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

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4
votes
2answers
206 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 ...
1
vote
3answers
119 views

The Name for $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ Exclusively Algebras?

The Wikipedia page for Normed Division Algebras has been redirected to Normed Algebras and the explanation given is that $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ algebras are not the ...
1
vote
1answer
33 views

Complex variable algebra mishap

One question on a problem set was the following: Show that $x^2 - y^2 = 1$ can be rewritten as $z^2 + \bar{z}^2 = 2$. (With $z = x + iy$) So I started working from the first expression based on ...
2
votes
3answers
61 views

Real matrices with non-real eigenvalues

I know this covers a lot, so perhaps someone could redirect me to a helpful website. for a) I have no idea where to start on the proof, as I don't understand why this is true. for b) I also have ...
0
votes
0answers
27 views

Find the set of $z$ which satisfies the given equation

Let $w \to w^{a}$ be the principal branch of the power function defined for $|\mathrm{Arg}(w)| <\pi$. Find the set of all values of $z\in \mathbb{C}$ such that the following identity holds for ALL ...
2
votes
0answers
40 views

Factorial of Complex Values

Since the gamma function is an analytic continuation of the factorial function, we can find the factorial of complex values. How does one go about doing so? I've looked far and wide on the internet ...
3
votes
5answers
134 views

Which one is correct for $\sqrt{-16} \times \sqrt{-1}$? $4$ or $-4$?

As we can find in order to evaluate $\sqrt{-16} \times \sqrt{-1}$, we can do it in two ways. FIRST \begin{align*} \sqrt{-16} \times \sqrt{-1} &= \sqrt{(-16) \times (-1)}\\ &= ...
2
votes
4answers
675 views

finding roots of complex equation

I have here a complex equation: $$z^2 - (7+j)z + 24 +j7 = 0$$ How do we get the roots of this equation? I started using the quadratic formula $-b \pm \sqrt{ b^2-4ac}\over 2$, but it yielded too much ...
11
votes
3answers
94 views

Prove that an expression is zero for all sets of distinct $a_1, \dotsc, a_n\in\mathbb{C}$

A while ago one of my professors gave the class a problem "to think about when lying on the beach." Well, I've been on the beach several times since then to no avail and my curiosity has finally ...
1
vote
0answers
39 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
-1
votes
2answers
55 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
5
votes
2answers
101 views

Finding non-negative integers $m$ such that $(1 + \sqrt{-2})^m$ has real part $\pm 1$.

I believe that the integers $m$ with $(1+\sqrt{-2})^m$ having real part $\pm 1$ are $0, 1, 2$ and $5$, but I'm having trouble proving it. Write $$a_m = \Re((1+\sqrt{-2})^m) = \frac{(1 + \sqrt{-2})^m ...
1
vote
0answers
49 views

Finding the number of elements in $\left(ℤ[i]\right)_m$

If $m$ = $m_1$ + $m_1i$ ∈ $ℤ[i]$, is there a general formula to find the number of elements in $\left(ℤ[i]\right)_m$?
4
votes
2answers
36 views

$\sum_j e^{i\phi_j}$ vs $\sum_j e^{ip\phi_j}$

Let $\phi_j$ be a collection of angles. If $p$ is a positive integer, how is the sum $\sum_je^{i\phi_j}$ related to $\sum_je^{ip\phi_j}$?
-3
votes
1answer
32 views

with out De Moivre’s Theorem find $x,y\in\mathbb R$ [on hold]

with out De Moivre’s Theorem find $x, y \in\mathbb R$ so that: $$(x+yi)^3=\frac{-6+2\sqrt7 i}{1-\sqrt7 i}$$
2
votes
1answer
44 views

The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong? Let's share the fun: $(\sqrt{x^2 +y^2}) i +2x ...
2
votes
5answers
66 views

Find $n$ for which $(1+i)^{2n}=(1-i)^{2n}$

Question: Find the values of $n$ for which $$(1+i)^{2n}=(1-i)^{2n}$$ wolfram alpha tells me that the answer should be : $$n=\frac{2i\pi m}{\log(1-i)-\log(1+i)}$$ $$n=-\frac{i(2\pi ...
2
votes
2answers
95 views

Why is $ i^2 \neq (1 + i)^4$?

Today I read that you can see the number $i$ as the rotation of 90° and therefore i^2 is the rotation of 180° or -1. I also learned that $1+i$ is 45° but if this would be true I should be able to ...
0
votes
1answer
35 views

Extension to the complex numbers for ex. 12 in ch. 6 of Axler's “Linear Algebra Done Right”

I'm wondering how the answer to Sheldon Axler's exercise 12 of chapter 6 "Linear Algebra Done Right" changes when the underlying field is extended from the reals to the complex numbers. The exercise ...
0
votes
1answer
39 views

Number theory proof regarding norms

How would you prove that if $x$ is a prime in $ℤ[i] \Longleftrightarrow$ $N(x)$ is a prime in $ℤ$ N(x) represents the norm of x.
1
vote
1answer
52 views

A simple complex inequality

I feel this is not hard, but no way to prove it $|\sqrt{z^2 -4}-z|\le 2$ Any body can help? Thanks! The total statement should be one of the branchs of square root should satisfy this ...
2
votes
1answer
35 views

What does taking the $n^{\text{th}}$ root of a complex number geometrically mean?

What are the geometrical implications of taking the $n^{\text{th}}$ root of a complex number, say $3+4i$. What is the geometrical implication of $\sqrt[n] {3+4i}$ in the complex plane?
1
vote
2answers
44 views

Evaluate expression in the form $a+bi$.

So, I have to evaluate $\sqrt{-3}\sqrt{-12}$ into the form $a+bi$. I know that $i^2 = -1$ so $i = \sqrt{-1}$ What I have done is: $$\begin{align}\sqrt{-3}\sqrt{-12} &= \sqrt{3(-1)}\sqrt{12(-1)}\\ ...
3
votes
1answer
53 views

Find a sequence

Find the function for the sequence $a_0 = 0, a_1 = 1$ and $a_{n}=a_{n+10}+a_n$ for all $n>0$.
1
vote
0answers
63 views

Integration Error

Sorry if this doesn't make any sense or if I did something obviously wrong, I was just playing around with taylor series' and then I got stuck. I know from the geometric series that: ...
1
vote
3answers
52 views

Show a complex equation has one or two roots

Let $a$ $\neq$ $0$, $b,$ and $c$ be complex constants. Show that the quadratic equation $az^2+bz+c=0$ has one or two roots. My thoughts: Let $a=a_1+ia_2,$ $b=b_1+ib_2,$ and $c=c_1+ic_2$. I also ...
1
vote
1answer
24 views

To use Vieta's formula for complex constant solution or not?

Let $b$ and $c$ be complex constants such that $z^2$ + $bz$ + $c$ = $0$ has two different real roots. Show $b$ and $c$ are real. I think I need to be using Vieta's formula, however I have solved it ...
2
votes
4answers
52 views

What is a complex constant and how do I use it?

I have a question I am trying to understand: "Let $b$ and $c$ be complex constants such that $z^2+bz+c=0$ has two different real roots. Show that $b$ and $c$ are real." My biggest problem here is ...
-2
votes
2answers
111 views

Error in proof that $1 = -1$ [duplicate]

I have created a proof that$ 1 = -1$ but I know that this is impossible. Could someone help me find the flaw in this proof... $i = \sqrt{-1}$ Given $i^2 = -1$ Given $i^4 = 1$ Given ...
4
votes
3answers
90 views

Why is this definition of complex numbers “informal”?

I'm reading the proofwiki page about complex number: https://proofwiki.org/wiki/Definition:Complex_Number According to proofwiki there is an informal and formal definitions of complex numbers. The ...
0
votes
1answer
18 views

weighted inner product of polynomials, can weight function be complex?

I am just learning about inner-products on polynomial space, where the coefficients of the polynomials may be complex: $P_m(\mathbf{F})$ The inner-product given by: $\langle p,q \rangle = \int_0^1 ...
0
votes
0answers
36 views

Plane geometry in the complex plane

i am asked to find the area of a triangle that has vertices $0, w_{1}, w_{2}$ in $\mathbb{C}$ by applying the transformation $z \rightarrow \bar{w_2}z.$ My attempt: since we are multiplying by the ...
1
vote
2answers
68 views

If a series converges then the power series converges for all z

How can I prove that if $\sum \limits_{n=1}^{\infty} c_n$ , $c_n\in \mathbb{C}$, converges then $\sum \limits_{n=1}^{\infty} c_n \frac{z^n}{1-z^n}$ converges for all z in $\mathbb{C}$ with ...
1
vote
1answer
22 views

Prove of complex numbers inequality

Is it true that for any two complex numbers, say $a, b$, the following inequality holds: $|a\bar{b}| \leq |a|^2 + |b|^2$ ? How can we prove this?
10
votes
6answers
11k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
0
votes
4answers
91 views

Shown that $f(z)=\left | z^2-4 \right |^2$ is holomorphic

Shown that $f(z)=\left | z^2-4 \right |^2$ is holomorphic. I need to prove that $f(z)$ is holomorphic or not. Well first I need to convert $f(z)$ in terms of $x,y$ but I dont understand how to do it. ...
1
vote
1answer
34 views

How find the minimum of the $|w^3+z^3|$,if $|z+w|=1,|z^2+w^2|=14$

let complex $z,w$ such $$|z+w|=1,|z^2+w^2|=14$$ find the minimum of the value $$|w^3+z^3|$$ My idea: let $$z=a+bi,w=c+di\Longrightarrow z+w=(a+c)+(b+d)i,z^2+w^2=(a^2+b^2+c^2+d^2)+2(ab+cd)i$$ then we ...
0
votes
1answer
19 views

Product of square of distances from vertices of a polygon of radius a

I want to find out the following product. $\prod_0^{n-1} (r^2 + a^2 -2ra\cos(2k\pi/n - \theta))$ I have been trying to use complex numbers but did not work out. The book I am reading the result is ...
0
votes
2answers
36 views

Is it appropriate to apply Euclidean Distance to Complex Numbers?

Would complex numbers be considered as part of Euclidean Space? Would this measurement give an accurate result? If not, what would be a more appropriate distance measurement/similarity measure with ...
0
votes
1answer
45 views

finding solutions to a complex number equation

Given that the square roots of $(-2+2\sqrt{3}\cdot{i})$ are $\pm(1+\sqrt{3}\cdot{i})$, find all solutions to $\{z:z^2+(\sqrt{3}-i)z+(1-\sqrt{3}\cdot{i})=0\}$ in Cartesian form. I'm unsure as to how ...
0
votes
3answers
43 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
0
votes
1answer
21 views

$GL_2(\mathbb{C})$ acting on extended complex numbers.

Let $GL_2(\mathbb{C})$ the general linear group of order two on complex. We can define a action from $GL_2(\mathbb{C})$ on $\mathbb{C}^*$ as ...
2
votes
4answers
110 views

$(-1)^{0.2}=0.8090 + 0.5878i$ how can this be?

I'm working on a numerical analysis project (working with matlab a lot) and I noticed that when I ask for matlab to compute the exponent of a negative number, it gives wrong output when the exponent ...
1
vote
2answers
45 views

Find $c$ if $a,b, \; c$ satisfy $c = (a+bi)^3 - 107i$

Find $c$ if $a,b, \; c$ are positive integers which satisfy $c = (a+bi)^3 - 107i$ I can try expanding the cube, but that seems too direct. What other ways are there to go about this?
2
votes
3answers
117 views

Computing complex number [duplicate]

"Compute $(1 + i)^{1000}$. So far I have: $(1+i)^{4 (2^2 5^3)} $ but I am not sure how to proceed. Ideas?
2
votes
2answers
62 views

Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
2
votes
1answer
55 views

Interesting examples of Cauchy's Integral formula [closed]

Question : What are some interesting and, albeit, counter intuitive examples of real integrals that are solved using Cauchy's integral Formula. Cauchy integral formula can magically transform some ...
0
votes
1answer
26 views

What is the best way to express a complex modulus squared in text?

I am looking for a way to describe the fact that I am taking a modulus squared in text. E.g. "inserting the potential (Eq. 1) into the expression for the wavefunction coefficients (Eq. 2) and taking ...
2
votes
2answers
28 views

2x2 inverse of a complex matrix with complex determinant

Firstly, my question may be related to a similar question here: Are complex determinants for matrices possible and if so, how can they be interpreted? I am using: $$ \left(\begin{array}{cc} a&b\\ ...