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

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-1
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2answers
60 views

Euler's Formula, Square root

I have been doing some work on Euler's formula. I need to come up with the formula for $e^{i\theta}$ where $i = \sqrt{-1}$. I know $e^{i\theta} = \cos \theta + i\sin \theta$, but I am not sure how ...
2
votes
1answer
24 views

Difference between the complex roots of $f(x)$ and $|f(x)|^2$

I suppose a basic question, but it's causing me more problems than I envisioned! I have some polynomial $f(x)$ for which the roots are complex, $x+iy$. How will these roots change if I now take ...
2
votes
4answers
48 views

Writing the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form

Now I can't finish this problem: Express the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form, where $0 < \alpha < \frac{\pi}{2}$. So the goal is to determine both ...
0
votes
2answers
72 views

Can I approximate a complex number by its imaginary part, if real part is small compared to imaginary part?

I have the following doubt. How do you explain this? Here $j$ means $\sqrt{-1}$.
0
votes
1answer
19 views

Calculating $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}$ for $\alpha \in \mathbb Q$.

I need to calculate $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}$ for $\alpha \in \mathbb Q$ and $r \in \mathbb R$. My Attempt: $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}=\sum_{n=0}^\infty r ...
0
votes
1answer
37 views

Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
2
votes
1answer
27 views

Equality of complex numbers

I'm currently reading some notes on Complex Numbers and came across this 'proof' regarding the equality of complex numbers. Claim: Two complex numbers $a+bi$ and $c+di$ are equal iff $a=b$ and $c=d$, ...
0
votes
2answers
24 views

Is ring of Gaussian rationals in unique factorization domain?

Instead of Gaussian integers, let us think about Gaussian rationals, where $a$ and $b$ in $a+bi$ are rational numbers. Then would ring of Gaussian rationals be in unique factorization domain?
1
vote
2answers
28 views

rationalize the complex number multiplication rule

For a middle school student without previous knowledge of complex number, how do one introduce the multiplication rules of complex number? i.e., if we have two real number pairs of $(a,b)$ and ...
0
votes
1answer
23 views

Complex numbers - locii

I have been asked to solve the following and represent the answer graphically: A) $| \arg z - (\pi/4) | < (\pi/2)$ I understand that this means the difference between the argument of $z$ and ...
-2
votes
1answer
51 views

Polar Coordinate usind De Moivre’s Theorem

I need the solutions for the following problem: Find all solutions over $\mathbb{C}$ to the equation $x^3=i^2$. I tried using De Moivre’s Theorem can't get around it. Note: The question originally ...
1
vote
1answer
51 views

square of complex numbers

I have this equation from here: but it is not equal to: $$(a + bi)^2 = a^2 + 2abi + (bi)^2.$$ could someone explain me what is the difference between this two calcultion?
1
vote
1answer
113 views

arg(z) vs. Arg(z)

What is the difference between the arg(z) and the Arg(z), where z is a complex number of the form a+bi for example z = -2 - 2i the angle from the positive ...
5
votes
0answers
57 views

Convergence of infinite series of complex numbers [duplicate]

This has been bugging me for some months since our lecturer, a fields medalist, mentioned that he couldn't solve it when he was our age, yet had had two students submit solutions to it (during our ...
2
votes
1answer
44 views

Factoring a complex polynomial

Factorize the polynomial : $$ p(x) = x^{5} - x^{4}+ 4x - 4 $$ In real factors in the lowest degree possible. So in previous questions I have been given at least one rot so that I can factorize it ...
1
vote
3answers
66 views

Complex Equations

The Equation: $$ z^{4} -2 z^{3} + 12z^{2} -14z + 35 = 0 $$ has a root with a real part 1, solve the equation. When it says a real part of 1, does this mean that we could use (z-1) and use ...
0
votes
1answer
30 views

Arg(z) from $z^n$

$z \in \mathbb{C}$. If the principal argument of $z^n$ is in the quadrant $q$, what is the complete set of values for $Arg(z)$? For example if $n = 3$ and $q = 2$, how could I find all values of ...
9
votes
2answers
206 views

History of the matrix representation of complex numbers

It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this ...
0
votes
1answer
33 views

Equilateral triangle from complex numbers

We know that $z_1+z_2+z_3=0$ and $|z_1|=|z_2|=|z_3|=1$ where $z_1,z_2,z_3$ are complex numbers. How can we show that the images of $w_1=z_1^4*z_2^3 , w_2=z_2^4*z_3^3 , w_3=z_3^4*z_1^3$ form an ...
5
votes
2answers
75 views

Complex Numbers - Finding Roots

Hi there I was wondering if someone could help me? I am struggling to find the roots of the polynomial $z^4+2z+3=0$ It is not a quadratic so can't use the quadratic formula so am not quite sure ...
10
votes
1answer
210 views

Interpret to a complex plane!

$\newcommand{\Re}{\operatorname{Re}}\newcommand{\Im}{\operatorname{Im}}$The question is: Interpret $$ \Re z + \Im z = 1 $$ geometrically in the complex plane. Writing $z = x + yi$, the condition ...
2
votes
2answers
52 views

Another way to solve this problem with complex expressions

The problem is this: Express $x$ and $y$ with $u$ and $v$, if $\dfrac{1}{x+iy} + \dfrac{1}{u+iv} = 1$ Where $x,y,u,v \in \mathbb{R}$, and $i^2 = -1$. I could solve it, but I used a hairy and ...
4
votes
2answers
222 views

Problem getting the real roots of this complex expression

I'm trying to get the real roots of this expression: $$\dfrac{1}{z-i}+\dfrac{2+i}{1+i} = \sqrt{2}$$ Where $i^2=-1$ and $z=x+iy$. I tried to simplify that with Algebra, and then separate the real ...
3
votes
3answers
99 views

Problems with trigonometry getting the power of this complex expression

I'm here because I can't finish this problem, that comes from a Russian book: Calculate $z^{40}$ where $z = \dfrac{1+i\sqrt{3}}{1-i}$ Here $i=\sqrt{-1}$. All I know right now is I need to use ...
5
votes
1answer
74 views

If $e^{i\theta}=e^{i\varphi}$, then $\theta-\varphi=2k\pi$

This is pretty easy I think but I am having a tough time trying to prove this in a satisfying way to me. I am trying to show that $$e^{i\theta}=e^{i\varphi} \Rightarrow \theta-\varphi=2k\pi,\, \text{ ...
3
votes
2answers
77 views

Find all complex number $z\in\Bbb{C}$ such that $\vert z\vert=\vert z^{-1}\vert=\vert z-1\vert$

Find all complex number $z\in\Bbb{C}$ such that $$\vert z\vert=\vert z^{-1}\vert=\vert z-1\vert$$ I tried to write $z=a+ib$, clearly $z=1$ is not a solution. I have to solve $$\left\{ ...
-4
votes
2answers
49 views

Let $z = 2 + 2i$ Find all complex numbers $w$ such that $w^4 = z$ [on hold]

Let $$ z = 2 + 2i$$ Find all complex numbers $w$ such that $w^4 = z$
0
votes
1answer
19 views

Factor complex equation

Im having some difficulty in factoring the following complex equation. The image bellow is taken from WolframAlpha, can anyone explain how I can factor this equation. In the task I am told one ...
0
votes
2answers
57 views

Can the cube of 2 different complex numbers be the same?

Can the cube of 2 different complex numbers be the same? I think it cannot be the same, but I don't really know how to prove it. I tried to expand it but it gives a very ugly result.
1
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0answers
18 views
+50

Computing the tangential and cross components of one quantity using gnomonic projection

I have a spin-2 field given called shape distortion of galaxies as $$\gamma=\gamma_1+i\gamma_2=|\gamma|e^{-2i\phi}$$ where $\phi$ is the orientation angle. If this quantity has been measured on ...
15
votes
9answers
1k views

Is there an interval notation for complex numbers?

Just as $$\{x \in \mathbb{R}: a \leq x \leq b\}$$ can be written in the more-compact form $[a,b],$ is there an analogous notation for $$\{z \in \mathbb{C}:z=x+yi, x \in[a,b], y \in[c,d]\} \quad ?$$ ...
1
vote
1answer
49 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
0
votes
2answers
38 views

How to draw the Bode diagram for a given transfer function?

With this transfer function: $$G(s)=\displaystyle\frac{10(s+1)}{s(0.1s+1)}$$ I need to do operations to draw the Bode diagram manually I have this: $G(jw)=\displaystyle\frac{10jw+10}{-0.1w^2+jw}$ ...
1
vote
2answers
69 views

Using complex solutions in a factorisation

I'm working through an assignment, and have become stuck understanding the question... In part (a) I am asked to solve the equation: $z^5 = -1$ I have done this, so I now have a set of solutions: ...
3
votes
3answers
282 views

The limit of complex sequence

$$\lim\limits_{n \rightarrow \infty} \left(\frac{i}{1+i}\right)^n$$ I think the limit is $0$; is it true that $\forall a,b\in \Bbb C$, if $|a|<|b|$ then $\lim\limits_{n\rightarrow ...
1
vote
2answers
32 views

Can I perform the quadratic formula on polynomial with complex coefficient?

2 weeks ago, we had a Math test on complex number. One of the question was: Let $z=x+iy$ be a non-zero complex number, where $x,y \in \mathbb{R}$. Given that $z+\frac{1}{z} = k$, where $k$ ...
1
vote
0answers
28 views

Cauchy-Riemann Equations - why $f'(z_o) = \frac{\partial f}{\partial x}(z_o)$ implies that f is differentiable at $z_o$

I'm trying to understand part b of this proof. The only line I don't understand is the sentence starting with "To prove the statement in (b)..." If someone could clarify why that line is true I ...
1
vote
1answer
23 views

Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$

Let $Ω=\{z=x+iy∈C : |y|<x\}.$ Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$ Okay. So I can find a conformal map from $Ω\rightarrow \mathbb{D}$. I used the map $f(z) = ...
0
votes
1answer
46 views

When $f=u+iv$ is a holomorphic function, the real part of $f'(z)$ is equal to $u_x(z)$

Suppose $f$ is holomorphic, and is written as $f=u+iv$ with $u,v$ real-valued. Why is the partial derivative $u_x(z)$ equal to $\operatorname{Re}(f'(z))$? Source This fact is used in the proof ...
5
votes
4answers
81 views

Question on definitions

I was going through some basic recap of complex numbers and in the book (M. Boas. Mathematical Methods in the Physical Sciences) she says we define $e^{ix}$ by the Taylor series with $x$ replaced by ...
6
votes
3answers
117 views

Towards a formula for the Euler $\phi$ function?

$\Phi_n(1)$ and $\Phi_n(-1)$ for the cyclotomic polynomials are well-known. I am now looking for $$\Phi_n(i)$$ and/or $$\Phi_n(-i)$$ with $i$ the complex unit. The reason is : I suppose it is ...
2
votes
2answers
62 views

Are numbers like $\left ( -2 \right )^{\sqrt{2}}$ real or complex?

I know that numbers with rational power can be converted to radicals and based on the degree of the radical we can say that whether they are real or complex. But what about numbers like $\left ( -2 ...
1
vote
1answer
27 views

complex functions inequalities plane

Given $w(z)=\frac{i-z}{i+z}$. Find the map w=f(z) of the part of the plane defined by inequalities: $|z|>1$ and $Im(z)>Re(z)$ so far: $|z|>1$ is this area from $Im(z)>Re(z)$ => ...
0
votes
1answer
34 views

Discontinuity of principal argument in nonpositive real axis

Let $\operatorname{Arg}(z)$ be principal argument function defined in branch $(-\pi, \pi]$. Prove that $\operatorname{Arg}(z)$ is discontinuous in every point in nonpositive real axis. "Solution": ...
1
vote
3answers
74 views

Are $i,j,k$ commutative?

I am trying to understand quaternions. I read that Hamilton came up with the great equation: A) $i^2 = j^2 = k^2 = ijk = −1$ In this equation I understand that $i,j,k$ are complex numbers. Later ...
1
vote
1answer
54 views

How to prove the formula for the argument of a complex number?

$\arg(x + iy) = 2 \cdot \arctan(\dfrac{y}{x + r})$ there is always the mark, this is derived from the 'Half-angle formula' How can I come from $\tan(\phi) = \tan(\phi + k \pi) = \dfrac{y}{x}$ to ...
0
votes
3answers
44 views

The modulus of complex number? [closed]

the modulus of complex number $\frac{3 + 4i}{1 - 2i}$ is: $A) -\pi \qquad B) - \pi/2 \qquad C) \pi/2 \qquad D) \pi$ explain procedure also with your answer.
0
votes
1answer
19 views

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function. In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and ...
6
votes
6answers
3k views

How can you find the cubed roots of $i$?

I am trying to figure out what the three possibilities of $z$ are such that $$ z^3=i $$ but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you ...
29
votes
9answers
4k views

Is “$a + 0i$” in every way equal to just “$a$”?

I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$). I say this is practically the case, so in every calculation you ...