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

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1answer
95 views

Rotation on the unit circle K

If $\{a^{n}:n\in\mathbb{Z}\}$ is dense on the unite circle K, then $\{a^{n}:n< 0\}$ is also dense on K. How to prove this result?
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4answers
212 views

What is the square root of $i^4$?

What is the $\sqrt{i^4}$? $i^4$ = $(i^2)^2$ So is $\sqrt{i^4}$ = $\sqrt{(i^2)^2}$ = $i^2$ = $-1$? Or is $\sqrt{i^4}$ = $\sqrt{1}$ = $1$? When I plug it into my TI-89 Titanium, I get $1$. Edit: I ...
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1answer
142 views

Find a complex number that satisfies the equation

Find one complex value of $x$ that satisfies the equation $\sqrt{3}\cdot x^7+x^4 +2=0$
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1answer
259 views

Complex plane sets - domain |Argz | < pi/4

Why is the following complex set a domain: |Arg z | < pi/ 4 if Arg z = 0 is not defined, so there is no polygonal path between the two quadrants.
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2answers
47 views

Estimate a complex modulus

I have to estimate the following quantity $$\vert e^{iz\vert x\vert}-e^{i\lambda\vert x\vert}\vert^2$$ where $x\in\mathbb{R}^3$, $\Im z>0$ and $\lambda>0$. So I write $$\vert e^{i\Re z\vert ...
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1answer
61 views

Put the following in rectangular form.

$$(\sqrt{3}+i)^7$$ My question: $r = 2$. For $\theta$, do I use $\dfrac{\pi}{6}$ or $\dfrac{\pi}{6} + 2n\pi$? The book uses the former but I thought the latter is more appropriate. Thank you.
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1answer
266 views

Continuous function on simple closed contour

Let $f$ denote a function that is continuous on a simple closed contour $C$. Using the Cauchy Integral formula, prove that the function $g(z)=\frac{1}{2\pi i}$ $\int_C$ $\frac{f(s)ds}{s-z}$ is ...
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1answer
278 views

Contour Integrals and positively oriented circles

If $C_0$ denotes a positively oriented circle $|z-z_0|=R$, then $\int_{C_0}$ $(z-z_0)^{n-1} dz$ = $\left\{ \begin{array}{lr} 0 & n=\pm1, \pm2, ...\\ 2\pi i & n=0\\ ...
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1answer
164 views

Connected set on complex plane

What's the numebr of connected components for the set of complex numbers $\{e^z:|z|=1\}$ on the complex plane? Remark: It represents a simple closed curve which intersects the real axis at points ...
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2answers
70 views

Is there a systematic way to solve the equation $x+yi=-y+xi=0$ where $x,y$ complex numbers?

Is there a systematic way to solve the equation $x+yi=-y+xi=0$ where x,y are complex numbers? Or is it simply solved by observation, and so the answer is $x=-i$ and $y=1$? Thanks in advance.
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0answers
222 views

If the product of two analytic functions is zero, then one must be identically zero.

I want to prove this statement: Let $f,g$ be analytic on $D(0,2)$. If $f(z)g(z) = 0$ when $z = 1/n$ for $n \in \mathbb{N}$, then either $f \equiv 0$ or $g \equiv 0$ in $D(0,2)$. My attempt: ...
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3answers
147 views

Complex Sine bounded when $|\text{Im } z| < 1$

Prove that $\sin z$ is bounded on $\{z \in \mathbb{C} : |\text{Im } z| < 1\}$. I know how to prove that it is unbounded, but I'm stuck on this.
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1answer
84 views

Why is the second equality wrong?

Here's a "proof" of $e^x=1$ for all $x$: $$\exp(x)=\exp\left(i2π⋅\frac{x}{i2π}\right)=\bigl(\exp(i2π)\bigr)^{x/(i2π)}=1^{x/(i2π)}=1$$ Why is the second equality wrong?
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3answers
599 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 ...
3
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1answer
85 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
333 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
85 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
100 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
34 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
157 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
181 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
238 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
208 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
85 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
88 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
60 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
48 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
776 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 ...
3
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2answers
159 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: $$a^2=1$$ $$b^3=1$$ $$c^4=1$$ $$d^6=1$$ $$a+b+c+d=0$$
3
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3answers
242 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
243 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
118 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
64 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
265 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 ...
3
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3answers
92 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?
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1answer
109 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
481 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
90 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
283 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
86 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
501 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
77 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
261 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
92 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
51 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. ...