^3=1$$ $$c^4=1$$ $$d^6=1$$ $$a+b+c+d=0$$
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3answers
259 views

A general proof of $f\left(\bar{z}\right)=\overline{f\left(z\right)}$

As a school student I have seen a striking property of functions . $$f\left(\bar{z}\right)=\overline{f\left(z\right)}$$ Where $z$ is a complex number and $\bar{z}$ it's complex conjugate. For eg: ...
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2answers
303 views

Sin inverse of a complex number

Is it possible to calculate the value of $\delta$ from the relation $\delta=\sin^{-1}(5.4i)$ ? where $i=\sqrt{-1}$
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1answer
119 views

$\int_{-\infty}^{\infty}i\cdot \sin(x)\sin(2{\pi}kx)\;dx$ during Fourier transform

I am trying to do a time-to-frequency domain transform using Fourier transform. My function is very simple: $$ f(x) = \sin(x) $$ By definition its Fourier transform should be: $$ F(k) = ...
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2answers
41 views

Help please on complex polynomials

I wanted to know if there's any good approaches to these questions a)By considering $z^9-1$ as a difference of two cubes, write $1+z+z^2+z^3+z^4+z^5+z^6+z^7+z^8$ as a product of two real factors one ...
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2answers
65 views

Finding roots of complex equation?

Determine all roots of the equation $x^6+(3+i)x^3 + 3i = 0$ in $\mathbb{C}$ Express answers in standard form
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2answers
304 views

Prove the direct product of nonzero complex numbers under multiplication.

Let $\mathbb{C}^{\times}$ be the group of nonzero complex numbers under multiplication. Then $\mathbb{C}^{\times}$ is the direct product of the circle group $T$ of unit complex numbers and the group ...
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3answers
97 views

Solve $z^5 + 16\bar z = 0$ for $z\in \mathbb{C}$

Solve $z^5 + 16\bar z = 0$ for $z\in \mathbb{C}$. Need some help figuring out this problem.
3
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1answer
55 views

Proof required that $\sum _{n=1} ^N(1-e^{(2n+1) \pi i/N})^{-1} = \frac N 2$

Numerical evidence suggests this is true, for all natural numbers $N$: $\sum _{n=1} ^N(1-e^{(2n+1) \pi i/N})^{-1} = \frac N 2$ Can anyone prove it?
4
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1answer
128 views

Roots of a polynomial and its derivative

All roots of a complex polynomial have positive imaginary part. Prove that all roots of its derivative also have positive imaginary part. It's not a homework. This issue has been proposed in the ...
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1answer
102 views

Two quick eigenvalues & complex numbers questions

A) For a vector $v\in\mathbb{C^n}$, is $Im(-v)=Im(\overline{v})$ ? ($Im(v)$denoting the imaginary part of the vector $v$) My understanding: since every row of the vector is a complex number (say ...
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6answers
514 views

If $A,B,C,D$ are complex numbers on the unit circle with $A+B+C+D=0$, then they form a rectangle

Let $A, B, C, D$ be points on a unit circle. Prove that if $A+B+C+D=0$, then $A,B,C,D$ make a rectangle. (Use complex numbers.) How do I prove this? I tried to use the dot product of 2 adjacent ...
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2answers
96 views

Real and Imaginary parts of Lebesgue integral in L1 (folland's real analysis)

I am working through Folland's Real Analysis and have a question about a proposition on functions in L1. Prop (2.22) states: given $f \in L^1$, then $|\int f| \leq \int|f|$ In the proof, for when f ...
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0answers
338 views

Determinant of a general circulant matrix

I'm dealing with a problem that is comparable to "How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?", yet slightly more difficult: I was asked to determine the ...
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0answers
87 views

convergence of a sum with zeroes of zeta function

Can it be proved that the sum of this series is smaller than $x$? $$ \sum_{\zeta(a+ib)=0}u_{a,b}(x)\lt x, $$ for all $x$, with $$ ...
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2answers
608 views

Locus of complex numbers $z$ with restricted $(z+1-i)/(z-1-i)$

Problem Describe the locus of the following points on the Argand diagram: $$\left|\frac{(z+1-i)}{(z-1-i)}\right| = 1$$ and $$\mathrm{arg}\left[\frac{(z+1+i)}{(z-1-i)}\right] = \pm ...
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3answers
79 views

Determine all $z \in\Bbb C$ such that $z^8 + 3iz^4 + 4 = 0$

Trying to study for my final, and this question came up. Any hints as how to how to begin would be greatly appreciated. -edit- thank you all for your help. I would have never thought of that in a ...
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3answers
290 views

Finite Union of proper subspaces of $\mathbb C^2$ can equal to $\mathbb C^2$? [duplicate]

My instructor for Linear Algebra gave us a problem to think about but am quite unsure on how to approach it: Let $V_1, V_2, ... V_{100}$ be $100$ proper subspaces of the complex vector space ...
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1answer
95 views

represent real number as case of complex number

my question is related one topic ,which i would like to clarify using your help.so please help me to clarify this topic.problem is following(actually it is just my interest to clarify it) in ...
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0answers
52 views

Subgroups of $P=\{z\in\mathbb{C} : z^{2n}=1 \;\mbox{for some}\; n\geq 0\}$

Investigate the subgroups of $P$ where $$P=\{z\in\mathbb{C} : z^{2n}=1 \;\mbox{for some}\; n\geq 0\}.$$ In particular, investigate the finitely generated subgroups and the infinite subgroups. ...
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1answer
116 views

What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?

Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.) In ...
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2answers
79 views

Modulus and moduli problem of complex number

The moduli of two complex numbers are less than unity. The the modulus of the sum of these complex number (a) less than unity (b) greater than unity (c) equal to unity (d) any of a,b,c Please ...
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2answers
85 views

Is $\mathrm{GL}_n(\mathbb C)$ divisible?

A group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ Is the group $\mathrm{GL}_n(\mathbb C)$ divisible? Or more precisely, ...
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3answers
1k views

Write the complex number on the polar form $r(\cos\theta + i \sin\theta)=re^{i\theta}$

As it said in the title. The number is $4-4i$ I did this $4-4i=\sqrt 2\,(1+i)=\cos(pi/4)+i \sin(pi/4)]=squareroot(2)e^(i)(pi/4)$ |4-4i|=squareroot(2) arg(4-4i)=(pi/4)+2pi but its wrong
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2answers
3k views

How to integrate complex exponential??

Consider $$\int^{\frac{1}{2}}_{-\frac{1}{2} } e^{i2\pi f} \,df = \int^{\frac{1}{2} }_{-\frac{1}{2} } \cos(2 \pi f)\, df$$ Why do we only look at the real part? What about the imaginary part ...
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1answer
82 views

Imaginary complex numbers

Let $z=x+iy$ and $v=2xy$, show that $v=Im[z^2]$ and find a harmonic conjugate of $v$ on domain $D$. Also find the largest domain $D$ on which $v$ is harmonic.
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1answer
187 views

Laurent series expansions, complex variables

Given $f(z)=\frac{1}{z^2(1-z)}$ I am to find two Laurent series expansions. There are two singularities, $z=0$ and $z=1$. So for the first expansion, I used the region $0<|z|<1$ and I got ...
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2answers
203 views

Show $f(z)=2u\left( \frac z2, \frac{-iz}{2}\right ) + \text{constant} $ if $f(z)=u(x,y)+iv(x,y)$ is an analytic function

Suppose $f(z)=u(x,y)+iv(x,y)$ be an analytic function. Show that $ \displaystyle (a)\; f(z)=2u\left(\frac z2,\frac{-iz}{2} \right ) +\text{ constant} \\(b) \; f(z)=2iv\left(\frac z2,\frac{iz}{2} ...
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3answers
103 views

Maximum value of $Z$

How to find maximum value of $| Z| $ if: $$ \Big| Z-\dfrac{4}{Z} \Big|=2; $$ Where $Z$ is a complex mumber
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2answers
445 views

Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$

Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$Given that $|1-zw^*|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$I think the first part can be proven by ...
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2answers
128 views

complex numbers modulus problems

If $|z_1+z_2|^2=|z_1-z_2|^2$ where $z_1$ and $z_2$ are non zero complex numbers, then which one is correct (a) Re$\left(\frac{z_1}{z_2}\right)=0$ (b) Im $\left(\frac{z_1}{z_2}\right)=0$ (c) ...
2
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1answer
445 views

Complex division: polar form vs complex conjugate

The original problem In an electricity course which I volunteered to help with, the students solve circuits using phasors. Using phasors requires a good knowledge of complex numbers arithmetics, ...
2
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3answers
99 views

complex number - modulus problem

If $|z_1+z_2|=|z_1|+|z_2|$ where $z_1 ; z_2$ are different non zero complex numbers, then (a) $Re(\frac{z_1}{z_2})=0$ (b) $Im(\frac{z_1}{z_2})=0$ (c) $z_1+z_2=0$ Please guide how to proceed...