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

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1answer
22 views

Roots of a quintic function

I need some pointers in the right direction for this question: Three of the roots of the equation $ax^5+bx^4+cx^3+dx^2+ex+f=0$ are $-2$, $2i$ and $1+i$. Find $a$, $b$, $c$, $d$, $e$ and $f$. I ...
1
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3answers
57 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 ...
2
votes
1answer
69 views

About the identity $e^{i\pi}=-1$

I have a question about the famous identity of Euler $e^{i\pi}=-1$. I opened the other day this question about the number of roots of a complex number with irrational exponent. Under this light and ...
3
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1answer
25 views

Every compact set $S\in \mathbb{C}$ is bounded

This is my proof for every compact set $S \subseteq \mathbb{C}$ is bounded. Let $S \subseteq \mathbb{C}$ be compact and assume that it is not bounded. Then for each $z\in \mathbb{C}$ and for each ...
0
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1answer
58 views

Given $w=e^{2i\pi/7}$, perform algebraic manipulations with complex numbers like $w+w^2+w^4$

It is not my own homework and I forgot how to solve this kind of things. Anyway, the following are the statement of the homework and my attempts: The homework: Let $w=e^{2i\pi/7}$, $u=w+w^2+w^4$, ...
2
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3answers
34 views

Cross Ratio is positive real if four points on a circle

Given four points $a, b, c, d \in \mathbb{C}$ on a circle I want to show that the ratio $ \frac{(a-c)(b-d)}{(a-b)(c-d)}$ is a positive real number. I have shown that it is real by expanding $ ...
2
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1answer
22 views

Evaluating $|a^b|$ when $a,b$ are complex

Here, $a^b=e^{b\log a}$ for some suitable (but fixed in advance) branch of the $\log$ function. What is the most general formula for $|a^b|$ when both $a$ and $b$ are complex, and what are the ...
2
votes
1answer
15 views

Variance of amplitude and phase from sin and cos regressors in polar coordinates

On a data set, I estimated the sine and cosine weights at a specific frequency, $\beta_{\sin}$ and $\beta_{\cos}$. I can extract the amplitude and phase from these regressors as follows: ...
0
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1answer
28 views

Finding which base number given operations

$$ (35_a + 24_a) * 21_a = 1081_a $$ Which base is the above number? Any advice on how to solve questions like these? I tried making it in to a polynomial: $(3a+5 + 2a+4) * (2a+1) = 108a + 1$ ...
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2answers
22 views

Calculating the argument of a complex number… something tends towards infinity?

A simple question, but I like to be clinical with my choice of words: I have a complex number, $z=-i$. If I were to calculate the argument of this complex number, $arg(z) = tan^{-1}( \frac{-1}{0}) ...
1
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1answer
38 views

$D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact

This is the proof I wrote for $D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact. $$ \bigcup_{n=2}^{\infty}D_{1-(1/n)}(0) $$ is clearly a open covering of $D_1(0)$. Consider the finite ...
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0answers
24 views

Complex trigonometric equation

Find a solution to the equation $\tan(z)=7i$ which satisfies the condition $0<\Re(z)< \pi$} We use the $\sin(z)=\frac{e^{zi}-e^{-zi}}{2i}$ and $\cos(z)=\frac{e^{xi}+e^{-xi}}{2}$. Here is ...
-1
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1answer
68 views

Prove that the sum of all distinct $n$-th roots of unity is zero? [on hold]

Prove that the sum of all distinct $n$-th roots of unity is zero. Interpret the fact geometrically.
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0answers
20 views

Proving Ptolemy Theorem using complex number

I am working on an assignment proving Ptolemy Theorem using complex number, and I am looking at a textbook Complex Numbers and Geometry by Hahn. Here is what I am working at this moment: THE ...
0
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4answers
56 views

Why has my prof worked out the multiplicative inverse of complex numbers this way?

She explains how to obtain the multiplicative inverse: In what follows, z=a+bi and w=c+di are complex numbers with a,b,c,d∈R. Is there a multiplicative inverse of z? If so, what is it? Note ...
1
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1answer
50 views

Definition of $z^0, z=a+bi, a,b \in \mathbb{R}, z \neq 0$

Today I found in a True-False the question; Does the equality $$z^0=1, z=a+bi, a,b \in \mathbb{R}$$ hold $\forall z \in \mathbb{C^*}$? The thing is, this was never clearly defined in the book, and ...
0
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3answers
33 views

How do you divide complex numbers in polar form?

A question in my textbook asks: Find $\frac{z_1}{z_2}$ if $z_1=2\left(\cos\left(\frac{\pi}3\right)+i\sin\left(\frac{\pi}3\right)\right)$ and ...
0
votes
1answer
38 views

How can the polar form of a complex number be$ r(\cos(x)-i\sin(x))$?

I don't understand how this can form can work: $r(\cos(x)-i\sin(x))$. I saw it in my textbook. Surely there should be a "$+$" rather than a "$-$" between the $\cos$ and the $\sin$. If the imaginary ...
0
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1answer
26 views

Prove that $|z||b-ad| \leq M $

I need to prove the following statement: $$ |z||\frac{az + b}{z+d}-a| <= M $$ with $a,b,c,z \in \mathbb{C}, |z| \geq 1 + |d|$ and $M\geq 0$. I have reduced this to $$ |z||b-ad| \leq M $$ Also $ad ...
1
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1answer
21 views

The annulus $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open

I want to prove that the set $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open. This is my attempt. Let $z \in A_{r,s}(z_0)$. Then $|z-z_0|-r>0$. Let $r'=[|z-z_0|-r]/2$. Then ...
0
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0answers
20 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...
1
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1answer
32 views

How many roots have a complex number with irrational exponent?

If a rational exponent on a complex number $z^q$ is the representation of a finite number of roots, then if the exponent is irrational this mean that there is infinite countable roots? If this is the ...
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0answers
15 views

Calculate $\frac{1}{2\pi}\int_{-\pi}^{\pi} {\rm{sinc}}(\alpha-t)\ e^{-\imath z t} dt$

Let us consider the $ {\rm{sinc}}$ normalized function: \begin{equation} {\rm{sinc}}(\tau)= \begin{cases} \frac{ \sin(\pi \tau)}{\pi \tau} \qquad &\tau \not= 0,\\ 1\qquad & \tau=0, ...
0
votes
2answers
50 views

How find this $a$ such $(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$

Question: find all the complex $a$,such for every complex $z_{1},z_{2}(|z_{1}|,|z_{2}|<1,z_{1}\neq z_{2})$,such $$(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$$ My idea: ...
2
votes
1answer
24 views

Quadratic formula with complex coefficients

Let $a,b$ and $c$ be complex numbers. I'm trying to prove that this version of the usual quadratic formula: $$z=\frac{-b+(b^2-4ac)^{\frac{1}{2}}} {2a}$$ solves the quadratic equation ...
2
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1answer
36 views

Ring homomorphism $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$

Exercise Find all the ring homomorphisms $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$ My attempt at a solution In a previous problem I've already proved that the only ...
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0answers
22 views

Can a complex number have two arguments

Now, the reason why I wrote two $\theta$s is because my answer is the answer we get from $\theta$2 and the answer in the book is given the value of $\theta$1. So, I was just wondering whether both ...
0
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0answers
34 views

Square root of complex number?

Real numbers usually have more than one square root, but when we write sqrt(x), we mean the principal square root of x. What about complex numbers? What root are we referring to when we write ...
2
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0answers
52 views

If the sum of absolute values of complex numbers is at least $1$, then some subset of these numbers has absolute value at least $C$

There is a challenging problem in a book of mine on complex analysis, and I seriously do not even know where to start. I'm more than sure I don't properly understand the problem. Prove that there ...
3
votes
3answers
128 views

Find solution of equation $(z+1)^5=z^5$

I attempt to solve the equation $(z+1)^5=z^5$. My first approach is to expand the left hand side but ı get more complicated equation. So I couldn't go further. Secondly, I write equation as, since ...
0
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2answers
28 views

A cute little system of nonlinear PDEs

I am wondering about the solutions to the following system of PDEs. Suppose we have functions a(x,y,z), b(x,y,z), and c(x,y,z) and the following equations: ...
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0answers
11 views

Complex Matrix Orientation

I recently learned about the fact that a linear mapping of a real vector space is orientation-preserving if the determinant of the matrix is positive. Now I was wondering if there exists a similar ...
1
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2answers
44 views

Quadratic Formula for complex variable with real coefficients

I have been upto proving the following $$(\forall x\in \mathbb C, ax^2 + b x + c = 0) \land(a\neq0)\Leftrightarrow (x = {\frac {-b \pm \sqrt{b^{2}-4ac}} {2a}})$$ Due to equality we need to proof bi ...
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3answers
36 views

Calculate the absolute value of $z=(10+5i)(1+10i)(4+2i)(5+2i)$

I am doing some repetition of complex numbers and I got to this question: Calculate the absolute value of $z=(10+5i)(1+10i)(4+2i)(5+2i)$ My approach has been to first multiply the imaginary numbers ...
0
votes
1answer
27 views

Is an absolute value acting on complex numbers a linear operator?

I just have to prove that it isn't with O(A+B)=O(A)+O(B) and O(kA)=k(OA) where O is the linear operator (i.e the absolute value), A+B and A would be a complex number, and k is some real constant. I ...
1
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2answers
36 views

Trignometric problem (using De Movier's Theorem)

Ok so this question, I started out writing tan as sin and cos in the right side of the equation, simplified as much as possible and ended up with a very (sort of) fascinating equation which is ...
4
votes
3answers
65 views

Why is Euler's formula defined for non-integer values?

Say that for some complex number $w$ $$e^{wi} = a$$ Now raise both sides to $1/4$. $$e^{wi/4} = a^{1/4}$$ Now $e^{wi/4}$ has a single defined value. Yet $a^{1/4}$ can have multiple values. So why ...
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0answers
40 views

Solving complex integral

I have the next integral: $$\int_{-\infty }^{0}\left |e^{-iw_1 x}+\frac{w_1-w_2}{w_1+w_2}e^{iw_1 x} \right |^2\mathrm{d}x$$ where $w_1$ and $w_2$ are real constants. After some algebraic process: ...
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1answer
24 views

Generating the special linear group of 2 by 2 matrices over the integers.

Our Number Theory professor claimed that the special linear group $\text{SL}_2(\mathbb{Z})$ is generated by just two matrices: $$ M_1=\begin{pmatrix} 0& -1\\ 1& 0 \\\end{pmatrix} $$ $$ ...
4
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1answer
73 views

Why is Euler's formula valid for all $n$ but not De Moivre's formula?

The Wikipedia page on De Moivre's Formula says the formula doesn't hold for non-integer $n$, since non-integer powers of a complex number can have multiple values. It then goes on to say that this ...
0
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1answer
52 views

How can I prove this equation?

My prof put this equation on the board, without any kind of explanation or proof. When I asked him for one, he didn't really give me a solid answer. $$w = r(\cos \theta + i\sin \theta)$$ Then $w^n ...
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0answers
21 views

hyperbolic series $\sum_{r=1}^n \cosh(rx)$

I have attempted to do this question on hyperbolic functions: Prove that $$\cosh x + \cosh 2x + ... + \cosh nx ...
0
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0answers
28 views

Guidance for complex numbers/analysis problem needed [duplicate]

I'm looking at this one problem in a book of mine, but I can't even seem to start it. Let $z_1,z_2,...$ be a countable set of distinct complex numbers. If $|z_j-z_k|$ is an integer for every ...
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0answers
45 views

Complex roots of equation $z^\mu=r$

Find the complex roots of equation concerning unknown complex number $z$ \begin{equation} z^\mu=r,\quad \mu>0,r\in\pmb{R} \end{equation} A solution given by a book is to only consider the ...
0
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3answers
28 views

Question regarding in periodic function

I have question I know that $\cos(x+2\pi)=\cos x$ and $\sin(x+2\pi)=\sin x$ but if we have $\cos(x+\pi)=?$ and $\sin(x+\pi)=?$ with explaination thanks
1
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2answers
51 views

Solve cos(z) + sin(z) = i, where z is a complex number and i the imaginary unit

So yeah everything is in the title, I tried the trigonometric identity with sin(a+bi) and cos(a+bi) and I tried changing sin(z) and cos(z) for their complex expression, but all to no avail EDIT: ...
3
votes
3answers
124 views

Proving $\arg(zw)=\arg(z)+\arg(w)$

This is my attempt I know this is incomplete or may even be wrong. Let $θ_1 \in \arg(z)$ and $θ_2 \in \arg(w)$. Then, $θ_1+θ_2 \in \arg(z)+\arg(w)$. Also, $θ_1+θ_2 \in \arg(zw)$. Is this sufficient ...
0
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2answers
32 views

Finding argument to complex number?

I'm reading a bit on complex numbers, but haven't deal with trigonometry a lot before, so here's my question; how do I calculate the argument of a complex number when the sin and cos of the argument ...
5
votes
6answers
145 views

What's the result? $1/i=?$, where $i=\sqrt{-1}$ [duplicate]

I just had my first math class in the university, and I understood everything pretty well, but I think I have misread this one because I read that the result is $-1$. Thanks for your answers!
0
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0answers
63 views

Given roots on the unit circle, find the complex reciprocal polynomial

Given "all" the $m$ zeros on the unit circle of a complex reciprocal polynomial of even degree $2N > m$, can we find the polynomial? The known conditions are: We have all the $m$ zeros on the ...