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

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2
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0answers
340 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
96 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: $D^\alpha e^{ix} = i^{\alpha}e^{ix} = e^{i\alpha \frac \pi2}e^{ix}$
-2
votes
2answers
127 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
237 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
185 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
252 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 + ...
6
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
41 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
77 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
587 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
337 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
114 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
207 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
272 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.
0
votes
1answer
248 views

making the domain of $z ↦\tan(z)$ injective

Given the following: $\sin(z)$ = ($e^i$$^z$ - $e^-$$^i$$^z$)/$2i$ $\cos(z)$ = ($e^i$$^z$ + $e^-$$^i$$^z$)/$2$ $\sin(z)\cos(w) - \cos(z)\sin(w) = \sin(z-w)$ $\sin(z) = 0$ has solution $z = kπ$ for ...
4
votes
2answers
109 views

Does the square root of $i$ necessitate quaternions?

The square root of i is $\frac{\sqrt{2} + i \sqrt{2}}{2}$. But how is it valid to use a number in expressing the square root of that number?
4
votes
4answers
1k views

Can one use complex numbers in probability?

I have never thought about using complex numbers in probability. I am examining Bayes Theorem, and attempting to relate it to projective geometry and this question came to mind. I am not talking about ...
0
votes
1answer
203 views

discrete subgroups of multiplicative non-zero complex numbers

Is it true that all discrete subgroups of the multipicative group of non-zero complex numbers $(\mathbb{C}\setminus \{0\},.)$ are cyclic?
1
vote
3answers
158 views

How is my textbook finding this rotation?

I have this transformation $\mathbf x\mapsto A\mathbf x $ which is the composition of a rotation and a scaling. I need to give the angle $\varphi$ of the rotation and give the scale factor $r$. Here ...
2
votes
1answer
138 views

Where is there a good introduction to hypercomplex numbers and calculus?

I'm looking for a good introduction to hypercomplex numbers that requires as little math knowledge as possible, yet covers hypercomplex numbers as thoroughly as possible. I'm interested specifically ...
2
votes
3answers
252 views

prove that : $ i = \sqrt {-1}\ $ [closed]

i have a pretty nasty question. i was glancing through a few olympiad papers and stumbled upon this question: prove that $ i = \sqrt {-1}\ $. i tried the conventional methods namely euler's formula ...
5
votes
1answer
108 views

A case where $z^z = 0$ where $z$ is complex number

Is there any case where $z^z = 0$ where $z$ is complex number? The case excludes the case where $z=0$.
8
votes
4answers
269 views

When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
1
vote
1answer
224 views

What is the odd Fourier extension of $\sin x \cos(2x)$?

odd half range extension of $$f(x) = \sin x \cos(2x)\text{ with limits $0$ to $\pi$}$$
2
votes
2answers
157 views

Roots of cubic polynomial lying inside the circle

Show that all roots of $a+bz+cz^2+z^3=0$ lie inside the circle $|z|=max{\{1,|a|+|b|+|c| \}}$ Now this problem is given in Beardon's Algebra and Geometry third chapter on complex numbers. What might ...
1
vote
1answer
167 views

Logical explanation of Euler's formula

This question is a about (if not proving) at least guessing the Euler's formula. I don't want the proof using the infinite sums. We can guess by logic that for example that the equation ...
2
votes
3answers
239 views

Simple Question on Roots of Unity

The question asks: Find integers $p$ and $q$ such that $(p + qj)^{5} = 4 + 4j$ The question prior to this was: Find the fifth roots of $4 + 4j$ in the form $re^{j\theta }$, where $r > 0$ and ...
1
vote
0answers
102 views

tangent function of rational angle

Can any ne help me to prove this problem? $x$ is called rational angle if $x=a\pi$ for $a\in \mathbb{Q}$. Let $0<x<\pi/4$ be a rational angle, prove that $\tan x$ is irrational. Let ...
1
vote
2answers
144 views

Understanding bicomplex numbers

I found by chance, the set of Bicomplex numbers. These numbers took particularly my attention because of their similarity to my previous personal research and question. I should say that I can't ...
7
votes
2answers
2k views

Simplest examples of real world situations that can be elegantly represented with complex numbers

Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
1
vote
2answers
108 views

How to write in polar form

To write in polar form you use this formula $$z=a+bi=r \left(\cos \theta+i\sin\theta \right)$$ I want the polarform for this rectangular function$$4\sqrt2(-1+i)$$ See this for more information ...
-2
votes
2answers
94 views

prove an equation of complex numbers

How to prove this equation: $$\sin\left(\frac{\pi}{n}\right)\cdot \sin\left(\frac{2\pi}{n}\right) \cdots \sin\left(\frac{(n-1)\pi}{n}\right)=\frac{2n}{2^n}$$ There's a hint: Consider the product of ...
2
votes
3answers
102 views

Show that $z^2=2i$ iff $z=\pm(1+i)$

I am reading Beardon's Algebra and Geometry. Show that $z^2=2i$ iff $z=\pm(1+i)$. For the problem in question, first I made the multiplication $(1+i)\times(1+i)$ which showed the result but I ...
0
votes
2answers
169 views

Real and Imaginary Parts of $\frac{\cos(z)}{(1-e^{ix})}$

Find $$\mathrm{Re}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$ and $$\mathrm{Im}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$ Please help I've been trying for some time now...
1
vote
1answer
985 views

A method for solving cubic equation

So I'm reading Beardon's Algebra and Geometry, and in chapter on complex numbers, author gives the following method for solving cubic equation: Suppose we want to solve cubic equation $p_1(z)=0$, ...
0
votes
2answers
138 views

What is the principal 12th root of one?

Let $w$ be the principal 12th root of 1. What is $w$, and what are the real and complex parts of the following: $w w^∗$ (* = complex conjugate) $w^9$
2
votes
2answers
48 views

Small inequality on unit open disc

For $|u|,|z|<1$, $u,z$ complex numbers, how to show the inequality: $|\frac{u-z}{1-\bar uz}|<1$?
1
vote
2answers
37 views

nasty exponentials

While trying to find the fourier transform of $\Large \frac{1}{1 + x^4} $, using the definition and the residue theorem has required me to evaluate nasty looking expressions like $$\large \rm ...
1
vote
2answers
156 views

Determine the integral $\int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$ using residues.

Determine the integral $$ \int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$$ using residues. This is from Section 79, Brown and Churchill's Complex Variables and Applications. In order to do this. We ...
1
vote
1answer
38 views

similarity : $z'=(1-i)z+1+i$ with the curve of $e^x-1-x$.

Let $S$ be the similarity defined by : $S(z)=(1-i)z+1+i$, for a complex number $z$ in the complex plane. What is the image of the curve : $y=e^x-x-1$ by the similarity $S$. My work : Let $z=x+iy$ ...
0
votes
1answer
39 views

convergence of complex series

Set that Re $z_n>=0$,$\forall$ n $\in$ N,Proof that if $\sum z_n$ and $\sum {z_n}^2$ are both convergent,then $\sum |z_n|^2$ is also convergent. Well I've no idea how to tackle it.
1
vote
2answers
64 views

Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist?

Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist? If yes, what is its value?
0
votes
2answers
83 views

Simplification of product of complex numbers

I look for a closed formula to the expression $$\prod_{k=1}^{n-1}\left(e^{\frac{2ik\pi}{n}}-1\right)$$ Any suggestion is welcome. Thanks.
-3
votes
1answer
96 views

$\mathbb{Z}[\sqrt{-23}]$: A uniquely written set?

I suspect that $\mathbb{Z}[\sqrt{-23}] \implies \forall~z=\sqrt{23b+a}~e^{i\arctan{\frac{23b}{a}}},~\text{where $z$ is uniquely written}~\forall~z\in \mathbb{Z}[\sqrt{-23}]$
3
votes
4answers
352 views

When does $az + b\bar{z} + c = 0$ represent a line?

$a,b,c$ and $z$ are all complex numbers. My idea was to show that it passes through the point $\infty$ in the extended complex plane, but I'm not quite sure how to execute that. Update: It says in ...
1
vote
1answer
53 views

Can we write $\sqrt[w]{z}=z^\frac{1}{w}$ when both $w$ and $z$ are complex numbers? [duplicate]

Let $w$ and $z$ be complex numbers defined in terms of real numbers $a$, $b$, $c$ and $d$ as follows: $$ w = a+bi \\ z = c+di $$ Can we analogically write $$ \sqrt[w]{z} = z^\frac{1}{w} \qquad ...