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

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1answer
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complex sequences

my series and sequence knowledge has gone a little rusty so I was wondering if you could help me on the right path here. The assignment is to calculate the sum of the series (1/8)^n * e^(j(npi)/8) as ...
2
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1answer
116 views

An inequality about complex numbers

How the prove the following inequality $|z-1|^r\ge|z^r-1|$ holds for some branch of $z^r$. where $0\le r<1$, and $z\in \mathbb{C}$ is a complex number. If such a branch exists will be fine. ...
4
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3answers
464 views

Which book on complex analysis is good for self study?

Which book on complex analysis is good for self study? I am an average student and have just a very basic knowledge of this subject.I want to cover up to Runge's Theorem. I heard about few books- ...
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1answer
170 views

How to correctly find the angle phi in de moivre's formula?

I am in high school, and we started learning De Moivre's formula. I had some problems with my homework concerning rooting of z. So far, this is what I know about the formula: $\sqrt[n]{z}= ...
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3answers
409 views

Given $\sin z=5$. Find $e^{iz}$ (Complex Trigonometric Function)

Given $sin$ $z=5$. Find $e^{iz}$. Here is what I have done: \begin{align} \sin z &= \frac{e^{iz}-e^{-iz}}{2i}=\frac{e^{i(x+iy)}-e^{-i(x+iy)}}{2i}\\ &=\frac{e^{-y}(\cos x+i\sin x)-e^{y}(\cos ...
2
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1answer
36 views

Let $a \in \mathbb{C}$, how many $n$'th roots of $a$ have nonnegative imaginary part?

Let $a \in \mathbb{C}$. It might as well be on the unit circle, so $a=e^{i \theta}$. I'm interesting in finding, for $n \ge 2$, how many $n$-th roots, $\omega_i$ (with $ 0 \le i \le n-1$), have $\Im ...
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2answers
971 views

Triangle Inequality with complex numbers: Prove that ||x|−|y||≤|x|-|y|.

Prove that $ ||x| - |y|| \le |x| - |y| $ for all $ x,y \in \mathbb{C} $. I fully understand the other inequality: $|x+y| \le |x|+|y| $ for all $ x,y \in \mathbb{C} $. But I have no clue how to start ...
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2answers
911 views

Total ordering on complex numbers

Show that there doesn't exist a relation $\succ$ between complex numbers such that (i) For any two complex numbers $z,w$, one and only one of the following is true: $z\succ w,w\succ z,$ or ...
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4answers
3k views

Determine the fourth roots of -16

Determine the fourth roots of -16 in the form $x +iy$ where $x$ and $y$ are not trigonometric functions. I do not even know what they really want from me in this question. My initial thought wass: ...
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1answer
75 views

an inequality about complex number

How to prove that $|a-b|^\gamma\ge||a|^\gamma-|b|^\gamma|$ where $0\le\gamma<1$ and $a,b$ are complex numbers. Is it a famous inequality?
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2answers
77 views

Convergence of the power of a complex number

Given any complex number $z$, I am interested in investigating the convergence properties of $z$ to some power $n$. We have three cases: i) If $|z| <1$, then we can write ...
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4answers
73 views

Find the values of $x$, where $x \in \Bbb C$, for which $x^4-1 =0$

Find the values of $x$, where $x \in \Bbb C$, for which $$x^4-1 =0$$ I can see that $x^4-1 = (x^2-1)(x^2+1)=0$ So one set of roots can be taken from $$x^2-1=0$$$$ \Rightarrow x=\pm1$$ However, ...
2
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6answers
172 views

Given $ai$ is a root of $x^3-bx^2+a^2x-a^2b=0$, prove it and find the other roots.

Given that $a$ and $b$ are real constants, prove that $ai$ is a root of the equation $$x^3-bx^2+a^2x-a^2b=0$$ Find the other roots of the equation in terms of $a$ and $b$. I realise the conjugate ...
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0answers
73 views

Zeroes of $s+\sum\limits_{n=2}^\infty \frac{(-1)^{n+1}}{n^s\ln n} $?

Where are the solutions of the equations $$s+\sum\limits_{n=2}^\infty \dfrac{1}{n^s\ln n}=0\quad \text{and}\quad s+\sum\limits_{n=2}^\infty \dfrac{(-1)^{n+1}}{n^s\ln n}=0 ?$$ Since the ...
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0answers
124 views

Simplifying a product over roots of unity

Let $\zeta_{n}=e^{2\pi i /n}$ be the nth root of unity. Now consider the product : $$\prod_{k=1}^{n-1} (1-\zeta_{n}^{k})^{\zeta_{n}^{k}}$$ Is there a simple formula for this product as a function of ...
2
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5answers
191 views

Proving $|e^{iθ}|=1$

How do I show that $|e^{iθ}|=1$? So I got that the length will be $\sqrt{\cos^2(x)-\sin^2(x)}$ and it can be written as the square root of $\cos 2x$ but I don't see how that equals 1.
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1answer
67 views

Complex inequality $\left|\dfrac{a-b}{1-\overline{a}b}\right|<1$

Prove that if $a,b$ are complex numbers such that $|a|<1$ and $ |b|<1$, then $$\left|\dfrac{a-b}{1-\overline{a}b}\right|<1.$$ So I assume $a=p+qi$ and $b=r+si$. Then $a-b=(p-r)+(q-s)i$ and ...
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1answer
101 views

Are the complex numbers best represented in 1 or 2 dimensions?

We note that all equations of the type $x+a=b$ can be solved using numbers in one dimension (such as all real numbers), and adding another composition rule, multiplication, we note that two ...
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6answers
1k views

How do I solve and plot the complex equation

I have the following complex equation: \begin{equation} z^6 + 1 = 0 \end{equation} I would like to be able to gain some intuition and understanding. I know from the fundamental theorem of algebra ...
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2answers
61 views

How to get to the first step of solving $z^2=i$?

How do I get from $z^2=i$ to $z=x+iy$? Is it a rule you use to solve the equation in general or specifically for this equation? (I don not understand the step from $z^2=i$ to $z=x+iy$) Thank you!
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4answers
3k views

How to solve a complex polynomial?

Solve: $$ z^3 - 3z^2 + 6z - 4 = 0$$ How do I solve this? Can I do it by basically letting $ z = x + iy$ such that $ i = \sqrt{-1}$ and $ x, y \in \mathbf R $ and then substitute that into the ...
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1answer
821 views

Complex analysis - Trigonometric conjugate

I'm having trouble with this assignment: Show that $ \overline {\cos(z)} = \cos(\bar z)$, where $\bar z$ represents conjugate of $z$. I know that the two are equal, but how to mathematically show ...
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1answer
77 views

Prove the following identities ..complex numbers

I found this question on my Algebra book but i couldn't answer it Can you please explain step by step Prove the following identities ..explain its geometric meaning $|1+z_1\bar z_2|^2 + |z_1-z_2|^2 = ...
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2answers
260 views

How would I approach finding the locus of these complex variable equations?

My textbook gives very little information on how to describe the locus of points for the following: $|z + 2i| + |z - 2i| = 6$ and $z(z^* + 2) = 3$. I was hoping someone could walk through it and ...
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3answers
1k views

limit point of set

general definition of limit point is following : A point $z_0$ is a limit point for a set of point if every neighborhood of $z_0$ contains points,other then $z_0$ of set. now definition of ...
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3answers
192 views

Proving that the limit of a sequence is $> 0$

Let $u$ be the complex sequence defined as follows : $u_0=i$ and $ \forall n \in \mathbb N, u_{n+1}=u_n + \frac {n+1-u_n}{|n+1-u_n|} $ . Consider $w_n$ defined by $\forall n \in \mathbb ...
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1answer
44 views

Results on magnitude of the column space of a complex matrix?

Let $y = Ax$ where $A \in \mathbb{C}^{n \times m},$ $x \in \mathbb{C}^m,$ $n>m,$ and $A$ has full column rank. Then $y$ is in a subspace of $\mathbb{C^n}$, the $m$-dimensional $C(A).$ Define $|y| ...
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1answer
264 views

Sum of distances for vertices lying on a circle

If the consecutive vertices $z_1,z_2,z_3,z_4$ of a quadrilateral lie on a circle, prove that $|z_1-z_3|\cdot|z_2-z_4|=|z_1-z_2|\cdot|z_3-z_4|+|z_2-z_3|\cdot|z_1-z_4|$. I know that when the four ...
4
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1answer
250 views

Cross ratios of permutations of four points

Express the cross ratios corresponding to the $24$ permutations of four points in terms of $\lambda=(z_1,z_2,z_3,z_4)$. So we have ...
2
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1answer
474 views

Cross ratio is real on image of real axis

Theorem: The cross ratio $(z_1,z_2,z_3,z_4)$ is real if and only if the four points lie on a circle or on a straight line. We need only show that the image of the real axis under any linear ...
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2answers
65 views

When $a$ is even, the difference between $(a/2) \mod N$ and $(a \mod N)/2$?

folks. Could I ask for your help? Let $N$ be a positive integer and $a$ be an even integer, i.e., $a=2x$ for an integer $x$. Then think of $W_N^{\frac{a}{2}}$, where $W_N=e^{j\frac{2\pi}{N}}$. ...
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2answers
5k views

Equation of ellipse, hyperbola, parabola in complex form

Write the equation of an ellipse, hyperbola, parabola in complex form. For an ellipse, there are two foci $a,b$, and the sum of the distances to both foci is constant. So $|z-a|+|z-b|=c$. For a ...
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1answer
112 views

$az+b\overline{z}+c=0$ represents a line [duplicate]

When does $az+b\overline{z}+c=0$ represent a line? All of $a,b,c$ are complex numbers. I know that a line in the complex plane is usually represented by $z=a+bt$, where the parameter $t$ runs ...
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0answers
130 views

Möbius tranformation

We define a Möbius transformation through: $$z\rightarrow \frac{az+b}{cz+d}, ad-bc\neq0, a,b,c,d\in \mathbb{C}$$ and extend to the Riemann sphere as follows: if $c=0$, $T(\infty)=\infty$, and if ...
2
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1answer
705 views

Finding center and radius of circumcircle

Find the center and radius of the circle which circumscribes the triangle with (complex) vertices $a_1,a_2,a_3$. Express the result in symmetric form. I'm not sure where to start in this ...
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1answer
197 views

Point symmetric to lines bisecting coordinate axes

Find the symmetric points of a complex number $a$ with respect to the lines which bisect the angles between the coordinate axes. Let $a=r(\cos\theta+i\sin\theta)$. The symmetric point of $a$ ...
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1answer
1k views

Equilateral triangle in complex plane [duplicate]

Prove that the points $a_1,a_2,a_3$ are vertices of an equilateral triangle if and only if $a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_3a_1$. I rewrite the equation as ...
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1answer
639 views

Solving exponential complex equation.

I need assistance in solving the following equation: $$ e^{z^2}=1, $$ where $z$ is a complex number.. I can't seem to get the answer of $z=\sqrt{k\pi}(1\pm i)$. Thank you in advanced for your ...
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2answers
804 views

Projections on the Riemann Sphere are antipodal

Prove that given two points $z,w\in\mathbb{C}$, we have that their projections on the Riemann sphere are antipodal if and only if: $z\bar{w}=-1$.
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Möbius tranformation are continuous?

We define a Möbius transformation through: $$z\rightarrow \frac{az+b}{cz+d}, ad-bc\neq0, a,b,c,d\in \mathbb{C}$$ and extend to the Riemann sphere as follows: if $c=0$, $T(\infty)=\infty$, and if ...
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2answers
119 views

Significance of the trace in isomorphic matrix fields

The field $\mathbb{Q}(\operatorname{i})$ has an isomorphic matrix field of degree two. The isomorphism is $$\varphi:x+\operatorname{i}\!y \longmapsto \left[\begin{array}{cc} x & -y \\ y & x ...
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3answers
325 views

Factorising complex equations

I do factorization by just plugging in simple numbers to check whether they are factors, using long division and also using synthetic division. But, how to do it in the case of complex equations. For ...
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2answers
65 views

Finding modulus of $\sqrt{6} - \sqrt{6}\,i$

I found the real part $=\sqrt{6}$. But I don't know how to find imaginary part. I thought it was whatever part of the function that involved $i$, with the $i$ removed? Therefore the imaginary part ...
3
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1answer
103 views

Maximum modulus principle.

Find the maximum value of $|f(z)|$ on the closed complex disk of radius 2, where $f(z)$$=$$z^4\over{z^2+10}$. Usually I approach these problems by calculating the modulus squared and simplifying, ...
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1answer
54 views

For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative?

Let $k>0$ be an integer. For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative? Since $13$ is prime, and for $\gcd(m,13)=1$, $P(2m)=P(2)=2^{-12}$ (can be shown by considering the ...
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1answer
590 views

Zero sum of roots of unity decomposition

It's known that sum of all $n$'th roots of some $z \in \mathbb C$ with $|z| = 1$ is zero (if $n \geqslant 2$). Is it true that any zero sum of roots of unity can be decomposed in this way? That is if ...
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2answers
156 views

Given that $z=2-i$ and $z^2=3-4i$ find the roots of the equation $(z+i)^2=3-4i$

Given that $z=2-i$ and $z^2=3-4i$ find the roots of the equation $(z+i)^2=3-4i$ How do you use the given properties to find the roots? I can only obtain them the long way by working through ...
0
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1answer
77 views

derivative of complex function

i want to ask question related to about derivative of complex function: if $f(z)$ is differentiable in a connected open set $R$ and if $f'(z)=0$, through $R$,then $f(z)$ is constant in $R$ i ...
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5answers
258 views

Don't know how to find all the roots

So i got this problem : Find all the roots of $r^{3}=(-1)$ i can only think to use : $\sqrt[n]{z} =\sqrt[n]{r}\left[\cos \left(\dfrac{\theta + 2\pi{k}}{n}\right) + i \sin\left(\dfrac{\theta + 2\pi ...
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3answers
278 views

How To Compute Args and Mods of Complex Numbers That Satisfy Inequalities

A complex number $z$ satisfies the inequality $$|z + 2 - (2\sqrt{3})i|\le 2$$ Find the least possible value of $|z|$ and the greatest possible value of $argz$ the answers given in the text book ...