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

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57 views

Application of Rouché's theorem to $e^{z-1}=z$

I am reviewing my complex analysis and I got stuck with an exercise about Rouché's theorem. It states: for $0 \leq C \leq \frac{1}{e}$, show that $Ce^z=z$ has exactly one root in the closed unit disc. ...
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1answer
27 views

Funny interconnection between a triangle and the ellipse inscribed

Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane. Consider the ellipse inscribed ...
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4answers
94 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
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1answer
32 views

Finding all z (complex) that satisfies an equation

I'm having a little trouble with this problem. It's asking to find all $z\in\mathbb C$ that satisfy $z^3 = -2(1+i\sqrt{3})\overline z$, and to keep the answers in standard form. I tried expanding ...
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2answers
65 views

How do I find the real and imaginary parts of $\dfrac{1}{z^2}$? [closed]

Find the real and imaginary parts of $\dfrac{1}{z^2}$ where $z = x + iy$
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1answer
28 views

Can't solve complex equation

Find all $z$ satisfying: $$e^z-2ie^{-z}=i-2$$ I jsut don't have any idea how can one solve it in a simple way. Please help.
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0answers
19 views

integration, anti- derivative, complex [duplicate]

Let $\gamma(w,R)$ denote the circular contour $t\mapsto w+Re^{it}$ where $0\lt t\lt2\pi$. Evaluate $$\int_\gamma\dfrac1{1+z^2}dz$$ when $\gamma$ is: ...
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5answers
68 views

Problem involving cube roots of unity

Given that $$\frac{1}{a+\omega}+\frac{1}{b+\omega}+\frac{1}{c+\omega}=2\omega^2\;\;\;\;\;(1)$$ $$\frac{1}{a+\omega^2}+\frac{1}{b+\omega^2}+\frac{1}{c+\omega^2}=2\omega\;\;\;\;\;(2)$$ Find ...
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0answers
27 views

complex logarithms

Using complex logarithms, how would I solve this $$\left.\frac12i\;\text{Log}\frac{1-i(1+e^{it})}{1+i(1+e^{it})}\right|_0^{2\pi}$$ would it equal; $$ \frac12i[ ln (\sqrt2) + I arg \frac{1-2i}{1+2i} ...
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2answers
31 views

Simplifying $z^3 e^{i\pi/3} +1 = 0 $

Given $$z^3 e^{i\pi/3} +1 = 0 $$ We have, $ z^3 = e^{i2\pi/3} $ I get $$ e^{i\pi/3}z^3 = -1 $$ $$ z^3 = \frac{-1}{e^{i\pi/3}} $$ $$ z^3 = -e^{i2\pi/3} $$ instead May I know how did we ...
4
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1answer
47 views

Complex analysis $\int^{2\pi}_0 \frac{d \theta}{(2-\sin \theta)^2}$

how do I compute $$\int^{2\pi}_0 \frac{d \theta}{(2-\sin \theta)^2}$$ I tried substituting $z=e^{i\theta}$ but it just got very messy..
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2answers
3k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
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1answer
165 views

How to solve: $x^4+x^2=1$

I solved $x^4+x^2+1=0$. But, the above one is hard. The equation is too hard for me to understand. Can anyone solve it? Please help.
1
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1answer
33 views

If $|\alpha|\leq 1$ and $|\beta|\leq 1$, prove that $|\alpha+\beta|\leq |1+\overline{\alpha}\beta|$

Note $\alpha$ and $\beta$ are complex numbers and $\overline{\alpha}$ is the conjugate of $\alpha$. I've tried using variations of the triangle inequality and I couldn't find anything to work.
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2answers
223 views

How to calculate $i^i$ [duplicate]

I've been struggling with this problem, actually I was doing a program in python and did 1j ** 1j(complex numbers) (In python ...
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1answer
21 views

Showing the Summation of $(\frac{w}{2})^k$ where w is a complex root

I got the correct answer for (i) and (ii) and the problem is with third part. I cant find my mistake. Since the third part is related to the second part, I will mention its answer. The ...
0
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0answers
26 views

removable singularity and injective function

Let $U \subset \mathbb{C} $ a conected open subset, $ a \in U $ and $ f:U- \{a\} \to \mathbb{C}$ a holomorphic function such that $ V=f (U-\{a\}) $ is a open bounded subset. (A) Show that $ f $ has a ...
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3answers
49 views

Complex number isomorphic to certain $2\times 2$ matrices?

I have been trying to prove this, but I am having trouble understanding how to prove the following mapping I found is injective and surjective. Just as a side note, I am trying to show the complex ...
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1answer
38 views

Open and closed complex sets

was wondering if someone could shine some light on the highlighted half of this question? Any help would be greatly appreciated. Please excuse me for the poor format of the question, I'm new to this! ...
7
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1answer
145 views

If $f(z)$ is real periodic and $g(z)$ is complex periodic , Can $g(z+f(z))$ be periodic?

Let $A$ be a nonzero real number and let $B$ be a nonreal complex number. Let $z$ be a complex number. Let $f(z)$ and $g(z)$ be non-constant functions defined for all complex numbers $z$ and satisfy ...
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1answer
73 views

Is this claim true$(\xi \circ k)(s)=(k \circ \xi )(s)=0$ $\implies$ $k(s)=\zeta(s)=0 $ is true if and only if RH is false?

It is well known that $\xi(s)=\xi(1-s)$ is a verified functional equation for all complex $s$, where $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. let $k(s)=\xi(1-s)$ and ...
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2answers
56 views

Proving an inequality with the Schwarz inequality

Given a vector space with a Hermitian dot product defined, prove the following inequality using the Schwarz inequality. Let $f$ be a complex value function that is continuous within $0 \le x \le 1$, ...
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1answer
66 views

Find exact value of $\cos (\frac{2\pi}{5})$ using complex numbers.

Factorise $z^5-1$ over the real field. Show that $\cos \frac{2\pi}{5}$ is a root of the equation $4x^2+2x-1=0$ and hence find its exact value. I have worked out that $$ ...
4
votes
3answers
138 views

Proof of $\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\frac12n\theta}{\sin\frac12\theta}\cos\frac12(n+1)\theta$

State the sum of the series $z+z^2+z^3+\cdots+z^n$, for $z\neq1$. By letting $z=\cos\theta+i\sin\theta$, show that $$\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos ...
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3answers
286 views

Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real.

Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real. Your answer should be expressed as a set of the form $S = \{z \in\mathbb C : \text{conditions satisfied by }z\}$. ...
3
votes
3answers
304 views

How to transform the complex number $\frac{(1+i)^{29}}{1-i}$ to the form $a + bi$?

My problem is I have to transform $\displaystyle \frac{(1+i)^{29}}{1-i}$ on its binomial representation $(a + bi)$. I was thinking about transforming that into it polar representation and then ...
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1answer
117 views

Write an expression for $(\cos θ + i\sin θ)^4$ using De Moivre’s Theorem.

Obtain another expression for $(\cos θ + i \sin θ)^4$ by direct multiplication (i.e., expand the bracket). Use the two expressions to show $$ \cos 4\theta = 8 \cos^4 \theta − 8 \cos^2 \theta + 1,\\ ...
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2answers
51 views

How does (cosx+isinx)^4 equate to 1-8 cos^2(x)+8 cos^4(x)-4 i cos(x) sin(x)+8 i cos^3(x) sin(x) [duplicate]

I can't figure out how (cosx+isinx)^4 expands to 1-8 cos^2(x)+8 cos^4(x)-4 i cos(x) sin(x)+8 i cos^3(x) sin(x) I got it equal to sin^4(x)+cos^4(x)+i (4 sin(x) cos^3(x)-4 sin^3(x) cos(x))-6 sin^2(x) ...
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3answers
62 views

Find the four complex zeros without given root. [closed]

$$f(x) = 3x^4-x^3+2x^2-x+3$$ Hint: set $= 0$ and divide each side by $x^2$, use identities equation. Please show me the work.
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1answer
49 views

Proving $(\mathbb{C},\mathbb{C})$ Is Not A Field [duplicate]

Let's $(\mathbb{C},\mathbb{C})$ be a ordered paired of elements form $\mathbb{C}$ when $\mathbb{C}$ is defined as (a,b). addition and multiplication is defined as in $\mathbb{C}$. How do I prove it ...
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1answer
19 views

Notation for a zeta function

What does this notation mean: To provide some context, here are some of the exercises related to it: I initially thought the notation was such that $cos(2pi/n^n)+sin(2pi/n^n)$. This doesn't ...
2
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1answer
32 views

Solve the equation $((x+y i)-\frac{1}{x+y i})/{(2 i)} = 2$ [duplicate]

Solve the equation $$\frac{\left((x+y i)-\frac{1}{x+y i}\right)}{2 i} = 2$$ So far, I got $(0, 2-\sqrt{3}i)$ and $(0, 2+ \sqrt{3}i)$ as solutions for $x$ and $y$. Do I require $2$ more solutions? ...
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1answer
33 views

Finding the argument of a complex number,

I'm trying to locate my four zeroes of a complex-valued function, in order to apply the Residue Theorem. After using the quadratic formula, I am left with $$z^2 = [-3 \pm i\sqrt7] / 2$$ writing the ...
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2answers
23 views

Simple explanation needed for exponential

While going through my professor's notes while calculating integral with branch cut, I came across this relation.It's basic,I guess.So,How this relation come from ...
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1answer
27 views

Some exact values of $\cos \theta$ using de Moivre's theorem

Presently I am faced with the following question: By showing that $$\cos5\theta = \cos\theta(16\cos^4\theta - 20\cos^2\theta + 5)$$ and then solving the equation $\cos5\theta = 0$, deduce that ...
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2answers
39 views

what are the problems with the followings “equations”?

what are the problems with the followings "equations"? A) In the complex number field consider the following: $-1=i^2=(i^4)^{\frac{1}{2}}=1^{\frac{1}{2}}=1$. B) In $\Bbb R$, ...
6
votes
1answer
183 views

$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$

$$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$$ i.e. an oscillation with frequency $3\Im(a)t^2 + 2\Im(b)t + \Im(c)$ and phase $0$, multiplied ...
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3answers
71 views

Finding conjugates of all z $\in$ C that satisfy $z^3$

$$ z^3 = \frac{16e^{i\frac{3\pi}4}}{(1-\sqrt3)+\sqrt6e^{i\frac\pi4}} $$ Anyone know of a good way to simplify this expression?
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1answer
27 views

Holomorphic functions in unitary disk

Let $f:D \longrightarrow{D}$ holomorphic, with $D$ is unitary disk. Show that if $f$ has two fixed point, then $f$ is identity in $D$ I've done: If $f(0)=0, f(a)=a, a\ne0$, as $|f(a)|=|a|$, per ...
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3answers
63 views

Find all the values of $w$ ∈ C that satisfy the equation.

Find all values of $\omega \in \mathbb{C}$ such that $\frac{\omega - \frac1\omega}{2i} = 2.$ So what I have is: Let $\omega$ be represented as $x + yi$ (by definition of complex numbers) Then ...
20
votes
4answers
377 views

How many $\mathbb R$s must a Mathematician walk down?

A mathematician is lost on the complex plane. He knows neither his position nor the direction he is facing. He wants to return to the main road, a strip of width $1$ around the real axis (that is, ...
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2answers
24 views

$\mathbb Q$-linear independence of unit vectors on upper half circle

Let $e(\theta)=(\cos(\theta),\sin(\theta))$ and let $C=\big\lbrace e(\theta) \ \big| \ 0\leq \theta < \pi\big\rbrace$. Trivially, $C$ is not linearly independent over $\mathbb R$ (for example, ...
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2answers
258 views

Complex numbers, polynomials

Let $a$ be complex number such that $a^5 + a + 1 = 0$. What are possible values of $a^2(a - 1)$? I have tried to find $a$. Is there any way to find it?
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1answer
47 views

Complex conjugate of polar form of $z \in \mathbb C$

Express in regular form the conjugates of $z \in c$ that satisfy $z^2 + 4i = 0$ Let $$ z=re^{i\theta} $$ Thus $z^2 = -4i = 4e^{i\frac{3\pi}2} $ $r^2=4, r=2$ $\theta_1 = \frac{3\pi}4, \theta_2 = ...
2
votes
3answers
44 views

Solving equation with complex numbers

My lecturer presented a equation with complex numbers that he simplified by completing the square to the following: $$(z + (i-1))^2 = -3+4i$$ Next he set $$w = z + (i-1) \\ w^2 = -3+4i$$ My first ...
4
votes
1answer
66 views

De Moivre's use to a complex number

I have this question which I'm stuck on, here's the question and what I did. Find the smallest positive integer m such that $\left(\sqrt{3}+i\right)^m=\left(\sqrt{3}-i\right)^m$. I expanded out each ...
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1answer
39 views

Find the principal value of $z=\ln\left(i\tanh\left(\frac{\pi}{2}\right)\right)$

Steps that I have taken: Substitute $\tanh(π/2)$ with $C$. Then we have $\ln(i\cdot C)=\ln|c| +i(\arg z)$. I am desperately stuck here. I also tried expressing $\tanh$ by definition using the powers ...
0
votes
2answers
589 views

Proving equations involving the powers of a complex cube root of unity ω

The question in this homework problem is to show $ω^4 + ω^5 = -ω^6$ given that $ω$ is a complex cube root of unity. I am also required to show that $(1 - ω)^2 = -3ω$, but if I am assisted with ...
1
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1answer
32 views

Finding roots in marginally stable system modeled by complex number

A system can be modeled by $(z + 3)(z + 2)(z + 1) + C = 0$, where $C > 0$, and $z = x + iy$. When it is marginally stable $Re(z) = 0$. What are the values of the roots in marginally stable ...
1
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1answer
85 views

Show that for $z $ a complex number, there exists a complex number $\alpha $, wiht $|\alpha |=1$ such that $\alpha z = |z |$

How can I show that for $z $ a complex number, there exists a complex number $\alpha $, wiht $|\alpha |=1$ such that $\alpha z = |z |$ Thanks in advance!