Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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1answer
54 views

Mean value theorem for harmonic

In Problems and Solution in Mathematics by Ta-Tsien, exercise 5123, the mean value theorem is used as: \begin{equation} \text{log} |F(0)| = \frac{1}{2 \pi} \int_0^{2\pi} \text{log}|F(re^{i\theta})| ...
2
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2answers
62 views

Is there any subset of Complex numbers that is algebraically closed?

That any polynomial that is allowed to have coefficients from that subset has also a root in that subset
1
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0answers
49 views

Complex numbers product and ratio, prove this relation.

Define a table $T$ as follows: $$\text{If}\; n\geq k \; \text{then} \; T(n,k) = (2+3 i) \sum _{i=1}^{n-1} T(n-i,k-1)+(5+7 i) \sum _{i=1}^{n-1} T(n-i,k) \; \text{else} \; T(n,k) = 0$$ Then take rows ...
1
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1answer
45 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 ...
1
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1answer
60 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 ...
3
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0answers
83 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
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5answers
146 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)}\\ &= ...
11
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3answers
145 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 ...
2
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3answers
92 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 ...
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0answers
74 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 ...
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0answers
65 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
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2answers
41 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 $\sum_je^{ip\phi_j}$?
2
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1answer
49 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
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5answers
72 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 ...
6
votes
2answers
165 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 ...
0
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1answer
54 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 $x$.
2
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2answers
103 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 ...
2
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1answer
42 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
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1answer
61 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 ...
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3answers
139 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 ...
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1answer
43 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 ...
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2answers
49 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)}\\ ...
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0answers
69 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: ...
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3answers
66 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 ...
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1answer
43 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
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4answers
416 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 ...
5
votes
3answers
139 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 ...
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2answers
133 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 ...
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0answers
48 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
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1answer
31 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?
0
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1answer
75 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 ...
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2answers
82 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
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1answer
37 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 ...
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1answer
28 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 ...
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2answers
276 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
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3answers
71 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
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1answer
55 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 ...
3
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1answer
58 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 $n>0$.
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1answer
29 views
0
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1answer
25 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
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4answers
122 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 ...
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2answers
58 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
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3answers
128 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
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2answers
74 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
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2answers
949 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\\ ...
1
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2answers
107 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
2
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2answers
132 views

Complex series radius convergence

How to find the values for which $z$ converges, $z\in\mathbb{C}$, in the serie $$\sum_{n=1}^{\infty}\frac{1}{(1+|z|^{2})^{n}}$$ I know I have to use the convergence radius expression, but what I ...
0
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4answers
74 views

Complex Numbers: Im$(\frac{12}{z-7})=1$

Sketch and describe the set of complex numbers satisfying $$Im(\frac{12}{z-7})=1$$ where $z=x+iy$ The answer should be in circle form. Here is what I have so far: $$Im(12)=z-7$$ $$Im(12)=x+iy-7$$ ...
2
votes
3answers
75 views

Square root of a squared number changes sign, which to apply first?

Heres something Ive always found interesting. Supose we have a variable $x$, and $x$ equals a negative number: Say: $$x=-17$$ Now, I can apply a square to both sides of the equation and preserve ...
0
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2answers
385 views

find the cube roots of -8i and plot them on a plane

I can't figure out the angle of this equation. I set it up like this: $$z^3=0-8i$$ I find that the r value is 2, but when I try to find the angle I'm stuck. I can't divide by 0, so where did I go ...