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

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5
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1answer
149 views

When does $\sqrt{wz}=\sqrt{w}\sqrt{z}$?

There exists a unique function $\sqrt{*} : \mathbb{C} \rightarrow \mathbb{C}$ such that for all $r \in [0,\infty)$ and $\theta \in (-\pi,\pi]$ it holds that ...
2
votes
2answers
145 views

trigonometric representation of a complex number.

Let $z=e^{it}+1$ where $0\leq t\leq \pi$, Find the trigonometric representation of $z^2+z+1$. (The trigonometric representation should be in the form of : $r(\cos \theta +i \sin \theta)$, where ...
1
vote
1answer
127 views

Help Understanding Complex Roots

I was reading a graphical explanation of complex roots, and between Figures 7 and 8 I became confused. The roots appear in the imaginary plane, but I don't understand why the original function ...
0
votes
1answer
163 views

Letters for complex numbers

Suppose that I am writing a proof or some other piece of mathematical writing, and wish to introduce $n$ distinct complex numbers, for some positive integer $n$. What are the complex numbers called? ...
1
vote
1answer
211 views

Describe the graph locus represented by this equation

I want to know the shape of the region described by $$ Im(z^2) = 4 $$ so I did the following: $$ z=x + iy $$ $$z^2 = x^2 + 2xiy -y^2 $$then $$Im(z^2) = 2xy $$ then the locus is $$ 2xy = 4 $$ ...
1
vote
2answers
67 views

Factorization in Gaussian integers

Let $p$ be a natural number, suppose $p$ prime. Show that the following conditions are equivalent: 1) the polynomial $x^2+1\in\mathbb{Z}_p$ has roots in $\mathbb{Z}_p$ 2) $p$ is reducible in the ...
0
votes
1answer
67 views

Why $\frac{1}{2i}(e^{i\omega t} - e^{-i\omega t}) = \frac{i}{2} (e^{-i \omega t} - e^{i\omega t})$

Let $i := \sqrt{-1}$, $f$ be the frequency ($\frac1p$), and $\omega := 2 \pi f$. From page 3 here, why does $\frac{1}{2i}(e^{i\omega t} - e^{-i\omega t}) = \frac{i}{2} (e^{-i \omega t} - e^{i\omega ...
0
votes
1answer
98 views

Solving complex linear congruences

Find $x \in \mathbb{Z}[i]$ such that: $(1+2i)x \equiv 1 \mod 3+3i$ How would you go about doing this? Best I can think of is keep guessing....
13
votes
3answers
144 views

$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$

Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost. Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying: $$x_1 + x_2 \dots + x_k = 0$$ $$x_1^2 + x_2^2 ...
3
votes
1answer
189 views

Find the flaw in my proof that $z^2 =1$ has more than $2$ distinct solutions.

Let $z \in \mathbb{C}$ be any number that satisfies the equation $z^2=1$. Certainly, $z=\pm1$ are two possible solutions to this equation. I claim that $z^k$ is also a solution to this equation for ...
2
votes
1answer
87 views

Complex numbers order property

As a complex number set/field isn't an ordered set/field. Now $1 \in \mathbb{C} $ & $2 \in \mathbb{C} $ . How is $2>1$ ?
0
votes
1answer
76 views

Some questions about complexification of a real vector space

Could you tell me how to prove that if $f:U \rightarrow U$ is $\mathbb{R}$-linear, then: 1) $U^{\mathbb{C}}$ is a vector space over $\mathbb{R}$ (should I check all eight conditions for a vector ...
0
votes
0answers
32 views

Does $\sum_{k=0}^n x^k = \prod_{k=1}^n \left(x - \mathtt{i}^\frac{2 k}{n+1}\right)$?

This seems to be true: $$\sum_{k=0}^n x^k = \prod_{k=1}^n \left(x - e^\frac{2\pi i k}{n+1} \right)$$ but I don't know how to demonstrate it, and definitely not neatly. I'd like to see why it should ...
0
votes
2answers
57 views

Complex number question

For any complex numbers $z_1, z_2$, is the quantity $S$: $$ S = 4 \left(| z_1|^6 + |z_2 |^6\right ) + 4 |z_1|^3 |z_2 |^3 + \left(2 |z_1|^2\times \overline{z_1}^2\times z_2^2\right) + \left(2 ...
3
votes
1answer
103 views

Behavior of $\Gamma(z)$ as $\text{Im} (z) \to \pm \infty$

In a paper I'm reading it states that $\displaystyle |\Gamma(z)| = |\Gamma(a+ib)| \sim \sqrt{2 \pi} |b|^{a-\frac{1}{2}} e^{-|b|\frac{\pi}{2}}$ as $\displaystyle|b| \to \infty$. How is that derived ...
1
vote
1answer
219 views

How can you prove Euler's phase angle formula for differential equations?

How can you prove this formula: $C_1 e^{(\alpha + i\beta) t} + C_2 e^{(\alpha - i\beta)t}=Ke^{\alpha t}\cos {(\beta t + \phi)}$ This gives $x(t)$ in the second-order differential equation for an ...
1
vote
4answers
153 views

Evaluation of a complex numbers partial sum

Let $w = e^{i\frac{2\pi}5}$. I would like to evaluate $$w^0 + w^1 + w^2 + w^3 +...+ w^{49}$$ Can anyone please give me an idea how to evaluate the expression? Thanks in advance
-1
votes
2answers
96 views

If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then is it a constant? [closed]

If a function $f: \mathbb{C}\to\mathbb{C}$ is bounded, then it is a constant. Is it true or false?
4
votes
3answers
1k views

Prove equality in triangle inequality for complex numbers

We need to show that $$ |z_{1}+z_{2}+\cdots+z_{n}|=|z_{1}|+|z_{2}|+\cdots+|z_{n}|$$ if and only if $z_{1},z_{2},\dots,z_{n}$ have the same argument (i.e. $z_{j}=r_{j}e^{i\theta}$ for $j=1,\dots,n$). ...
0
votes
2answers
131 views

What exactly is the complex plane, and how is it useful?

A lot of functions are defined on the complex plane, like the Gamma function: the Lambert W function, etc. But I have no idea about what the complex plane means and how it's useful, or just ...
1
vote
0answers
18 views

simplification of a complex expression

I am collecting proofs for certain integrals. To simplify certain proofs, I use $e^{Ax}cos(Bx)=\mathcal{Re}[e^{(A+iB)x}]$, where $A$, $B$, and $x$ are real. Is there an analogous simplification for $ ...
2
votes
3answers
187 views

The negative square root of $-1$ as the value of $i$

I have a small point to be clarified.We all know $ i^2 = -1 $ when we define complex numbers, and we usually take the positive square root of $-1$ as the value of "$i$" , i.e, $i = (-1)^{1/2} $. I ...
43
votes
16answers
5k views

What is $-i$ exactly?

We all know that $i$ doesn't have any sign: it is neither positive nor negative. Then how can people use $-i$ for anything? Also, we define $i$ a number such that $i^2 = -1$. But it can also be seen ...
2
votes
1answer
44 views

Formula for the sum of the value of a rational function over roots of unity

Let $n,k$ be integers, and let $U$ be the set of all $n$-ths roots of unity (so there are exactly $n$ elements in $U$). Let $U'=U \setminus \lbrace 1 \rbrace$. Are there simple formulas (in terms of ...
1
vote
2answers
61 views

Trigonometric problem

I'm trying to get the roots for a complex number $x^2+1$ $x^2+1=0\rightarrow x^2=-1 \rightarrow x = \sqrt{-1} \rightarrow i$ So, $w^2 = 0 + 1i$ $p = \sqrt{0^2+1^2} = 1$ $\theta = \tan^{-1} \left( ...
2
votes
3answers
263 views

Proving an inequality: $|1-e^{i\theta}|\le|\theta|$

We have been using this result without proof in my class, but I don't know how to prove it. Could someone point me in the right direction? $$|1-e^{i\theta}|\le|\theta|$$ I believe this is true for ...
0
votes
3answers
6k views

Cube roots of the complex numbers 1+i?

I cant get any good reference in my books regarding cube of complex numbers. Please help me find cube roots of the Complex number i+1??
6
votes
5answers
148 views

Strong characterization of $\mathbb C$ with respect to $\mathbb R$

$\mathbb R^2$ is not a field, but $2$-tuple arithmetic rules like $(a,b)(c,d)=(ac-bd,ad+bc)$ coupled with $\mathbb R^2$ make it a field, but are there other rules than $(a,b)(c,d)=(ac-db,ad+bc)$ ...
1
vote
1answer
71 views

How can I practice Jean-Robert Argand idea of the rotation of a square root of -1

I am studying complex numbers and I really need an intuition on how they work. I found the following video of Mathematician named Adrien Douady https://www.youtube.com/watch?v=2kbM96Jr4nk He ...
2
votes
1answer
87 views

Is there a geometric relationship between plane geometry and polynomials?

It is well known that the complex plane is algebraically closed: Every polynomial has a zero. The relationship seems, to me, to run deeper: For every complex-differentiable function, there exists a ...
-1
votes
1answer
349 views

Showing that the area of a triangle is $\frac{1}{2}|Im[w_1\bar{w_2}]|$?

I'm reading Beardon's Algebra and Geometry. Let $T$ be a triangle in $\mathbb{C}$ with vertices at $0$, $w_1$, $w_2$. By applying the mapping $z\mapsto \bar{w_2}z$, show that the area of $T$ is ...
2
votes
2answers
169 views

$i^{th}$ root(s) of unity

If we define $S:=\{z\in\mathbb{C}:z^n=1\}$ (i.e. the $n^{th}$ roots of unity), then $|S|=n$ (i.e. we have $n$ of them). We can even go as far as to say: $$S=\{z_k:k\in\mathbb{N}\cap[1,n]\}=\{e^{2\pi ...
2
votes
0answers
302 views

Probability distribution of the product of two independent complex gaussian random variables

I have to calculate the pdf of $Z = X*Y$, where $X \in \mathcal{C}(\mu_x,\Sigma_x)$ and $Y \in \mathcal{C}(\mu_y,\Sigma_y)$, where $\mathcal{C}$ is a complex distribution. It can be assumed that ...
0
votes
1answer
89 views

Showing that the segment joining $0$ to $z$ is perpendicular to the segment joining $0$ and $w$ iff $Re[z\bar{w}]=0$

I'm reading Beardon's Algebra and Geometry. Suppose that $zw\neq0$. Show that the segment joining $0$ to $z$ is perpendicular to the segment joining $0$ to $w$ if and only if $Re[z\bar{w}]=0$. ...
0
votes
1answer
79 views

Introduction to fractional calculus: problem with identity

I can't see the next step: $D^\alpha e^{ix} = i^{\alpha}e^{ix} = e^{i\alpha \frac \pi2}e^{ix}$
-1
votes
2answers
123 views

contradicting identity theorem?

the identity theorem for holomorphic functions states: given functions $f$ and $g$ holomorphic on a connected open set $D$, if $f = g$ on some open subset of $D$, then $f = g$ on $D$ Let $f(z) = \sin ...
0
votes
1answer
236 views

Addition in polar form

$$u_{1}(t) = 120\sqrt{2}e^{j5000t}$$ $$u_{2}(t) = -j160\sqrt{2}e^{j5000t}$$ I need to add these two values, so: $u(t) = u_{1}(t) + u_{2}(t) = (120 - j160)\sqrt{2}e^{j5000t} = ...$ What next? How ...
-4
votes
1answer
184 views

Root of a quadratic equation that has modulus $1$

Let us suppose $\alpha \in \mathbb C$ and $|\alpha|=1$ and $\alpha$ satisfies a monic quadratic equation. Then prove that $\alpha^{12} =1$. Show me the right way to solve this. Thanks in advance.
1
vote
1answer
210 views

extended Euclidean (xgcd) in quadratic integer rings

Given a discriminant $D < 0$, I have the quadratic imaginary field $\mathbb{K} := \mathbb{Q}(\sqrt{D})$. And the quadratic integer ring is given by $\mathcal{O} = \mathbb{Z} + \mathbb{Z} \frac{D + ...
5
votes
2answers
2k views

Prove that the zeros of an analytic function are finite and isolated

Let us assume that the zeros of $f = \{Z_1,\ldots,Z_n,a\}$ are infinite and converge towards $a$. The book which I am reading says that any neighborhood of $a$ will contain infinite zeros. Since $f$ ...
1
vote
2answers
26 views

(A,B) regular => there is a scalar s such that A+s*B is regular ??

Given two matrices $A,B \in \mathbb{C}^{n \times n}$, is it true that $rank([A,B])=n \implies \exists s\in \mathbb{C}: rank(A+sB)=n$ It seems to me this could be easily proved by writing both in ...
0
votes
1answer
71 views

Find the residue of $\frac{1 - \cos z}{z^{3} (z-3)}$

Is my solution correct? Also, are there removable singularities? Im having trouble classifying singularities
4
votes
1answer
39 views

Complex number equivalency

I'm a bit confused over the solution to a complex ode: $i\alpha y = \beta y''$ The solution to the characteristic polynomial is $r = \pm \sqrt{i\alpha/\beta}$. Somehow my book is getting the ...
1
vote
1answer
73 views

What is the easiest way to define a complex number in exponential form in maple?

What is the easiest way to define a complex number in exponential form in maple? Is there a built-in function? eg: $\underline{Z} = 600 \cdot e^{-j45^\circ}$
4
votes
3answers
556 views

Solve $\sin(z) = z$ in complex numbers

Show that $\sin(z) = z$ has infinitely many solutions in complex numbers. Little Picard theorem should help, but using big Picard theorem is undesirable. Thanks a lot!
5
votes
0answers
319 views

Natural extension of the divisor function $\sigma$ to the complex integers $\mathbb{Z}[i]$

The divisor function $\sigma$ of a positive integer $n$ is defined as the sum of the positive divisors of $n$. It turns out that this definition is equivalent to ...
0
votes
1answer
106 views

Complex numbers and absolute values

If i have equation: \begin{align} P = \left|\psi\right|^2 \end{align} where $P$ is a probability and we know there is no negative probability. This means $P$ must belong to $\mathbb{R}$. If i want ...
4
votes
3answers
206 views

Is $\sqrt{-1}$ positive or negative?

Do the concept of positive or negative make sense in this case? I remember that $\mathbb{R}^2$ has four quadrants thus ordered pairs of numbers could be $(+,+),(+,-),(-,-)(-,+)$, I presume that ...
7
votes
2answers
257 views

How to show that $\overline{zw}=\overline{z}\,\overline{w}$?

I thought about first multiplying two complex which aren't in the conjugate form: $$zw=a c+i a d+i b c-b d$$ Then multiply two complex conjugates: $$\overline{z}\,\overline{w}=a c\color{red}{-}i a ...
2
votes
1answer
84 views

I am puzzled with which one is right.

I am puzzled with which one is right.If my work is wrong.please give me a right explanation in detail.