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

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86 views

Minimizing rounding error on calculation of Complex Atanh

I am trying to implement the calculation of the complex inverse tangens hyperbolicus function in my program in VB.NET. I am using the formula : $\arctan(z)=\dfrac{1}{2} {\rm ...
0
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1answer
2k views

How do I find transformation matrix with respect to standard basis?

I know that in order to find transformation matrix with respect to a basis, I need to apply the transformation to said basis and the result is the column of the transformation matrix. But what ...
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2answers
854 views

Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
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0answers
44 views

Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
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4answers
671 views

Integers $n$ such that $i(i+1)(i+2) \cdots (i+n)$ is real or pure imaginary

A couple of days ago I happened to come across [1], where the curious fact that $i(i-1)(i-2)(i-3)=-10$ appears ($i$ is the imaginary unit). This led me to the following question: Problem 1: Is $3$ ...
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2answers
495 views

Laurent series for $(\sin 2z)/z^3$

I have to find the Laurent series for $(\sin2z)/z^3$ in $|z|>0$, but I really don't know how to start. And I thought that in this area it's a Taylor serie because the singularity isn't in the area, ...
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1answer
59 views

Calculate the inverse of a complex matrix

I am trying to calculate the inverse of a given matrix but somewhere I have an error in my calculation that I cannot find $$\begin{array}{ccc} && \left( \begin{array}{ccc|ccc} 1-i & 2 ...
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5answers
196 views

Find all real values $a$ and $b$ such that $a+ib=i^{i^{i}}$?

Find all real values $a$ and $b$ such that $a+ib=i^{i^{i}}$ ? My Try : using the fact that $z^w=e^{w \log z}$, First I compute $i^i$ $$i^i=e^{i \log i}= e^{i (\log |i| + i arg (i))}=e^{- ...
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1answer
37 views

Calculating j - a complex number

Using a shortcut to get the magnitude for a filter, to get the magnitude at 0.25Fs you can replace z (from the transfer function) with 'j' - a complex number derived from: $$ {e^{j(Pi/2)}}$$ (where ...
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1answer
77 views

area of rectangle whose vertices are roots of equation $z\overline{z}^3+ \overline{z}z^3=350 $

We have a complex number $z=x+iy$ where $i=\sqrt{-1}$ and $\overline{z}$ represents conjugate $$z\overline{z^3}+ \overline{z}z^3=350 $$ so i proceeded by taking $z\overline{z}$ common thus ...
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2answers
213 views

z^4-16z*^2=0 in complex numbers

I'm a huge newbie with complex numbers, I ran this through WolframAlpha: z^4-16z*^2=0 and got this Real Solutions (I don't even know what that means) $z=-4$, $z=0,z=4$ and Complex Solutions (again, ...
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9answers
3k views

Why is $i^3$ (the complex number “$i$”) equal to $-i$ instead of $i$? [duplicate]

$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i $$ Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?
3
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1answer
49 views

prove statement. cyclic sum complex numbers

If $(a+b)^3 = (b+c)^3 = (c+a)³\,\,\,\,a,b,c\,\,\in\mathbb{C},\,\,a\neq b\neq c$, show that $a^3 = b^3 = c^3$. Some hints would be great.
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2answers
121 views

Polar form of z = 12i

I only know a dab about complex numbers so I was reviewing here for group theory. I don't understand the first paragraph which isn't explained in detail. Second, how does the sentence marked with the ...
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2answers
55 views

Complex Matrix Limit

If $A$ is an $n \times n$ complex matrix, show that if $\lim_{k\rightarrow\infty}||A^kv||=0$ for every vector $v \in \Bbb C^n$, then $|\lambda|\leq1$ for every eigenvalue $\lambda$ of $A$.
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2answers
39 views

Algebra of the complex plane

Suppose $a$,$b$,$c$,$d$ be the points of set of all complex no$(C)$. with $c$ not equal to $0$ and $ad$ not equal to $bc$.$f$ be a function such that $f(z)$=$(az+b)$/$(cz+d)$. how to prove that $f$ ...
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2answers
125 views

Laurent Series Difficulty

Hello all at StackExchange, I'm having some trouble understanding computing the Laurent series for different domains. Here's my approach to finding the Laurent series for $\dfrac{3}{(z+1)(z-2)}$ for ...
3
votes
2answers
129 views

Existence of holomorphic function with a sequence of zeros in the unit disc

The question is : Prove that there exists a holomorphic function $f$ on the open unit disc $\{z \in \mathbb{C} : |z| <1\}$ with the properties that $f(0) = 0$ and $f(1-1/n)=1$ for every integer $n$ ...
3
votes
3answers
108 views

Finding the Sum of the square of two positive integers.

Write the following equation as the sum of the square of two integers, $a^2 +b^2$. $$(8^2+5^2)(13^2+7^2)$$ I remember that you are supposed to do something with complex numbers or at least that ...
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1answer
32 views

For which values ​​of $z$ the inequality $|e^{z-1}|<2$ holds

I want to find for which values of $z$ the following inquality holds $$|e^{z-1}|<2$$ what I tried to do is: $$|e^{z-1}|=|e^{x-1+y\mathbb{i}}|<2$$ $$=e^{x-1}\cdot(\cos(y) + \mathbb{i} \sin(y))$$ ...
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3answers
119 views

Find all values of $z=\ln(\sqrt{3}+I)$ where $|z|<6$

I want to find all the values of the following: $$z=\ln(\sqrt{3}+i), \quad |z|<6$$ I understand that is all the values that found inside the circle $|z|<6$ what I did so far : If $\zeta = ...
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1answer
54 views

Multiply complex numbers to show trigonometric addition formulas

Use the rules for multiplication of two complex numbers written in the form $r(\cos\theta +i\sin\theta)$ to show that $\sin(\theta_1 +\theta_2)=\sin\theta_1\cos\theta_2 +\sin\theta_2\cos\theta_1$ ...
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2answers
218 views

Factor the polynomial $x^3 − 27$ using De Moivre's theorem (Please explain solution)

I was reading the book A First Course in Linear Algebra by Ken Kuttler (link to nearly identical page http://librarum.org/book/312/11) and I did not understand this part: Q: Factor the polynomial ...
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4answers
233 views

What are the complex solutions of $z^4+16=0$?

What are the complex solutions for $z^4+16=0$? I know that one solution is $z=a+bi=-2$. How can I figure out the other solutions?
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6answers
227 views

What is $(-1)^{\frac{2}{3}}$?

Following from this question, I came up with another interesting question: What is $(-1)^{\frac{2}{3}}$? Wolfram alpha says it equals to some weird complex number (-0.5 +0.866... i), but when I try ...
0
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1answer
42 views

Describe $\text{Re} \frac{1}{z}=1$, $z$ Complex Number

I would like to describe this equation on the XY Plane, where $z$ is a Complex Number $$\text{Re}\left( \frac{1}{z}\right)=1$$ What I did is to get the Real part: $$z=x+y\mathbb{i} ...
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1answer
75 views

Solve for $z$ in equation $|z-a|+|z+a|=2|c|$

I'm trying to solve for $z$ in the equation $$|z-a|+|z+a|=2|c|.$$ My idea is to square both sides $$|z-a|^2+|z+a|^2+2|(z-a)(z+a)|=4|c|^2$$ Using $|z|^2=z\bar{z}$, this becomes ...
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1answer
455 views

To calculate residue of the function $f(z) = \frac{z^2 + \sin z}{\cos z - 1}$.

I was trying to find the residue of the function $$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$ Here is the my attempt: The given function has a pole of order two at $z = 2n\pi$. So, we use the ...
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2answers
123 views

The roots of the equation $z^n=(1+z)^n$…

The complex roots of the equation $$z^{n}=(1+z)^{n}$$ $A.$ are vertices of a regular polygon $B.$ lie on a circle $C.$ are collinear $D.$ none of these Don't know where to start.. Please ...
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3answers
71 views

Minimum of |z|+|z-cos a -i sin a|…

For a complex number $z$, the minimum value of $|z|+|z-\cos(a)-\iota \sin(a)|$ is: $A. 0$ $B. 1$ $C. 2$ $D.$ None of these The answer is option $B$. I see that the given expression is the sum of ...
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3answers
97 views

f(x) and g(x) are two polynomials, then choose the right option…

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=xf(x^{3})+x^{2}g(x^{6})$ is divisible by $x^{2}+x+1$, then choose the correct option: $A. f(1)=g(1)$ $B. f(1) $ is not equal ...
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4answers
107 views

Prove that for vectors $v_1,…,v_n$ in $\mathbb C^n$, $\{v_1,…,v_n\}$ is a basis for $\mathbb C^n$ iff its conjugate is a basis for $\mathbb C^n$

Prove that for vectors $v_1,...,v_n$ in $\mathbb C^n$, $\{v_1,...,v_n\}$ is a basis for $\mathbb C^n$ if and only if $\{\bar v_1,..., \bar v_n\}$ is a basis for $\mathbb C^n$. I know intuitively that ...
1
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1answer
93 views

$\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$

I want to prove: $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$. Is there any direct way to prove? I have computed that the splitting field of $x^7-12$ ...
1
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1answer
147 views

Integral of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$F(z)=\sum_{n=1}^\infty a_nz^n$$ converges in $|z|<1$. How can I ...
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3answers
337 views

Help with hard complex numbers

We had the topic of complex numbers for my senior math team meet this week and i wasn't able to get two of the problems. 1.) $z=i^{\displaystyle \left(i^{\displaystyle ...
3
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1answer
120 views

problem about complex integration

The question is to find $\displaystyle\int \frac{z^2-z+1}{z-1}dz$ over $|z|=1$. My solution is : Using cauchy's integral formula we have $\displaystyle f(1) = \frac{1}{2\pi i}\int ...
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3answers
44 views

Set with complex numbers

Let $\alpha \in \mathbb{R}$ and $ c \in \mathbb{C} \backslash \{0\}$. Determine the set $$M := \{z \in \mathbb{C} | Re (\overline c z) + \alpha = 0 \}$$ and sketch it for $\alpha = 2, c=1+3i$ and ...
3
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2answers
37 views

Convergence of $\sum_{n=2}^{\infty}n^2\left(\frac{1-i}{2+i}\right)^n$

Does this sequence converge/diverge and if so, does it in a (not)absolute way? $$\sum_{n=2}^{\infty}n^2\left(\frac{1-i}{2+i}\right)^n$$
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1answer
260 views

Complex Number , square root

Given $(x+iy)^2 = 8+6i$, find the values of $x$ and $y$. Hence find $\sqrt{8+6i}$. My question is when we solve we get $x = 3$ and $x = -3$, which give and $y = 1$ and $y = -1$ then Why $-3-i$ is ...
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2answers
285 views

What Number Set Contains The Subset of Complex Numbers? Is there even such a set?

Basically what I'm asking is what set are complex numbers inside of? Surely there must be a set that encompasses complex numbers and so on. In my pre-calculus book from senior year high school the ...
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1answer
69 views

A question on Harmonic functions.

$f$ and $g$ are two given continuous functions on $\overline {D(0,1)}$) and both $f$, $g:\overline {D(0,1)}\to\Bbb R$ are harmonic on unit disc $D(0,1)$. Now, If $f\equiv g$ on some $C=\{e^{i\theta} ...
2
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2answers
45 views

question related with cauch'ys inequality

We have a statement that if f(z) is analytic in a domain D and $C = \{z : |z-a| = R\}$ is contained in D. Then, $|f^n(a)|\leq \frac{n!M_R}{R^n}$ where $M_R = \max|f(z)|$ on C. What if the given D ...
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1answer
35 views

How can I find the complex solutions to a polynomial?

Given a polynomial such as: $x^3 + 8i$ How can I solve this? The first obvious step is to move the $8i$ over, so you get: $x^3 = -8i$ From there, I need to create a complex number of some sort. ...
3
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1answer
68 views

Convergence of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ If $z=re^{i\theta}=x+iy$, $$F(z)=\sum_{n=1}^\infty r^na_ne^{in\theta}$$ is a harmonic ...
3
votes
2answers
57 views

doubt in complex integration

I was doing problems on complex integration and got stuck at one question. The question is $$ \int_{\gamma}{{\rm e}^{{\rm i}\pi z}\left(z + i\right)^{2}\cos\left(nz\right) ...
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2answers
35 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
4
votes
2answers
220 views

Use $\alpha, \beta, \gamma $ roots of a polynomial to construct another polynomial [duplicate]

Let $\alpha, \beta, \gamma $ be roots $\in \mathbb{C}$ of $x^3-3x+1$. Determinate a monic polynomial, degree $3$, witch roots are $1- \alpha^{-1},1-\beta^{-1},1-\gamma^{-1}$ The catch is that i can't ...
5
votes
1answer
2k views

Is L'Hopitals rule applicable to complex functions?

I have a question about something I'm wondering about. I've read somewhere that L'Hopitals rule can also be applied to complex functions, when they are analytic. So if have for instance: $$ \lim_{z ...
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2answers
110 views

Why is this result of the Cauchy-Goursat theorem true?

One of the results of the Cauchy-Goursat theorem is that for any simple closed countour $C$ that contains the point $z_0$: $$\oint_C{\frac{dz}{(z-z_0)^n}} = \begin{cases}2\pi i & n=1 \\ 0 & n ...
4
votes
2answers
215 views

Evaluating $\sum_{j\geqslant1}\sum_{k\geqslant1}(-1)^{k+j}\frac{(2k-1)+i(2j-1)}{((2k-1)^{2}+(2j-1)^{2})^{3/2}}.$

After a test I've taken, I considered an infinite grid of eletric charges and wondered the resultant force at the origin. The origin has a charge $+1$ and every gaussian integer $a+bi$ in the first ...