Tagged Questions
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 ...
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
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
1
vote
1answer
51 views
$\int_{-\infty}^{\infty}i\cdot \sin(x)\sin(2{\pi}kx)\;dx$ during Fourier transform
I am trying to do a time-to-frequency domain transform using Fourier transform. My function is very simple:
$$
f(x) = \sin(x)
$$
By definition its Fourier transform should be:
$$
F(k) = ...
2
votes
2answers
48 views
Calculate $\lim_{n\to\infty} ((a+b+c)^n+(a+\epsilon b+\epsilon^2c)^n+(a+\epsilon^2b+\epsilon c)^n)$
Calculate $\lim_{n\to\infty} ((a+b+c)^n+(a+\epsilon b+\epsilon^2c)^n+(a+\epsilon^2b+\epsilon c)^n)$ with $a,b,c \in \Bbb R$ and $\epsilon \in \Bbb C \setminus \Bbb R, \epsilon^3=1$.
Since $a+\epsilon ...
2
votes
1answer
75 views
Find limit of a complex function
Does it exist? if it exists, how to find the below limit:
$$\lim_{z\rightarrow 0}z\log\left(\sin \pi z\right)=?$$,where $z \in \Bbb{C}$
2
votes
1answer
51 views
Is this estimation correct?
I have to estimate the following quantity
$$\frac{|e^{i\sqrt{\lambda+i\varepsilon}|x|}-e^{i\sqrt{\lambda}|x|}|^2}{|x|^2}$$
in $\mathbb{R}^3$ ($\lambda>0$) where
...
0
votes
3answers
83 views
A not too simple complex number inequality
Prove the following inequality $\forall n>0$ $\forall z \in \mathbb{C}$ such that $|z|=1$:
$$\vert z+\frac{1}{z} \vert <\vert z^{n} + i \vert + \vert \overline{z}^{n} + i \vert \leq 2\sqrt{2} ...
2
votes
4answers
70 views
rewrite $2ie^{i\pi}+i^3$
i am asked to rewrite $2ie^{i\pi}+i^3$ into $x+iy$ form. i just tried all what i know so far, but couldnot come to solution. i said: $2ie^{i\pi}+i^3=2ie^{i\pi}-i$ but further i am stuck really. i am ...
1
vote
0answers
44 views
show this summation hold in term of integral
i can show $\sum |x|^2=\int_a^b|f(x)|^2dx$ in term of integral, or this one $|\sum x\overline y|^2=|\int_a^bf(x)\overline {g(x)}dx|^2$ but i don't know how to show this one ...
1
vote
1answer
57 views
What is more compact equation of this relationship?
What is more compact equation of this relationship?
$\sum |x_i|^2\sum |y_j|^2+\sum |x_j|^2\sum |y_i|^2-2|\sum x_i \overline y_i||\sum x_j \overline y_j|$
Remark:
Euclidean space
$\sum x_i^2\sum ...
2
votes
1answer
73 views
Evaluate a certain derivative
Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $\{l_1,\dots,l_n\}$ a vector of natural numbers such that $l_1+l_2+\dots+l_n=N$. Let
$$ h_j(x)=\prod_{i\neq j,i=1,\dots, n} ...
2
votes
3answers
189 views
Are there five complex numbers satisfying the following equalities?
Can anyone help on the following question?
Are there five complex numbers $z_{1}$, $z_{2}$ , $z_{3}$ , $z_{4}$
and $z_{5}$ with ...
1
vote
1answer
96 views
Complex Numbers geometry question
on each edge of a quadrilateral ABCD you build a square such that the points H, G, F, E are the centers of these squares (the intersection of the diagonals).
I need to use complex numbers to prove ...
1
vote
2answers
65 views
Exponential Power Identity
Let $A_1,A_2,A_3$ and $\alpha_1,\alpha_2,\alpha_3$ be real constants.
Suppose the equation $A_1e^{i\alpha_1x}+A_2e^{i\alpha_2x}=A_3e^{i\alpha_3x}$
holds $\forall x\in\mathbb{R}$. (where $i^2=-1$)
...
0
votes
1answer
56 views
Find $\int_{C}\frac{e^{\pi z}}{\frac{(z-4i)^2}{z-i}}dz$
Let $C$ be $x^2+y^2=9$, oriented counterclockwise.
Find
$$\int_{C}\frac{e^{\pi z}}{\frac{(z-4i)^2}{z-i}}dz$$
It is easy to find the parameterization of $C$. However, when it comes to the integral, I ...
2
votes
1answer
124 views
Does $\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$ exits?
For$$\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$$
Does $\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$ exits? If the answer is no, why?
Does $\bar{z}$ represents $a-bi$?
1
vote
4answers
66 views
Complex functions
I need help solving this problem:
For $z$ a complex number, let $g(z) = \frac{1 + 2z }{1 + z}$. Find a function $h_1(z)$ such that $h_1(g(z)) = z$ and another (possibly the same) $h_2(z)$ such that ...
4
votes
2answers
261 views
Does $\sqrt{x}$ have a limit for $x \to 0$?
I am taking a calculus course, and in one of the exercises in the book, I am asked to find the limits for both sides of $\sqrt{x}$ where $x \to 0$.
Graph for sqrt(x) from WolframAlpha:
This is how ...
0
votes
0answers
88 views
analytic map between two triangles
Consider the two dimensional real place $\mathbb{R}^2$. We have a triangle (lets call it $\triangle$) with the vertices $(\frac{\sqrt{3}}{2},\frac{3}{2}),~(-\frac{\sqrt{3}}{2},\frac{3}{2})$ and ...
0
votes
3answers
528 views
Evaluate $\int_0^{2\pi} |x \cos(\theta)+y \sin(\theta)|\, d\theta$
I am required to prove that $\displaystyle \int_0^{2\pi} |x \cos(\theta)+y \sin(\theta)|\, d\theta= 4\sqrt{x^2+y^2}$, $\ x$ and $y$ are real.
I let $\sin\theta = \frac yz$, $\cos\theta=\frac xz$, ...
1
vote
1answer
157 views
Real part of an integral with complex argument
This is a paper about Fourier cosine series approximation to option pricing problem.
The coefficient $A_k$ is defined as $$A_k=\frac{2}{b-a}\int_a^bf(x)\cos\left(k\pi\frac{x-a}{b-a}\right)dx$$ Then ...
0
votes
1answer
93 views
Show that the function $g: S \rightarrow \mathbb{C}$, given by $z\mapsto z^3$, is surjective but not injective.
Let $S$ denote the closed sector $0 \leq \arg (z) \leq 2\pi/3$, in the complex plane, including the vertex at $z = 0$. Show that the function, $g: S \rightarrow \mathbb{C}$ , given by $z\mapsto z^3$, ...
3
votes
3answers
372 views
Factorize the polynomial $P(z) = z^4 - 2z^3-z^2+2z+10$, into linear and/or quadratic factors with real coefficients
$2+i$ is given to be one of the roots of the polynomial.
I am doing this as a practice for exam prep.
Since $2+i$, is a root, then $(z-2-i)$ is a factor?
So I have:
$(z-2-i)(z^3-Az^2-Bz+C) = ...
1
vote
0answers
15 views
Optimal bounding constant for partial sums of a signed sum of numbers in the unit disk.
This recent question received several answers, and GenericHuman's answer and the comments below provide a good synthesis of all the other answers in my opinion.
In this synthesis, only one related ...
2
votes
5answers
260 views
Complex number: calculate $(1 + i)^n$.
I have to solve the following complex number exercise: calculate $(1 + i)^n\forall n\in\mathbb{N}$ giving the result in $a + ib$ notation.
Basically what I have done is calculate $(1 + i)^n$ for some ...
1
vote
1answer
123 views
Trigonometric result concerning DeMoivre's formula.
Given this question is rather long to answer, and I'm losing hope it'll ever be, I just want an answer to this particular claim:
Working on the unitary circle, let $x=1-\cos \theta$ and $t=1-\cos n ...
1
vote
0answers
140 views
How to choose a proper contour for a contour integral?
When analyzing real integrals with contour integrals, how does one choose a proper contour integral?
Many cases can be solved by integrating around the top half of a circle with radius of infinity ...
4
votes
6answers
610 views
Solving $z^4 + 2z^3 + 6z - 9 = 0$
I'm trying to solve $z^4 + 2z^3 + 6z - 9 = 0$.
$z$ is a complex number.
I usually can solve those equations when they are of second degree.
I don't know what to do, breaking out $z$ doesn't help...
...
0
votes
1answer
106 views
Trigonometric Input to an First Order Differential Equation, Exponentials
In an ODE class, the differential equation is given
$y' + ky = kq_e(t)$
where the input $q_e(t)$ is given as $cos \ \omega t$. The teacher "complexifies" the problem by using the real part of ...
5
votes
5answers
513 views
Are there any calculus/complex numbers/etc proofs of the pythagorean theorem?
I have been looking for proofs for the pythagorean theorem that don't use area calculation but calculus, complex numbers or any other interesting ways to proof it.
I would love to see any interesting ...
2
votes
3answers
350 views
How to show $\arcsin{x} = \frac{\pi}{2} + i \ln{(x+\sqrt{x^2-1})}$?
Is the following identity correct
$\arcsin{x} = \frac{\pi}{2} + i \ln{(x+\sqrt{x^2-1})}?$
Here, $x < 1$. How can we show that it is true? One way to see it is by differentiating, since
...
