Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.
1
vote
0answers
42 views
Improper integration (log, exp)
Can any one, please, help me with this problem,
$$\int_0^{\infty}\!\frac{ p (x)}{ q( x)} {(\ln(x))^{n}} e^{imx}\,dx = ?$$
where $n=1,2,\ldots,m$ and $ m \in \Bbb Z^+$, $$p (x), q(x) \in \Bbb R[x]$$
...
20
votes
10answers
924 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 ...
1
vote
1answer
22 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
44 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
70 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
54 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??
1
vote
1answer
37 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
50 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 ...
-2
votes
1answer
65 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
133 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: $$S=\{z_k:k\in\mathbb{N}\cap[1,n]\}=\{e^{2\pi ...
1
vote
0answers
24 views
Probability distribution of the product of two independent complex gaussian random variables
I have to calculate the pdf of $Z = X*Y$, where $X \in \mathcal{C}(\mu_x,\Sigma_x)$ and $Y \in \mathcal{C}(\mu_y,\Sigma_y)$, where $\mathcal{C}$ is a complex distribution. It can be assumed that ...
0
votes
1answer
41 views
Showing that the segment joining $0$ to $z$ is perpendicular to the segment joining $0$ and $w$ iff $Re[z\bar{w}]=0$
I'm reading Beardon's Algebra and Geometry.
Suppose that $zw\neq0$. Show that the segment joining $0$ to $z$ is perpendicular to the segment joining $0$ to $w$ if and only if $Re[z\bar{w}]=0$.
...
0
votes
0answers
26 views
Introduction to fractional calculus: problem with identity
I can't see the next step:
$D^\alpha e^{ix} = i^{\alpha}e^{ix} = e^{i\alpha \frac \pi2}e^{ix}$
0
votes
2answers
30 views
contradicting identity theorem?
the identity theorem for holomorphic functions states: given functions $f$ and $g$ holomorphic on a connected open set $D$, if $f = g$ on some open subset of $D$, then $f = g$ on $D$
Let $f(z) = \sin ...
0
votes
1answer
20 views
Addition in polar form
$$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 ...
-5
votes
1answer
119 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
33 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
79 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
22 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
32 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
19 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
36 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}$
3
votes
3answers
96 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!
4
votes
0answers
81 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
29 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 ...
5
votes
3answers
167 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
78 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
79 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
139 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
82 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?
1
vote
2answers
90 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
23 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
51 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 ...
0
votes
1answer
39 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 ...
1
vote
3answers
107 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
83 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 $z=0$.
7
votes
5answers
239 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 ...
3
votes
6answers
50 views
Motivation for creation of complex exponentiation
I am curious how mathematicians came to develop complex exponentiation. How is the rule for complex exponentiation derived?
0
votes
0answers
26 views
complex equation: inequality of complex numbers
Let $a$ and $c$ be two complex numbers. Then there is at least one complex number $z$ such that
$|z-a| + |z+a| = 2|c|$
if and only if
(1) $|c| < |a|$
(2) $|c| <= |a|$
(3) $|c| > |a|$
...
1
vote
1answer
50 views
What is the odd fourier extention of sin x cos(2x)
odd half range extension of
f(x) = sin x cos(2x) with limits 0 to pi
2
votes
2answers
36 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
60 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
55 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 ...
0
votes
0answers
21 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
45 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 ...
5
votes
0answers
57 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
29 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
65 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
78 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
46 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...
