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

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Principal value of complex number

Let $z=-1-i$, find the principal value ? Here $x=-1,y=-1$ therefore $\arg(z)=\tan \alpha=|\frac{y}{x}|=|\frac{-1}{-1}|$ Therefore, $\alpha =\tan^{-1}=\frac{\pi}{4}$ which lies between $0$ and ...
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5answers
443 views

Complex numbers and trig identities. I've heard this question is easy but I don't know how. Help?

Using the equally rule $a + bi = c + di$ and trigonometric identities how do I make... $$\cos^3(\theta) - 3\sin^2(\theta)\ \cos(\theta) + 3i\ \sin(\theta)\ \cos^2(\theta) - i\ \sin^3(\theta)= ...
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3answers
155 views

How can complex polynomials be represented?

I know that real polynomials (polynomials with real coefficients) are sometimes graphed on a 3D complex space ($x=a, y=b, z=f(a+bi)$), but how are polynomials like $(1+2i)x^2+(3+4i)x+7$ represented?
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2answers
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Sketch the Set in Complex Plane

I don't understand how to sketch a set in the Complex Plane. So i would appreciate if someone could explain it to me. What do i have to know ? Which formulas do i need ? Can someone explain how a open ...
3
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2answers
277 views

Write in polynomial in factored form in complex number

Write the following polynomial in factored form(in complex number): $$1+z+z^2+z^3+z^4+z^5+z^6$$ Also, is there general solution of factoring for $1+z+z^2...z^n$ types of polynomial?
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2answers
43 views

Showing series expansion

How would I show $$\frac{\sinh z}{z^2}=\frac{1}{z}+\sum_0^\infty \frac{z^{2n+1}}{(2n+3)!}$$ I have $\sinh z=\sum_0^\infty\frac{z^{2n+1}}{(2n+1)!}$ and if I multiply by $\frac{1}{z^2}$ then ...
20
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8answers
687 views

Infinite powering by $ i$ [duplicate]

Find the value of: $i^{i^{i^{i^{i^{i^{....\infty}}}}}}$ Simply infinite powering by i's and the limiting value. Thank you for the help.
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6answers
227 views

De Moivre's Theorem Related - Complex number

According to de Moivre's Theorem: If $n$ is any positive integer then: $(\cos\theta + i\sin\theta)^n=\cos n\theta +i\sin n\theta$ Also $(\cos\theta +i\sin \theta)^{\frac{1}{n}} = \cos \frac{2r\pi ...
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2answers
71 views

number of roots of polynomial of order n

from theorem of algebra,it is well know that polynomial of order n has exactly n roots,for exmaple quadratic equation like $ax^2+bx+c$ has three cases let $D=b^2-4ac$ ,so we have ...
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1answer
293 views

How can I show that circles in the complex plane correspond to circles on the Riemann sphere? How about lines?

Suppose $ T \subset \mathbb{C} $. Show that the corresponding set $ S \subset \Sigma $ is a. a circle if $ T $ is a circle. b. a circle minus (0, 0, 1) if $ T $ is a line. Here we are defining $ ...
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2answers
69 views

Holomorphic function $f$ such that $f'(z_0) \neq 0$

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an holomorphic function such that $f'(z_0) \neq 0$ for some $z_0 \in \mathbb{C}$.Prove that there is $r>0$ such that, if $|z-z_0|<r$ and $z \neq z_0 ...
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2answers
7k views

How to determine if a matrix is positive/negative definite, having complex eigenvalues?

I am trying to deal with an issue: I am trying to determine the nature of some points, that's why I need to check in Matlab if a matrix with complex elements is positive or negative definite. After ...
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4answers
500 views

Find all of the solutions of $z^4=2i$

I have to find all the solutions of this complex equation. I am trying to do: $z = r^{1/4} e^{i(\theta +2\pi k)/4}$ but I don't know how to find the angle because is 2/0 so any hints are welcome . ...
3
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3answers
108 views

Find $ \lim_{n \to \infty } z_n $ if $z_{n+1} = \frac{1}{2} \left( z_n + \frac{1}{z_n} \right) $

How do I approach the problem? Q: Let $ \displaystyle z_{n+1} = \frac{1}{2} \left( z_n + \frac{1}{z_n} \right)$ where $ n = 0, 1, 2, \ldots $ and $\frac{-\pi}{2} < \arg (z_0) < ...
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2answers
65 views

Is this set region?

$$ |z^2 - 1| < 1 $$ Hint: use polar coordinates. the answer is not a region. I don't know how to start. Whenever I am trying to do, it failed. *. z is complex number.
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2answers
85 views

complex number power

I have question related to power of i,which is determined by equality $i=\sqrt{-1}$ actually from complex number book I know that $i^2=-1$, as much as i know if we compare physical ...
2
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2answers
65 views

Calculate $\lim_{n\to\infty} ((a+b+c)^n+(a+\epsilon b+\epsilon^2c)^n+(a+\epsilon^2b+\epsilon c)^n)$

Calculate $\lim_{n\to\infty} ((a+b+c)^n+(a+\epsilon b+\epsilon^2c)^n+(a+\epsilon^2b+\epsilon c)^n)$ with $a,b,c \in \Bbb R$ and $\epsilon \in \Bbb C \setminus \Bbb R, \epsilon^3=1$. Since $a+\epsilon ...
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3answers
2k views

Find the principal argument of a complex number

I have a text book question to find the principal argument of $$ z = {i \over -2-2i}. $$ I know formulas where we find using $$ \tan^{-1} {y \over x}$$ but I am kinda stuck here can somebody please ...
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3answers
532 views

Proving that a complex number $z$ is real.

A problem I have in my book is to prove that $z$ is real if and only if $\bar{z} = z$. So far I have got that for $z = x + iy$, if $z$ is real, $y = 0$ and thus $z = x = \bar{z}$ as $\bar{z} = x - ...
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2answers
99 views

Complex numbers

I am a newbie to complex numbers so please bear with me if i ask some very naive question., So i was trying to solve my class tutorials and the very first question is, Show that ...
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2answers
669 views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
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1answer
88 views

Prove that one of the roots of a poly lies outside the unit circle in the complex plane

please could you help me with the following problem. I need to show that a particular multistep method in numerical analysis satisfies the root condition. I have reduced the problem to showing that ...
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1answer
38 views

Complex Variables…contour integral

For what values of $m$ and $n$ does $\int_C z^mz^{-n}dz=0$ and for what values does $\int_C z^mz^{-n}dz=2i\pi?$ I am stuck on this problem, any hint? Thanks
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1answer
36 views

Need help interpreting an equation from an article (related to quaternions).

At this link, about half way down the page, there is an equation I don't understand http://physicsforgames.blogspot.com/2010/02/quaternions-why.html This is the equation. $$VV† = -x^2I^2 - y^2J^2 - ...
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1answer
130 views

What are the Complex (Non-Real) Eigenvectors of $3\times 3$ Rotation Matrices?

A $3\times 3$ rotation matrix $R$ that rotates $\mathbb{R}^3$ around the unit vector $v\in\mathbb{R}^3$ by angle $\theta$ (as defined by Rodrigues' rotation formula) satisfies the ...
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3answers
384 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
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2answers
2k views

How to find the real part of this fraction?

I don't understand how to get the real part of the following fraction. $\quad\dfrac{1}{a + jw}$ Here, "j" is the imaginary unit. What's the process of retrieving the real part? Thanks.
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1answer
383 views

How to compute the sum of every $k$-th binomial coefficient?

My teacher was discussing binomial expansions of $(1 + x)^n$ and he gave as an interesting example with $x = i$ whereby you could obtain the sum of all the odd coefficients ($C_n^1+ C_n^3+ C_n^5 ...$) ...
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1answer
59 views

Intuition for generalized complex exponentials (like $i^i$)

I understand complex exponential function $e^z$ and its geometric meaning, but when we expand complex exponentiation to $z^w$ for arbitrary complex z and w, $z \neq 0$, I have no intuition what that ...
2
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1answer
145 views

Find limit of a complex function

Does it exist? if it exists, how to find the below limit: $$\lim_{z\rightarrow 0}z\log\left(\sin \pi z\right)=?$$,where $z \in \Bbb{C}$
3
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1answer
243 views

The multiplication of 2D vectors produces what?

I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication. To avoid confusion with other types of multiplication, this is the basic form I ...
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1answer
59 views

$\lim_{t \rightarrow \infty}\frac{t^n z^n}{|t^nz^n + \cdots+tz +c|} $?

How to find the limit $$\lim_{t \rightarrow \infty}\frac{t^n z^n}{|t^nz^n + \cdots+tz +c|} $$ where $c \in\Bbb C$? is the answer $z^n$? please help :)
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2answers
52 views

maths problem with absolute numbers and complex conjugates

With the following expression with complex conjugates $$ -xx^{*}+yy^{*} = -|x|^2 + |y|^2$$ can it be represented as $$|x|^2 - |y|^2$$ or since $ |x|^2 + |y|^2 = 2$ is it true that $|x|^2 - |y|^2 = ...
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2answers
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Need help with matrix multiplication: $ (aI + bJ)(cI + dJ) $.

Consider the matrix $$ A = \left[ \matrix{a & -b \\ b & a} \right], $$ and write this as $ A = aI + bJ $, where $$ I = \left[ \matrix{1 & 0 \\ 0 & 1} \right] \quad \text{and} \quad J = ...
3
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2answers
63 views

What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension?

In this article that talks about some history of hamilton http://plus.maths.org/content/curious-quaternions There is a snippet that says this: Multiplication is very sneaky. You can only set up ...
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1answer
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Using Cauchy's integral formula to evaluate a function

This problem is from Brown/Churchill Complex Variables and Applications, 8th edition 2009. Section 52, exercise 2, subsection (a) How do I show that the integral of the function $g(z) = ...
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1answer
64 views

$z= \frac{u-\overline{u}v}{1-v}$ is real is equivalent to $|v|=1$.

Let $u,v$ be complex numbers such as $u,v\notin \mathbb{R} $, and : $$z= \frac{u-\overline{u}v}{1-v}$$ Prove that : $z\in\mathbb{R} \Longleftrightarrow |v|=1$.
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0answers
67 views

Inequality with complex numbers and a power

I am working on a problem in nonlinear analysis, and I would like to estimate a term that I can write, in abstract form, as follows: $$ \left||z|^{2p-2}(\Re (z \overline{h}))^2 - |w|^{2p-2} (\Re (w ...
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1answer
54 views

Is this estimation correct?

I have to estimate the following quantity $$\frac{|e^{i\sqrt{\lambda+i\varepsilon}|x|}-e^{i\sqrt{\lambda}|x|}|^2}{|x|^2}$$ in $\mathbb{R}^3$ ($\lambda>0$) where ...
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1answer
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Complex roots of $z^6 + z^3 + 1 = 0$

The equation I'm trying to solve is $f(z) = 0$ where $$f(z) = z^6 + z^3 + 1$$ I already tried the following: randomly throwing in complex numbers and real numbers, rational root theorem, banging my ...
3
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2answers
992 views

How to show $\sin(-iy)=i\sinh(y)$?

How to show $\sin(-iy)=i \sinh(y)$? I get: $\sin(-iy)=\frac{1}{2i}(e^{-iy}-e^{iy})=\frac{1}{2i}(\cos(y)-i\sin(y)-\cos(y)-i\sin(y))=...=-\sin(y)$. I don't get it. $-sin(y) \neq i sinh(y)$ - look at ...
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2answers
209 views

Comparing square roots of negative numbers

If we have for instance $\sqrt{-25}$, that is, a square root of $-25$, I know the answer can be $5i$ (Is $-5i$ also correct? Sorry not professional in mathematics). My main question here is how to ...
2
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1answer
370 views

Solutions to $(z+1)^n = z^n$ using conformal maps.

I'm doing a homework problem where I have to find all roots of $(z+1)^7 - (z)^7 = 0$ using the roots of unity for $z^7$ I noticed that if $a$ is a root of unity for $z^7$, then $1/(a-1)$ maps the ...
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4answers
690 views

How would you explain why $e^{i\pi}+1=0$ to a middle school student?

Hi I was asked by a friends child who is in middle school why $e^{i\pi}+1=0$. Now I couldn't think of a way to explain it so he would understand. Albert Einstein once said “If you can't explain it ...
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3answers
535 views

Accumulation Points of a Complex Sequence

Let $z$ be a complex number of absolute value 1: $ z=e^{i\theta}, 0 \le \theta \lt 2\pi$. What are the accumulation points of the sequence $\lbrace z^n \rbrace$? Distinguish between the case where ...
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0answers
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About growth rate of the iterated exponential on the complex plane.

Let $n$ be a positive integer. Let $f(n,x) = exp(f(n-1,x))$ and $f(0,x)=x$. Let $Q(f(n,x)) =1$ if $Re(f(n,x))<2$. Let $S(n,x) = \Sigma_{1}^{n} Q(f(n,x))$. How to estimate $S(n,2+i)$ efficiently ? ...
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2answers
257 views

The roots of the derivative $P'(z)$ of the polynomial $P(z)\in\mathbb C[x]$ lie in the convex hull of the set of roots of $P(z)$.

Assume $S=\{z_1,z_2,...,z_k\}, z_i\in \mathbb C$$, C(S)$ and define $$C(S):=\{z=a_1z_1+a_2z_2+...+a_kz_k | a_i\ge0 ,a_1+a_2+...+a_k=1\}$$ where $$A:=\{z\in \mathbb C:f(z)=0 ...
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2answers
272 views

Triangle inequality complex analysis

Using Triangle Inequality, prove if $|z−c| \le |c|/2$, then $|z| \ge |c|/2$.
2
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2answers
47 views

what's the conjugate of $i^{-\frac{1}{2}}$?

If a complex number is $A=a+bi$, then its conjugate is $\bar{A}=a-bi$. What's more, the conjugate of $e^{i\theta}$ is $e^{-i\theta}$. Well, it is known to us. Now, if a complex number is ...
0
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0answers
57 views

Limit points of a sequence contained in $S^1$

Let $\theta\in (0,2\pi)$ be a real number such that $\displaystyle\frac{\theta}{\pi}\notin\mathbb{Q}$. We define $z:=\cos(\theta)+i\sin(\theta)\in S^1\subseteq\mathbb{C}$ and let ...