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

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3answers
65 views

Find all $z \in \mathbb{C}$ that satisfy $z^3 = −2(1 + i\sqrt{3})\bar{z}$

Find all $z \in \mathbb{C}$ that satisfy $z^3 = −2(1 + i\sqrt{3})\bar{z}$. You must express your answers in the standard form. So far, I'm thinking of writing $z = a + bi$, but then I have to ...
0
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3answers
55 views

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ in my lecture notes somehow my lecture got from$-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ to ...
1
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1answer
57 views

Estimating the modulus of the roots of $\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$

If $θ_1,θ_2,θ_3,θ_4$ are four real numbers, then any root of the equation $$\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$$ lying inside the unit circle $\vert z\vert$=1, satisfies which inequality? ...
3
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2answers
75 views

Determine whether ${\dfrac{2+i}{2-i}}$ is a root of unity

I need to determine whether ${\dfrac{2+i}{2-i}}$ is a root of unity. At first, I expressed this number as ${\dfrac{3}{5}+\dfrac{4}{5}i}$. Then I tried to use a formula for $\sin{nx}$, where x = ...
1
vote
1answer
33 views

equation draws a circle in C

It be $a\in\Bbb C , b ∈\Bbb R$ and $|a|^2 > b$. Also, $a'$ is the conjugation of $a: a' = x - iy$ when $a = x + iy$ (and equally for $z$). It needs to be shown, that the solutions of the equation: ...
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0answers
35 views

What's the name of the form (123i + 321)

Okay, so $0.5$ can be written as a fraction $\frac {1}{2}$. Is there an official name for writing a number in the form of $ai + b$? Complex numbers could be written in this form $z = a\ e^{i ...
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1answer
14 views

what will be the PDF of the magnitude of this random variable x+j y?

if we have a complex random variable [x+j*y] where (j :sqrt(-1)) and x,y both have Gaussian distribution and statistically dependent , so what will be the distribution (PDF) of the magnitude of this ...
4
votes
2answers
147 views

the first $2k$ terms of the power series of $\sec x + \tan x$ at $x=-\pi/2$

We know the power series of $\sec x+\tan x$ is as follows, $f(x)=\sum_{n\geq 0}\frac{E_n}{n!}x^n$, where $E_n$ is Euler Zigzag numbers and clearly the radius of convergence of $f(x)$ is $\pi/2$. ...
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1answer
46 views

Path integral in the complex plane

Evaluate $\int_Tz\,\mathrm dz$ and $\int_T\overline z\,\mathrm dz$ where $T$ is the triangle with vertices $0,1,-i$ oriented clockwise. I am trying to solve this question, but I'm unsure how to ...
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2answers
53 views

Find 2 imaginary numbers that have a cosine of 4, using $\cos z =\frac{e^{iz}+e^{-iz}}{2}$

Use the definition $$ \cos z =\frac{e^{iz}+e^{-iz}}{2} $$ to find $2$ imaginary numbers having a cosine of $4$. I tried two approaches, both of which ended in failure: $$ 8=e^{iz}+e^{-iz}\\ ...
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0answers
22 views

Solve complex equation with exponentials

I have to solve: $e^z+2i=2e^{-z}$ I multipply both sides by $e^z$ and have: $(e^z)^2+2ie^z-2=0$ Now substitute $x=e^z$. $x^2+2ix-2=0$ $\Delta = -4+8=4$ so $x_1=-i-1$ and $x_2=-i+1$ Now go back ...
3
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2answers
54 views

Solving $(1-x)^3 = -1$ over the complex field

What are the solutions of: $(1-x)^3 = -1$ over $\mathbb{C}$? We have one real solution which is $2$ so there are two complex solutions.
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2answers
48 views

If $(1-i)^n = 2^n$ , then find $n$.

If $$(1-i)^n = 2^n$$ then find $n$. If anything raised to $0$ is $1$, but according to my book $ n \ne 0$. Is the print wrong?
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0answers
33 views

Quadratic formula and complex numbers

Let $az^2+bz+c=0$ be a complex quadratic equation. We know that it has $2$ roots: $z_1=\frac{-b-\sqrt{b^2-4ac}}{2a}$ and $z_2=\frac{-b+\sqrt{b^2-4ac}}{2a}$ If $b^2-4ac=1+i$ for example we have to ...
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1answer
42 views

synthetic division with $i$ in divisor

I divided $x^3-4x^2+4x-16$ by $-2i$ using synthetic division and got a remainder of $-8i-8$. Is that right? I'm not sure I'm doing this right.
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0answers
14 views

How to express vectors with more than 2 components in complex coordinates

It is straightforward to extend the notion of a 2D vector in the Cartesian x,y plane to 3D (x,y,z) or to any D. Sometimes it is useful to express vectors in the complex plane, where the 2D vector has ...
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1answer
23 views

Changing exponent sign

Sorry for the bad title, I am not sure how do I name it. Find all the roots that satisfy $z^4$ $$z^4 =\frac 12 e^{-i{\frac π7}} $$ $$z^4 = \frac 12 e^{i{\frac {13\pi}7}} $$ Therefore, the roots are. ...
2
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2answers
65 views

What does it mean for a function to be holomorphic?

I am trying to wrap my head around the definition of holomorphic. Wikipedia tells me that: A holomorphic function is a complex-valued function of one or more complex variables that is complex ...
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1answer
14 views

$e^z=-3i$ find $z\in \mathbb C$ check my answer

I am unsure of my solution to this question, since the definition of the complex logarithm is somewhat complex. Since $-3i = 3e^{i\frac{3}{2}\pi}$ we get that $e^z=3e^{i\frac{3}{2}\pi}$ So if we use ...
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1answer
37 views

Null Space of Transformation

I am given that $V$ is n-dimensional vector space over $\mathbb{C}$ and $T \in L(V)$. And $T$ has least $m$ distinct nonzero eigenvalues. How do I show that $\text{null}(T^{n-m}) = ...
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2answers
49 views

How to solve $|z^2-1|<|z|^2$ where $z$ is a complex number?

How to solve $|z^2-1|<|z|^2$ where $z$ is a complex number? I have tried it both with cartesian and polar coordinates but did not get a solution. I got that far: $z=x+yi$ and then I got: $$\pm x ...
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4answers
75 views

Ordering of the complex numbers

The complex numbers as a whole cannot be ordered but could you order the complex numbers of the form ai where a is a real number?
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0answers
51 views

What are the cases when the power result is a complex number? [duplicate]

Reference question - Make $a^b$ to have a complex answer Considering I have $a ^ b$ where both are real numbers and that the complex result is achieved in case when $$a<0 \wedge b = ...
5
votes
2answers
64 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
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0answers
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
23 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 ...
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0answers
24 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 $f$ ...
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0answers
15 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 ...
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2answers
45 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 ...
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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: ...
0
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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} ...
2
votes
1answer
111 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
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 ...
2
votes
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 ...
5
votes
4answers
95 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
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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 ...
1
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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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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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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 ...
0
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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! ...
1
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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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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 ...
0
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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 $$ ...
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
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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votes
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) ...
1
vote
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,\\ ...
-2
votes
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.