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

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-2
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1answer
120 views

Find all z $\in \mathbb{C}$ that satisfy $z^3 = -2\left(1+i\sqrt{3}\right)\bar{z}$ [closed]

Find all z $\in \mathbb{C}$ that satisfy $z^3 = -2\left(1+i\sqrt{3}\right)\bar{z}$. You must express your answer in standard form. Hi, I tried letting $z = x+yi$ and $\bar{z}=x-yi $ but the steps ...
5
votes
2answers
63 views

Solve $e^{z-1}=z$ with $|z| \leq 1$

I'm looking for solutions to $$e^{z-1}=z$$ when $z \in \mathbb{C}$ with $|z| \leq 1$. The obvious solution is $z=1$, but I don't know how to show that there aren't any others. This question is ...
1
vote
0answers
56 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. ...
0
votes
1answer
18 views

How can this equation be simplified this way? Transmission line: Zin

I thought of putting this on the Electrical Engineering Exchange but I thought since this seems more mathematical than related to engineering I thought I should place it here instead. Question: Why ...
1
vote
0answers
19 views

Holomorphic funtions

Let $U$ be an open connected subset of $\mathbb{C}^n$, and $O(U)$ the ring of holomorphic functions on $U$. Prove that $O(U)$ is an integral domain. I have done If $fg\equiv0$ in $U$, then $f$ ...
0
votes
0answers
14 views

Converting sum of complex exponential to sum of cosine

So I am trying to convert the equation $$\sum_{k=-2}^2 \alpha_k e^{i \frac{2 \pi}{T_0} kt}$$ Where $\alpha_0 = 1$, $\alpha_1 = 2 \angle30^\circ$, $\alpha_{-1} = 2 \angle{-30^\circ}$, $\alpha_2 = 1 ...
0
votes
2answers
40 views

Sum function operation: coefficient.

I have problem with the sum: $$ \sum_{k=0}^n \dbinom{n}{k}(\cos \alpha)^k(i\sin \alpha)^{n-k}\,\, $$ Apparantly, I have an imaginary unit therefore I need to distinguish even and odd powers of $i$ to ...
1
vote
2answers
56 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 $z = x + iy$
0
votes
1answer
27 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.
0
votes
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: ...
0
votes
0answers
25 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} ...
2
votes
1answer
110 views

Make $a^b$ to have a complex answer [closed]

Considering I have $a ^ b$ where both are real numbers, for which values of $a$ and $b$ I will have a complex answer $(m+n*i)$. I figured out that one case is when $a<0$ and $b \in (0, 1)$. Any ...
3
votes
5answers
58 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 ...
2
votes
2answers
28 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 ...
5
votes
4answers
84 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 ...
1
vote
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 ...
1
vote
1answer
23 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 ...
1
vote
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.
0
votes
0answers
21 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 ...
0
votes
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! ...
1
vote
3answers
46 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 ...
7
votes
2answers
211 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 ...
0
votes
1answer
63 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 $$ ...
3
votes
1answer
53 views

Find the possible values of |A + B + C |

$ |A |= |B | = |C | = 1 $ ,where A B and C are complex nos $$ \frac{A^2}{BC}+ \ \frac{B^2}{ \ {CA}} \ +\ \frac{C^2}{ \ {AB}} + 1 = 0$$ Find the possible values of $ |A + B + C |$ Tried ...
4
votes
1answer
44 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..
-1
votes
2answers
49 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) ...
1
vote
1answer
114 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,\\ ...
-2
votes
3answers
54 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.
3
votes
3answers
303 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 ...
1
vote
1answer
152 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.
0
votes
1answer
44 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 ...
1
vote
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 ...
1
vote
3answers
283 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 $S = \{z \in\mathbb C : \text{conditions satisfied by }z\}$. ...
2
votes
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? ...
2
votes
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 ...
0
votes
1answer
53 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 ...
1
vote
1answer
210 views

The image of a map $H^2 \setminus \Delta \to \mathbf{R}^4$ where $H$ is the upper half-plane and $\Delta$ is the diagonal

Let $H = \{z \in \mathbf{C}: \operatorname{Im} z > 0\}$ be the upper half-plane, and let $D = H^2 \setminus \Delta$, where $\Delta = \{(u,u) \in H^2\}$ is the diagonal. Define $\varphi: D \to ...
1
vote
1answer
23 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 ...
0
votes
2answers
36 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 $\Bbb R$, ...
1
vote
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?
0
votes
1answer
25 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 ...
1
vote
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 ...
0
votes
3answers
61 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 ...
1
vote
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, ...
18
votes
4answers
355 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, ...
1
vote
1answer
44 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
39 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 ...
1
vote
1answer
31 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 ...
14
votes
2answers
251 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?
1
vote
1answer
63 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!