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

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Polynomial divisibility

Given $p(x) \in \mathbb Q[x] $ an irreducible polynomial, and $\alpha \in\mathbb C $ root of $p(x)$. Prove that if $q(x) \in \mathbb Q[x]$ it's a polynomial, such $q(\alpha) = 0$ then $p(x) \mid ...
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1answer
37 views

Complex proof problem

Let be P(z) a complex polynomial with a degree of n>=1 and |(p(z)| <= a|z| then there exist a complex number c such that p(z) = cz.
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1answer
119 views

Graphing Complex Number on Argand Diagram

Can someone please answer me, HOW does Im(z^2) = 4 get graphed like this? and not like a normal parabola? Like Re(z^2) = 4 But of course on the Imaginary axis. It has been eating my mind up - ...
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152 views

Complex number ( equation) Find the number of solutions of $z^3+\overline{z}=0$ …

Problem : Find the number of solutions of $z^3+\overline{z}=0$ Solution : $z^3 =-\overline{z} \Rightarrow |z|^3 = |(-\overline{z})|$ $\Rightarrow |z|^3 = |z| \Rightarrow |z|^3-|z| =0$ $\Rightarrow ...
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2answers
44 views

Why does this have a complex component?

Why does: $$(-2)^{\frac{2}{3}}$$ have a complex component? I thought it would be equal to: $$((-2)^2)^{\frac{1}{3}}$$ $$= 4^{\frac{1}{3}}$$ which doesn't have a complex component. But Wolfram ...
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1answer
46 views

Representing complex numbers as matrices, show that $A(z)+A(z')=A(z+z')$

I am doing a task where in which I am representing complex numbers as matrices, so $z=x+iy \in \Bbb C$ is represented by: $A(z)=\begin{bmatrix} x & -y \\ y & x \end{bmatrix}$ Now I have to ...
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4answers
92 views

Is posible proof that there is not a holomorphic function $f:\Delta\rightarrow\bar{\Delta}$, such that $f(0)=0$ and $f(z)=i$ for some $z\in\Delta$.

I have problems with this: Proof that there is not a holomorphic function $f:\Delta\rightarrow\bar{\Delta}$, such that $f(0)=0$ and $f(z)=i$ for some $z\in\Delta$. My problem is that I am getting ...
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1answer
17 views

Step in finding $\sin^{-1}z = w$ for a fixed complex $w$ and unknown complex $z$

This is in the section of the book preceding a general formula but I don't know how the author arrives to the second equation in the picture. The closes I have gotten to it is $$2iz = ...
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2answers
60 views

The calculation of roots of complex numbers.

How to calculate the roots of $x^6+64=0$? Or how to calculate the roots of $1+x^{2n}=0$? Give its easy and understanble solution method. Thank you. In general, the results of "exp" are obtained.
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1answer
33 views

Find the minimum value of $\operatorname{Im}(z^5)/(\operatorname{Im}(z))^5$ [closed]

If $z$ is a complex number, then find the minimum value of $$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5},$$ where $\operatorname{Im}(z)$ denotes the imaginary part of z.
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46 views

How to show that all roots of $(11+v)q^3-18q^2+9q-2$ have their absolute value less than 1.

The equation is $(11+v)q^3-18q^2+9q-2=0$, where $v>0$ I need to show that either absolute value of all the roots is not greater than one or there exists a root $q: |q|>1$. Using Weierstrass ...
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2answers
54 views

Polynomial with complex coefficients

I can't solve the following questions: Let $a,b$ be real numbers, $Z= a + ib$. How much polynomials with complex coefficients $q(x) = x^3 + b_2 x^2 + b_1 x + b_0$ there are so that $Z$ is a root of ...
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1answer
54 views

Geometry using complex e powers

The question is to expresss $\cos(4\theta)$ and $\sin(4\theta)$ in terms of $\cos(\theta)$ and $\sin(\theta)$. This in itself is not that hard using geometrical rules. But my problem is that you need ...
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1answer
65 views

Solution check for $z^2 + |z|^2 = 2+ 2i$ where $z^2=t$

I solved this equation, $$z^2 + |z|^2 = 2+ 2i$$ placing $z^2=t$ at the end remains: $$\begin{align} t&=(2+2i)/2 \\ t&=1+i \\ z^2&=i+1 \\ (z+1)(z-1)&=i \\ z_1&=1+i \\ ...
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3answers
71 views

Solve $x^2-(6+7i)x-4+20i=0$

Solve $x^2-(6+7i)x-4+20i=0$ I probably do too many substitutions, but here is my attempt: https://www.dropbox.com/sh/pqze4iagbo3nics/QhvkBcuZsH
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1answer
100 views

Factor the polynomial completely and find all its zeros. $P(x)= x^4+2x^2+1$

Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $P(x)= x^4+2x^2+1 $ I'm stuck on this problem, can anyone show me the steps on how to factor it out ...
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1answer
39 views

Writing $ h(z) =\frac{ z + i}{z^2 +1} $ in standard form

My attempt: $ h(z) =\frac{ z + i}{z^2 +1}$ Let $z = x + iy$ Then $$ \frac{ z + i}{z^2 +1} = \frac{x + iy + i}{x^2 - y^2 + 2ixy + 1}$$ This is approximately where I get stuck. I'm supposed to ...
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relating complex number to real no. using eulers formula.

$$e^{i*\theta}=\cos\theta+i\sin\theta$$ for $\theta=\pi/2$ $$e^{i\pi/2}=i$$ raising index by -i on Lhs and Rhs $$e^{\pi/2}=i^{-i}$$ does this means i^(-i) is real number. Or am I doing something ...
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3answers
49 views

Finding the point on the real axis collinear with $i$ and $-1+2i$

Let $A = i$ and $B = -1+2i$ be two points on the complex plane. Find the point $D$ on the $x$-axis such that $A$, $B$ and $D$ are collinear. I don't know from where to start, and what does "$D$ ...
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41 views

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 ...
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210 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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141 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 ...
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1answer
39 views

Complex conjugate as ordered pair

let $(a,b) \in \mathbb{C}$, the complex conjugate of $(a,b)$, as ordered pair, is $(a,-b)$... is correct?
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1answer
285 views

determine if pole is inside unit circle

i would like to know how to determine if pole of given function is inside unit circle contour? for example let us take this function $f(z)=(i-1)/(z+i)$ and we have contour ...
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2answers
99 views

Negative square roots and complex numbers

I am doing a quadratic equation which has given me; $\frac{2\pm\sqrt{-4}}{2}$ I know that $\sqrt{-1}=i$ Is it then okay to assume $\sqrt{-4} = 4i$ ? Which would make the next step; $\frac{2\pm ...
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2answers
172 views

The square root of $1$ [duplicate]

I know this is wrong but I don't know why. In the set of complex numbers: $\sqrt1 = \sqrt{i^2\cdot i^2} = i\cdot i = -1$ What is wrong with this?
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Square root of complex number in polar or rectangular form

I am trying to find how to simplify: $$\sqrt{\frac{A+jb}{C+jd}}$$ My calculator errors out, giving a math error, and I don't know how else to solve this.
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161 views

Difficulty in understanding integrals of complex numbers

I understand what integration of real numbers is. I know how the definition of it is made. I have trouble in understanding how it works for complex numbers. I am referring to the notes here: ...
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1answer
74 views

If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Is $f(x+iy) = \exp(x+iy)$?

$(1)$ If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Then, is $f(x+iy) = \exp(x+iy)$ for every $x$ and every $y$? $(2)$ If $f$ is an entire function such that ...
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199 views

Remainder of a complex function

Dividing $f(z)$ by $z-i$, the remainder is $1-i$ and by dividing $z+i$ the remainder is $1+i$, then what is the remainder when $f(z)$ is divided by $z^2+1$? I just started solution using division ...
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184 views

Sides of the Right angled Triangle in Complex notation.

If $z=a+ib$ is a complex number, then $z, iz, z+iz$ represents sides of the right angled triangle. I got this result through Cartesian form, i,e. $(a,b),(-b,a) and (a-b,a+b)$ are the vertices of the ...
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1answer
106 views

Solving complex linear congruences

Find $x \in \mathbb{Z}[i]$ such that: $(1+2i)x \equiv 1 \mod 3+3i$ How would you go about doing this? Best I can think of is keep guessing....
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84 views

Some questions about complexification of a real vector space

Could you tell me how to prove that if $f:U \rightarrow U$ is $\mathbb{R}$-linear, then: 1) $U^{\mathbb{C}}$ is a vector space over $\mathbb{R}$ (should I check all eight conditions for a vector ...
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1answer
96 views

Showing that the segment joining $0$ to $z$ is perpendicular to the segment joining $0$ and $w$ iff $Re[z\bar{w}]=0$

I'm reading Beardon's Algebra and Geometry. Suppose that $zw\neq0$. Show that the segment joining $0$ to $z$ is perpendicular to the segment joining $0$ to $w$ if and only if $Re[z\bar{w}]=0$. ...
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1answer
115 views

Complex numbers and absolute values

If i have equation: \begin{align} P = \left|\psi\right|^2 \end{align} where $P$ is a probability and we know there is no negative probability. This means $P$ must belong to $\mathbb{R}$. If i want ...
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1answer
73 views

A rather ugly limit [duplicate]

Evaluate $$\lim_{n \rightarrow \infty} n \sin (2\pi e n!).$$ I wanna ask what's wrong with my method: Define $C_n= n \cos (2\pi e n!)$ and $S_n=n \sin (2\pi e n!)$, then $C_n+iS_n=ne^{i2\pi ...
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1answer
573 views

Complex Numbers - Omega

I'm told that $ω^n$ = $ω^{(n+3k)}$ Also and $k = 0,1,2.$ How does $ω^{(-1)} = ω^{(2)}$, in this equation? Taking $ω^{(-k)}$, $k = 1$ $= ω^{(n + 3(-1))} = ω^{(n -3)}$ Also what does $n$ ...
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109 views

Complex expression for periodic binary sequences

We have infinite binary sequences of type $$\langle g_n \rangle_{j=4}=\{0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,...\} \,;\, n\to\infty$$ where $j$ indicates the length of a period that starts/ends with a $1$; ...
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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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1answer
35 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
1k views

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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2answers
164 views

Two circles with different center points as the radius tends to infinity

If you have two circles that always have equal radius on a plane, and the circles have different center points, both on the x axis to make it simpler. Such that they intersect twice. As the radius ...
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94 views

What is the inverse $z^{-1}(z)$ of $z(\varphi)=e^{i\varphi}$ with $\varphi\in\Bbb N_0$?

Suppose I am given a complex number $z=x+iy\in\Bbb C$, with $\left|z\right|=1$, and I am told that $z=e^{i\varphi}$ for some integer $\varphi\in\Bbb N_0$ (the value of which is not given to me). How ...
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3answers
496 views

prove that the definition of $\log(z)$ is not everywhere continuous

I would like to prove by contradiction that $\log(z) = \ln(r) + i\theta$ is not continuous at $\alpha = \theta$, for $r > 0$ and for $\alpha \le \theta \lt \alpha + 2\pi $. It seems pretty ...
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3answers
101 views

Solving complex equations

Suppose the equation $(E):z^2-2\sin(\alpha)z+2(1+\cos(\alpha))=0$ / $z\in \mathbb{C}$. I tried to calculate the discriminant but I could determinate it's sign(there is a hint $\Im (z_{1})\ge ...
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1answer
32 views

Equivalence between real part of two complex numbers

Suppose $\forall z \in \mathbb{C}\setminus\{i\}$ we set $\displaystyle f(z)=\frac{z+i}{1+iz}$. How can I prove that: $\displaystyle \Re (f(z))=\frac{1}{2} \Longleftrightarrow \Re (z) = ...
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2answers
96 views

Euler's Formula Question.

Find $\cos\theta + \cos3\theta + ... + \cos((2n+1)\theta$, and $\sin\theta + \sin3\theta + ... + \sin((2n+1)\theta$. Where $\theta \in$ Reals.
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1answer
80 views

Finding the largest set where a complex function is analytic

f(z) = e^z / (sinz - cosz) So I solved for sinz - cosz = 0 and got pi/4. But why is it Pi/4 + kpi and not Pi/4 + k2pi for the part of the complex plane where this function is not analytic. Thanks.
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2answers
42 views

Complex sets: factoring into circle

How must $|z|=3|z-1|$ be factored so I end up with a circle, plugging in $z=x+iy$ seems to just up with square roots everywhere. Detailed steps is much appreciated, thanks!
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1answer
44 views

Need help with understanding exponential form.

I'm looking at this example in my book: $$z = -1 - i$$ The book doesn't explain how it got to the exponential form, which is: $$\sqrt2e^{-i3\pi/4}$$ I understand how $3\pi/4$ was found, but I don't ...