Domain Definition and 704 Threads

Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex? For example, given \begin{equation} \begin{split} \hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...

The integral looks like Y_{mut, mn} = -j^{m+n}nm \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{2 ab (k^2 - k_x^2) \sin^2(\frac{k_yb}{2}) \cos^2(\frac{k_xa}{2})}{\omega \mu k_z (\frac{k_yb}{2})^2 [(n\pi)^2 - (k_xa)^2][(m\pi)^2 - (k_xa)^2]} dk_x dk_y Here, k_z = -1j \sqrt{(-(k_0^2 -...

In this example, We need to covert e2 & e5 to a form with imaginary number . we will obtain e2=j10 & e5=20 Can anyone explain how we got this?

I was just thinking about this and couldn't decide whether it was a silly question or not, so naturally I thought I might ask. It was partly prompted by one of the questions asked in the homework section. Every physical law I can think of is "self-correcting" if you substitute negative values...

I need to perform a phase shift for a measured time domain signal. The signal consists of around 1000 points. I wrote an algorithm for this in Python a while ago and it seems to work as intended, but now I noticed that the algorithm is actually unexpectedly slow. Is it so that phase shifting is...

So we'd like to find leg C. But we can't use Law of Cosines yet so we will use Law of Sines. We can easily find the length of A and this is ##\sqrt{13}##. With some geometry we can see that ##\angle a = 53.1##. We can now use Law of sines. $$\frac{\sin(a)}{A} = \frac{\sin(b)}{B}$$ We want to...

D={(x,y)∈ℝ2: 2|y|-2≤|x|≤½|y|+1} I am struggling on finding the domain of such function my attempt : first system \begin{cases} x≥2y-2\\ -x≥2y-2\\ x≥-2y-2\\ -x≥-2y-2 \end{cases} second system \begin{cases} x≤y/2+1\\ x≤-y/2+1\\ -x≤y/2+1\\ -x≤-y/2+1\\ \end{cases} i draw the graph and get the...

hey there I'm struggling on finding the domain of the following function log (xy2)+x2y) I then do xy(y+x)>0 but then i don't know what to do with xy one attempt \begin{cases} y+x>0\\ x>0\\ y>0 \end{cases} union \begin{cases} y+x<0\\ x<0\\ y<0 \end{cases} but this doesn't lead to the...

There are an infinite number of natural numbers. Why is that? Well this follows from the following facts: (i) There is at least one natural number. (ii) For each natural number there is a distinct number which is its successor, i.e., for each number $x$ there is a distinct number $y$ such that...

## Let~~f(x)=h(x)+g(x) , where~~h(x)=10^{\sin x}~~and~~g(x)=10^{\csc x}## ##Then,~~D_f = {D_h}\cap {D_g}## ##Clearly,~~D_h=ℝ~~and~~D_g=ℝ-\{nπ|n∈ℤ\}## ##∴~~D_f =ℝ-\{nπ|n∈ℤ\}## After considering the new domain, the range of ##\sin x## in ##10^{\sin x}## is ##[-1,1]-\{0\}## Therefore, the range of...

In the derivation of energy conservation, there is the transformation ##q(t)\rightarrow q'(t)=q(t+\epsilon)##, whose end points are kind of fuzzy. The original path ##q(t)## is only defined from ##t_1## to ##t_2##. If this transformation rule is imposed, ##q'(t_2-\epsilon)=q(t_2)## to...

$\textsf{6.2.15 Find the domain of each function.}$ (a) $f(x)=\dfrac{1-e^{x^2}}{1-e^{1-x^2}}$ set the denominator to zero and solve $1-e^{1-x^2}=0$ then $x=1,-1$ from testing the domain is $(-1,1)$(b) $f(x)=\dfrac{1+x}{e^{ \cos x}}$ set $e^{\cos x}=0$...

Consider a one to one transformation of a ##3##-##D## volume from variable ##(x,y,z)## to ##(t,u,v)##: ##\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv## ##(1)## Now for a particular three dimensional volume, is it...

Total no. of amino acid in each transmembrane domain = 4 nm/ 0.15 nm +1 =27.66 Total no. of amino acids in 7 transmembrane domains = 27.66 x 7 = 193.62 ≈ 190 Hence option d. Is my approach correct?

To check if it is injective : ##h'(x) = 3(x^2-1)## ##\implies h'(x) \geq 0## for ##x \in (-\infty, -1]## Thus, ##f(x)## is increasing over the given domain and thus is one-one. To check if it is surjective : Range of ##f(x) = (0, e^4]## but co-domain is ##(0, e^5]## thus the function is into...

Consider the function sqrt(x). What is the domain of this function? Is it all real positive numbers? This is what I was taught in high school, but I was also taught that plugging in -1 would give an answer of i. So if the function takes negative inputs, shouldn’t they be part of the domain?

Hello, I have a signal and got the FFT result of that. I have shown them both below along with the MATLAB code. May I ask if there is any method to find the time zone(s) in the signal that a specific frequency has(have) happened? The reason I'm asking this is that I want to specify the time...

Homework Statement Find a function where the domain is integers, codomain is real numbers, and image isn't equal to codomain. Homework EquationsThe Attempt at a Solution I know that it means that when I plug in an integer I will obtain a real number, but how do I make it so that the image is...

Hello everyone, I am new here, and wondering if anyone would have some knowledge on a current problem I have been working on. I am trying to create time- and frequency domain 3D models of an object hanging using a spring from a fixed point within a shell. I am trying to figure out how the...

Hi. I'm trying to parallelize my code. I am new at MPI, I'm learning the basics. I want to use a domain decomposition strategy in my code. I've been reading a bit, and wanted to use this subroutine to exchange points between neighbors. I've started by trying to modify a code presented by Gropp...

There is some controversy as to why the spectral content of vinyl records contain frequency "information" above what the analog tape recorder was capable of capturing. A thought experiment: If I have a piano that has a key which plays 15khz (primary freq. of that key's piano string), what...

ArsTechnica on Copyright expiration and the legal pitfalls around it: https://arstechnica.com/tech-policy/2019/01/a-whole-years-worth-of-works-just-fell-into-the-public-domain/

1. A process can be represented by the first order equation ##4\frac{dy(t)}{dt}+y(t)=3u(t)## Assume the initial steady state is steady (y=0 at t=-0) (a) Determine the transfer function of this process in the s domain (b) If the input is a ramp change in u(t)=4t, determine the value of...

Hi PF! I type the following into Mathematica: domain = -10; ListPlot[{0.3}, PlotMarkers -> {{"\[LongDash]", Large}}, PlotRange -> {{0, 5.5}, {-10, 200}}] And notice we see the plot 0.3. But, now let domain = 0. Notice now we can't see 0.3. Does anyone know how to fix this?

Homework Statement The operating cost of a truck is 12+(x/6) per km, when the truck travels x km/hr. If the driver earns 6 rupees (*or whatever currency your country has) per hour, what is the most economic speed to operate the truck on a 400 km road? Also, due to weather, the truck can travel...

can anyone help me on how I can map an isosceles trapezoid onto a rectangular/square domain.Actually I need to solve Laplace equation(delta u = 0) over this isosceles trapezoidal domain. Schwarz Christoffel mapping may help me. But can anyone give me any hint on this mapping procedure?

Find the inverse function $f^{-1}$ of $f$. Find the range of $f$ and the domain and range of $f^{-1}$ $$f(x)=5 \sin x +2; \quad -\frac{\pi}{2}\le x \le \frac{\pi}{2}$$ exchange x for y and solve for y \begin{align*}\displaystyle x&=5\sin y + 2 \\...

Homework Statement Find the domain of f(x,y)=sqrt(5xy-2y^2) Homework EquationsThe Attempt at a Solution As far as I understand it's all about solving inequalities when you have this kind of problem. The problem doesn't state it is in ℝ2 but I'm pretty sure it is. So no negative numbers under...

I just asked a similar question, but I got help for that one, and now I am stumped again. I need to find the domain for $$f(x) = ln(x^2-5x)$$ What's confusing me is how to deal with the exponent. I can't think of a way to get around it.

I need to find the domain of f(x) = log5(8 - 2x). I am not sure if I could find this by doing a inequality equation, but I think my professor wants us to do it on a number-line anyway. I'm sorry, but I am not sure where to start.

How do you find the domain and range of a graph?

In this DOS video, he talks about 5 common misconceptions on QUantum COmputing:

In the book "The Basics of Abstract Algebra" Bland defines a Euclidean Domain using two conditions as follows: In the book "Abstract Algebra"by Dummit and Foote we find that a Euclidean Domain is defined using only one of Bland's conditions ... as follows:What are the consequences of these...

Apologies if this is a stupid question, but is the fundamental domain unique? And what exactly is the definition of a cusp- a quick google tells me it is 'where two curves meet', so looking at the fundamental domain,I would say ##\omega=\exp^{\frac{2\pi i}{ 3}} ## and ##\omega*## are? thanks

Domain of Science reviews science in sci fi movies: The reviewer talks about how physics is misused in movies and cringes at some of the funny movie moments where they got physics wrong.

Hi everyone. My book (Smith's "An Introduction to Goedel's Theorems") defines the numerical domain of an algorithm as the set of naturals that, when input individually to the algorithm, result in its "working", that is to terminate and output some result. In the book it is also stated that any...

Homework Statement Consider the function ##f\left(x\right)=\sqrt {x+2}##. Determine if the function is a one-to-one function, If so, find ##f^{-1}\left(x\right)## and state the domain and range of ##f\left(x\right)## and ##f^{-1}\left(x\right)## Homework Equations N/A The Attempt at a...

Hi, in a text provided by DrDu which I am still reading, it is given that "the momentum operator P is not self-adjoint even if its adjoint ##P^{\dagger}=-\hbar D## has the same formal expression, but it acts on a different space of functions." Regarding the two main operators, X and D, each has...

Hi everyone, I'm reading this paper about the solution of the heat equation inside an infinite domain: https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/lecture-notes/fourtran.pdf 1) Please let me know if the following discussion is correct. The...

Hi, I have a strange nonlinear operator which yields non-Hermitian solutions when treated in a simple ODE, ##H\Psi##=0. It appears from a paper by Dr Du in a different posting, that an operator can be non-self-adjoint in one domain, but be self-adjoint in another domain defined by the interval...

So, I'm a bit confused. The thing is, basically, all elementary functions are of the form ƒ:ℝ→ℝ. So the domain is ℝ and so is the codomain. However, if we have a function ƒ:ℝ→ℝ, given with f(x) = √x, it's domain is now x≥0. So, is the domain of this function ℝ or [0,+∞>? Also, let's say we have...

Recently when I reviewed something about the composite function for my calculus exam, I remembered I had been thinking a question for quite a long time (maybe I was going into a dead end) since I was in high school. I was thinking whether f(x)=1/(1/x) and g(x)=x are the same function or not. It...

Hi, I'm struggling to understand something. Does domain restriction work the same way for composition of inverse functions as it does for other composite functions? I would assume it does, but the end result seems counter-intuitive. For example: If I have the function f(x) = 1/(1+x), with...

Find the domain of f(x) = 2/(x - 3). 1. Are we looking for the domain of f or f(x)? 2. Solution Set x - 3 = 0 and solve for x. x - 3 = 0 x - 3 + 3 = 3 x = 3 Let D = domain D = ALL REAL NUMBERS except for x = 3. Yes? P.S. Does x = 3 mean there is a hole at the point (3, 0) for this...

I found in a book that the domain of tan x was {(2n+1)π/2 , n∈I} The graph however shows that for every value of x , the function takes on a value .So, why is the domain like this?

Homework Statement The piece wise function is -x-2, x<-1 x^2-3x, -1≤ x ≤5 3x+5, x>5 The problem is to transform the function with...

Hi, I would like to know how one can simultaneously measure the thickness and refractive index of a sample using Fourier Domain OCT. I have a glue layer on a glass microscope slide and I've calculated the thickness of the glue layer, but I'm unsure of how to the refractive index of the glue. Any...

1) Given that the range of function f(A) = \frac{1 + sinA}{1 - sinA} is { 0,1,3} . Find the domain of the function .

Hi, I need to solve Laplace equation ## \nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ## r_1<r<r_2 ##, ## 0<z<L ##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ &\Phi(L,r)=V_P \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial...

The parameterized post-Newtonian formalism is an approximation of general relativity that is easier to calculate with that applies in "weak" gravitational fields where objects are "slow" moving. My question is: "What is the approximate domain of applicability of the post-Newtonian formalism?"...