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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0
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1answer
41 views

How to find the General expression of $\sum_{k=0}^ {\lfloor n/3\rfloor} {n \choose 3k}$ [duplicate]

Well as the title says I'm having problems trying to derive a general expression for this sum which involves cubic roots of unity $$\sum_{k=0}^ {\lfloor \frac n 3\rfloor} {n \choose 3k}$$ Need help ...
1
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3answers
73 views

$e^{a 2\pi i} = (e^{2\pi i})^a$.

When $a$ is any real number , Is it true $e^{a 2\pi i} = (e^{2\pi i})^a$ ? The reason why I ask this question is that I met this situation wheter this equality hold in Calculating Integral in Complex ...
0
votes
2answers
53 views

Complex numbers inside determinant

Let $ \begin{vmatrix}6\iota & -3\iota & 1\\ 4 & 3\iota & -1\\ 20 & 3 & \iota \\ \end{vmatrix}= x +\iota y$, then what are the values of $x$ and $y$?
0
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1answer
164 views

Is there a way to prove that i²=-1? [duplicate]

I have 4 questions regarding the imaginary and complex numbers. (And some ideas) My questions are about the way that I’m trying to come up with a proof to the equation i²=-1 (and from there maybe ...
0
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2answers
17 views

Complex Conjugate roots with non real coefficients

I understand that a polynomial with real coefficients must have complex conjugate roots (if complex roots exist) Is it possible for a polynomial with non-real coefficients to have complex conjugate ...
2
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1answer
31 views

Visualizing a complex function

Ever since I learned about complex valued functions I've been wondering if there was a better visualization for them. Obviously we can't visualize four dimensions, but I was wondering if it would be ...
0
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3answers
76 views

Solve in $\mathbb{C}$ : $|z-i| = |z-1|$

I just had that question in my final exam Solve in $\mathbb{C}$ : $|z-i| = |z-1|$ and I couldn't do it. I found a similar thread here : Showing that $\{z\in\mathbb{C}:|z-1|<|z+i|\}$ is an open ...
1
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0answers
25 views

Bounded real parts of the solutions of an equation

I'd like to show that, with $a,b>0$, the real parts of the solutions $z_n$ of the equation $$ az+\sqrt{z^2-ib}\tanh\sqrt{z^2-ib}=0 $$ are bounded. An indication for that can be found if we ...
0
votes
1answer
26 views

biholomorphic on unit disk

Let $D$ be the unit disk and $f: D\rightarrow G$, $\; p_1$ the maximum value of $dist(f(z),f(0))\;$ and $p_2$ the minimum value of $dist(f(z),f(0))$ for $z\in \partial \bar G$ Prove that : $|f(z)-...
1
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1answer
57 views

Evaluate $\cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7}$

Evaluate $$\cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7}.$$ The first thing i noticed was that $$\cos \frac{\pi}{7}=\frac{\zeta_{14}+\zeta_{14}^{-1}}{2},$$ where $\zeta_{14}=e^{2\pi i/14}$...
0
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1answer
29 views

Sine and cosine solutions of a differential equation

I have to solve a differential equation with constant coefficient such as$$ay'''+by''+cy'+dy=f(x)$$ which has for a characteristic equation$$P_c(\lambda)=a\lambda^3+b\lambda^2+c\lambda+d=0$$First I ...
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0answers
20 views

Newton's method for nth roots of complex numbers

Is it possible to use Newton's method to compute roots of complex numbers, say $\sqrt[n]{a+ib}$ to any desired accuracy? If yes,for what initial values will converge?
1
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2answers
43 views

Proving $(w-1)^m$ is purely imaginary.

I'm having trouble trying to prove this: Let $ m\in \mathbb Z$, m even and $w\in\mathbb C$ a primitive $2m$-th root of unity. Prove that $(w-1)^m$ is purely imaginary. What I've tried to do so ...
2
votes
1answer
42 views

Least value of complex expression

If $z_{1},z_{2},z_{3},z_{4}$ are $4$ points on a circle $|z| = 1$ such that $z_{1}+z_{2}+z_{3}+z_{4}=0\;,$ Then least value of expression $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{4}|^2+|z_{4}-...
-1
votes
1answer
90 views

Why is De Moivre's theorem not generalised for $(\sin x+i\cos x)$?

A representation of the form $(\sin x+i\cos x)^n$ can be reduced as follows $$( \sin x + i \cos x )^n$$ $$( \cos (90-x) + i \sin(90-x) )^n$$ $$( \cos (90n - nx) + i \sin(90n - nx) )$$ Now for all ...
2
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4answers
66 views

Linear algebra : Solving $i \cdot\bar{z} = 2 +2i$

$i\cdot\bar{z} = 2+2i$ I know that $\bar{z} = a-bi$ so then i get $i(a-bi)=2+2i$ Then $ai+b=2+2i$ (because $i^2=-1$) When 2 complex numbers are equal you usually can equal their parts Ex: $2+2i=a+bi$...
1
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1answer
37 views

How does analytic continuation lets us extend functions to the complex plane?

I'm trying to understand analytic continuation and I noticed on wolfram that it allows the natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic ...
1
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0answers
47 views

Write complex number $z^5$ in the form $r(\cos θ + i\sin θ)$

Let $z = 4(\cos \frac 8 7 π + i\sin\frac 8 7π)$ be a complex number. Find $z^5$ in the form $r(\cos θ + i\sin θ)$ with r being a positive real number, and with $0 ≤ θ < 2π$. My attempt: $z^5 = ...
1
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1answer
46 views

The identity $ \sqrt[n]{z}\sqrt[n]{w} = \sqrt[n]{zw}$ for complex numbers

In the general case, when $z$ and $w$ are two complex numbers, we have that $ (1) \sqrt[n]{z}\sqrt[n]{w} \neq \sqrt[n]{zw}$ For example, $\sqrt{-1}\sqrt{-1} \neq \sqrt{-1.-1} = 1$. However, there ...
1
vote
1answer
20 views

Holomorphic functions and complex conjugation

Suppose I have given two holomorphic functions $g,f:\mathbb{C}\backslash(-\infty,1]\rightarrow \mathbb{C}$ and I know that $\overline{ g(z)}=f(z)$ for all $\vert 2-z \vert <1.$ I am wondering if ...
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0answers
23 views

discrete logarithm with complex numbers

let $z = a + bi$ where $a,b$ are integers on $[0,N)$ let $a + bi \mod t = (a \mod t) + (b \mod t) \cdot i$ Consider the problem of finding $e$ where $z^e \mod N = c$ and $c, N$ and $z$ are known. Is ...
0
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1answer
17 views

inequality by taking reciprocal or other way to check if pole lies inside unit circle

If $ a^2$ <1 is given in the problem then how do we prove that the pole z=1/a lies outside the unit circle ?
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0answers
16 views

Finding a branch of the complex logarithmic function $\log(1-z).$

I have a question that asks me to find the holomorphic branch $L(1 − z)$ of $\log(1 − z)$ valid in the cut-plane $z \in \mathbb{C}\setminus [1, ∞)$ and such that $L(1) = 0.$ We have defined the ...
0
votes
1answer
41 views

Sufficient condition on open subsets to be equal

Let $U,V\subseteq\Bbb C$ connected open non empty, such that their closure in $\Bbb C$, say $\overline U,\overline V$, be simply connected. Then, is it true that, if $$ U\cap V\neq\emptyset\\ \...
2
votes
2answers
118 views

An apparently new method to compute the $n$th root of any complex number

I found  a series of articles (in Portuguese) by a Brazilian mathematician named Ludenir Santos, where presents a series of iterative methods, he said new, to extract nth roots of any complex number ...
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0answers
14 views

Infinite exponential sum doubt

Hello! I have a couple of doubts regarding a formula seen here : $$\sum _{k=1}^{\infty } \frac {e^{kz}}{k}= -\log (1-e^{z}) /; Re(z)<0$$ What would happen if the real part of z Re(z) were equal ...
0
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1answer
16 views

Determine $w + \overline w + (w + w^2 )^2- w^{38}(1-w^2)$ for each $w \in G_7$.

I'm starting to see complex numbers in algebra. I've missed a few classes and I have exercises similar to this one: Determine $w + \overline w + (w + w^2 )^2- w^{38}(1-w^2)$ for each $w \in G_7$. ...
-1
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4answers
48 views

Show that $\sum_{k=1}^{n-1} \sin(\frac{2 k \pi}{n})$ is equal to $0$ [closed]

Proof of $ \sum_{k=1}^{n-1} \sin(\frac{2 k \pi}{n})= 0 $. How can I prove this statement without dividing into cases of odd $n$ and even $n$?
0
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2answers
65 views

Geometric interpretation of a complex set

These usually aren't too bad but I had difficulties thinking of what the set $$\{z\in\mathbb{C}:|z+i|=2|z|\}$$ looks like in the complex plane. I got as far as $$|z+i|=2|z|\Rightarrow \sqrt{(z+i)(\...
3
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1answer
29 views

Complex Numbers and Binomial Expansion

I was able to show the above by equating the solution of $cos \ 5\theta=0$ (which gives $\pi/10$) and the solution of ${16cos}^5\theta - {20cos}^3\theta+5cos\theta = 0$ (which is what you get when you ...
2
votes
1answer
48 views

Find on which $z=x+iy\in\mathbb{C}$ the function $f(z)=(\overline{z}+1)^3 - 3\overline{z}$ is differentiable

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. If it's any indication of difficulty, the exercise is ...
2
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4answers
75 views

$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$

In wikipedia it says, $$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$$ I want to now why is this true and what does a logarithm of a complex number even mean. I'm guessing that if I use the Taylor ...
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0answers
10 views

Decomposition of the transformations of the Hilbert curve into real and imaginary parts

I have the set of transformations to generate the Hilbert curve. The complex representation is $\zeta_0 z=\frac{1}{2} \bar{z} i$ $\zeta_1 z=\frac{1}{2} z+ \frac{i}{2}$ $\zeta_2 z=\frac{1}{2}z + \...
0
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0answers
14 views

Looking at direction in complex numbers and vector analysis as a cartesian products of 2 imaginary and real total orders.

Can we abstract the idea of direction into an a cartesian product of two total orders? for example:$T_R = \{a,b,...\}$ and $T_I =\{ai,bi,...\}$ where ${T_R}×{T_I}$ is all possible directions and the ...
3
votes
4answers
206 views

Solution to this complex number equation

Solve $z^5 +32 =0$ My attempt : $$z^5 = -32$$ Multiply the powers on both sides by $\frac{1}{5}$ we get $$z = 2 * (-1)^\frac{1}{5}$$ Now I'm stuck at this step I don't know how to ...
1
vote
1answer
40 views

Write $(z-w)$ as $a+bi$, where $z = 8 - 4i$ and $w = 8 + 4i$

If $z = 8 - 4i$ and $w = 8 + 4i$, then write the expression $(z-w)$ in the standard form $a + bi$. However when I do that I get $0$, because from what I assume $8-4i$ and $8+4i$ cancel each other out....
1
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1answer
49 views

Five roots on an ellipse in the complex plane [closed]

What is special with an originating fifth order polynomial with one real and four complex roots lying on an ellipse when plotted as vectors in complex plane? If three in number in the special case ...
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0answers
29 views

Extension of Scalars of Complex Numbers

If we consider the complex numbers as a $\mathbb{R}$-module (vector space in this case), then its natural extension of scalars to $\mathbb{C}$ seems to be the complex numbers themselves, with the ...
7
votes
2answers
177 views

Compute complex integral resulting from FT

I obtain the following integral after doing a FT of a function $$\int_{-\infty}^{\infty} e^{-\pi(x + i\xi)^2}dx$$ I am not sure how to evaluate it. I tried change of variable $y = x+ i\xi$. but what ...
0
votes
1answer
71 views

Finding the centre of the circle on the complex plane

The centre of the circle represented by $|z+1|$=$2|z-1|$ on the complex plane is (a)0 (b)5/3 (c)1/3 (d)None of these What I've tried so far in attached in the pic below. Please refer to it
1
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1answer
26 views

Sketching Complex region

This is the question: Conisder the points in the region $R$ shown in the Argand diagram of Figure 2, consisting of all points in a right-angled sector of radius $1$, except for the point $ z ...
0
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2answers
80 views

Solve $z^5=-32$ and draw its solutions in complex space, then describe their characteristic geometrical property.

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. If it's any indication of difficulty, the exercise is ...
1
vote
1answer
38 views

A fallacy in the imaginary numbers. [duplicate]

$$\sqrt{-5}*\sqrt{-3}=\sqrt{-1*5}*\sqrt{-1*3}$$ $$\sqrt{-1*-1}*\sqrt{5*3}=\sqrt{5*3}$$ $$=\sqrt{15}$$ But we all know that this below is right, $$\sqrt{5}i*\sqrt{3}i=-\sqrt{15}$$ So, please explain ...
1
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5answers
111 views

Difference between real and complex solutions of cubic equations [closed]

Take for an example, this equation. $$x^3+15x+4=0$$ This equation has two complex solutions and a real one. $$x≈0.1327-3.8798i$$ $$x≈0.1327+3.8798i$$ $$x≈-0.26542$$ What's extra in the complex ...
0
votes
3answers
53 views

Finding the number of roots of the given equation

Number of roots of the equation $z^{10}-z^5-992=0$ where real parts are negative is (a) 3 (b)4 (c)5 (d)6 What I've tried so far Let $z=x+iy$ Now, putting the value of $z$ in the equation, we ...
0
votes
1answer
60 views

Computing Complex Line Integrals

I'm having trouble understanding exactly how to compute a complex line integral in $\mathbb{C}$. With my understanding of multivariable calculus, I view the line integral of a vector field $F: \mathbb{...
12
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1answer
1k views

Does an iterated exponential $z^{z^{z^{…}}}$ always have a finite period

Let $z \in \mathbb{C}.$ Let $t = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $...
0
votes
0answers
16 views

How do I determine and sketch the images $g(\mathbb{R})^2$ as sets and as geometric objects?

It's given function $g(x, y) = \begin{pmatrix}e^x \cos y\\ e^x \sin y\end{pmatrix}$. How do I determine and sketch the images $g(\mathbb{R})^2$ as sets and as geometric objects?
0
votes
1answer
33 views

Find a linear map $f_{\theta}: \mathbb R^2 \to \mathbb R^2$ which describes rotation by $\theta$ in counterclockwise direction

$\theta \in [0, 2\pi).$ Hint: for a given angle $\theta$, find $a, b, c, d \in \mathbb R$ such that $f_{\theta}(x_1, x_2) = (ax_1 + bx_2, cx_1 + dx_2).$ This problem occurs at the end of a ...