Transform Definition and 1000 Threads

Consider \(u_t = -u_{nxxx} - 3(u^2)_{nx}\). The Fourier Transform is linear so taking the Inverse Fourier transform of the Fourier Transform on the RHS we have \begin{align} -\mathcal{F}^{-1}\left[\mathcal{F}\left[u_{nxxx} - 3(u^2)_{nx}\right]\right] &= -\mathcal{F}^{-1}...

Homework Statement I'm stuck trying to find out the inverse Laplace of F(s) to get y(t) (the solution for the differential equation): Y(s) = 1 / [ (s-1)^2 + 1 ]^2 The Attempt at a Solution I tried using a translation theorem and then apply the sine formula, but the denominator...

Homework Statement Can people help me on these two questions, please. Q1) Does f(t) have a Laplace transform F(s) for sufficiently large real value of s, where f(t) = et/(t4-1). Q2) Either find a function f(t) for which F(s) = L{f(t);t→s} = es, or explain why no such function f(t)...

Today I found a program, which does Fourier transforms on pictures and tried it on some basic patterns. One of those was a lattice of dots and I have attached this and its Fourier transform to the thread. I would very much like if someone in basic details could explain what is going on. Why...

In my differential equations book (Edward and Penny) there are many examples of Laplace transforms being applied to linear differential equations with constant coefficients and no examples of them being applied to linear differential equations with variable coefficients. My question is, can this...

Finite Fourier Transform on a 2d wave How does the finite Fourier transform work exactly? The transform of f(x) is \widetilde{f}(\lambda_{n}) =\int^{L}_{0} f(x) X_{n} dx If I had a 3d wave equation pde and I applied Finite Fourier transform on the pde for z(x,y,t)=X(x)Y(y)T(t)...

Homework Statement Given a certain poincare map, show that it is area preserving for all values of rHomework Equations \binom{x_{n+1}}{y_{n+1}}=\begin{pmatrix}e^{r} & 0 \\ 0 & e^{-r} \end{pmatrix}\begin{pmatrix}cos(\phi+I_{n}) & -sin(\phi+I_{n}) \\ sin(\phi+I_{n}) & cos(\phi+I_{n})...

Hi all, as a physics student, I seldom use Fourier transform but from my understanding, given a periodic function you can decompose the function into sine function with different frequencies. Also, to get a ultra short pulse in time domain, this would require mixing many frequencies. I would...

Homework Statement \psi (x) = Ne^{ \frac{-|x|}{a}+ \frac{ixp_o}{/hbar}} Compute Fourier transform defined by ##\phi (p) = \frac{1}{ \sqrt{2 \pi \hbar}} \int \psi (x) e^{ \frac{-ipx} {\hbar}} dx## to obtain ## \phi (x) ## Homework Equations Fourier transform = ##g(x)= \frac {1}{2 \pi} \int...

Homework Statement The problem can be found in Jackson's book. An infinitesimal Lorentz transform and its inverse can be written under the form ##x^{'\alpha}=(\eta ^{\alpha \beta}+\epsilon ^{\alpha \beta})x_{\beta}## and ##x^\alpha = (\eta ^{\alpha \beta}+\epsilon ^{'\alpha \beta})...

Homework Statement Derive he Laplace Transform of the third derivative of f(t). Homework Equations The Attempt at a Solution So, I'm not at all sure how to do this. I think I can start with: L{f'''(t)} = But I'm honestly not sure how this works. Any guidance would be...

I am reviewing some material on Laplace Transforms, specifically in the context of solving PDEs, and have a question. Suppose I have an Inverse Laplace Transform of the form u(s,t)=e^((as^2+bs)t) where a,b<0. How can I invert this with respect to s, giving a function u(x,t)? Would the inverse...

Hi, I have the following question: A signal x(t) which is band-limited to 10kHz is sampled with a sampling frequency of 20kHz. The DFT (Discrete Fourier Transform) of N= 1000 samples of x(n) is then computed. To what analogue frequency does the index k=120 respond to? I'm trying to...

Hi all, I want to calculate \int_0^{\infty}e^{-a t^2}\cos(2xt)dt=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{\frac{-x^2}{a}}. The answer is known from the literature, but I don't know how to do it step by step. Any one has a clue? Thanks. Jo

Homework Statement I must show that the one dimensional wave equation ##\frac{1}{c^2} \frac{\partial u}{\partial t^2}-\frac{\partial ^2 u}{\partial x^2}=0## is invariant under the Lorentz transformation ##t'=\gamma \left ( t-\frac{xv}{c^2} \right )## , ##x'=\gamma (x-vt)##Homework Equations...

Homework Statement The Attempt at a Solution I know that u(t) is a unit step function and holds a value of either 0 or 1. In laplace transform, when we integrate f(t) from 0 to infinity, we take u(t) to be 1. In this case, since u(t) is u(-t), does this mean it holds a value of 0? Does not...

I took f(t) = SIN(10*t) +SIN(5*t) and got this f(0) = 0 f(1) = -1.5 f(2) = 0.4 f(3) = -0.3 now I tried to do the DFT Fs = 4Hz N = 4 samples 3 f[r] = Ʃ x[k]ε^(-j(2πkr/4) k=0 f[r] = 0 -1.5ε^(-j(2πr/4) + 0.4ε^(-j(2π(2)r/4) -0.3ε^(-j(2π(3)r/4) f[0] = 0 - 1.5 +...

Hi all, I am trying to hard to understand integral's transform. While an interpretation of Fourier transform is relatively easy to furnish in terms of signal decomposition and harmonics, it seems the "meaning" of Laplace transfrom is more subtle (in spite of the similarities between the two)...

Homework Statement Find the inverse Laplace transform of e^(-3pi*s)/(s^2+2s+3). Homework Equations I know that you're supposed to factor out the e^(-3pi*s) and the other part becomes 1/(s+1)^2+2 but how do you get the answer? I'm confused. The Attempt at a Solution The answer is...

Hi there, I am trying to get some practice with Fourier Transforms, there is a long way to go. For example, let me consider the function $$ \gamma (t) = \int_{-\infty}^{t} C(t-\tau) \sigma(\tau) \mathrm{d}{\tau}$$ Defining the Fourier Transform as $$ \gamma(\omega) = \frac{1}{2 \pi}...

Homework Statement Find the Inverse Laplace Transform of \frac{1}{(s^{2} + 1)^{2}} Homework Equations The Attempt at a Solution I tried using partial fractions but it didn't work. It looks like a cosine transform, but I don't know what else to do. Help please :(

Hi All, in a previous post on the physical meaning of Laplace's Transform I found the following statement " The fundamental Laplace transform pair is H(t), the Heaviside step function, and 1/s, its spectrum of damped sinusoids. Note that the spectrum is weighted towards low frequencies...

Homework Statement Find the inverse Laplace transform of F(s)=(8s^2-4s+12)/(s(s^2+4)). Homework Equations A/s+(Bs+C)/(s^2+4) 8s^2-4s+12=A(s^2+4)+(Bs+C)(s)=As^2+4A+Bs^2+Cs=s^2(A+B)+Cs+4A 8=A+B C=-4 A=3 B=5 L^-1 (3/s)+L^-1 ((5s-4)/(s^2+4)) =3+ (Now I'm stucked.) The Attempt...

I was looking through some examples which applied the duality principle while studying for an up and coming exam when it hit me that the transform applied 4 times gives you back the same function. So is there some theory that uses this? perhaps some sort of operator? I thought it...

Homework Statement Find the inverse Laplace transform of F(s)=(2s-3)/(s^2-4). Homework Equations I don't want to find the answer by looking at the Table. F(s)=2s/(s^2-4)-3/(s^2-4) The Attempt at a Solution The answer is f(t)=2 cosh 2t - (3/2) sinh 2t.

Homework Statement Compute the Fourier transform of a function of norm f(\norm{x}). Homework Equations \mathbb{F}{\frac{1}{1+\norm{x}} The Attempt at a Solution Attempt at using Cauchy theorem and the contour integral with the contour [(-R,R),(R,R+ip),(R+ip,-R+ip),(-R+ip,-R)] does...

Homework Statement The problem is tough to type out correctly. Pasting problem statement image http://postimg.org/image/a0r92a0wl/ http://postimg.org/image/a0r92a0wl/ The Attempt at a Solution I just need to know how to proceed with the problem. Not the answer. This is the scan...

I attached the problem as a word document. I'm stuck trying to determine the laplace transform for t-tU(t-1). I know I'm supposed to work with 1/s^2(s+2) and solve for A, B,C. I got B=1/2, A=-1/4, and C=1/4 when 1=(As+B)(s+2)+Cs^2. The answer to the problem is y= 1/4 + 1/2t +1/4 e^-2t -[1/4...

Why do we say that t'=t for Galilean transformation, when the low velocity limit of the Lorentz transformation is t'=t+vx/c2? If x is really big, then doesn't time cease to be absolute, no matter how small v/c is?

Checked around a buch and could not find any help. But I needed help with: Understanding that if I get the Inverse FT of K-space data, what is the scaling on the X-space (object space) resultant image/data i.e. for every tick on the axis, how do I know the spatial length? More detailed...

Having difficulty with L-1 {1/(s^2+s-20)}: Should I make it L-1 {1/(s+5)(s-4)}? I'm stuck.

what is the inverse laplace transform of (2s)(1/(s-2))? could i use the identity ∫f(T)g(t-T)dT=F(s)G(s)? i was hesitant so i figured i'd just ask before i continue..

Hello everyone: I have some question using the FFT in MATLAB for data interpolating. I don't know what the relation between the normal Fourier series and the real, image number. For example, given a set of measurement data, I can use the curve fitting toolbox to fit a curve. The general...

Homework Statement Hi, this is not a homework question per se, but something I'm wondering. Let C be a circulant n x n matrix, let x, b, be vectors such that C x = b. We would like to find a solution x. One way is to use the DFT: According to section 5, In Linear Equations, in the wikipedia...

I have been studying Fourier Optics and I have a basic conceptual question. I understand the mathematics of how to perform Fourier Transforms however the part of this topic I seem to have missed is why the action of a lens on light is the same as performing a Fourier Transform on the functional...

In the attachment that I added I highlighted the portion I am questioning. I will define L[f(t)](s) to be the laplace transform of the function f(t). f(t) = e^t L[f(t)](s) = 1/(s-1). The laplace transform is defined for all values s≠1. L[f(t)](2) = 1. Question: "What do they mean by...

Homework Statement y(t) solves the following IVP y''(t) + 2y'(t) + 10y(t) = r(t) y(0) = 2 y'(0) = 3 r(t) = 0 if t < 0 t if 0 ≤ t ≤ 1 0 if t > 1 Demonstrate that the laplace transform of y(t) is Y(s) = \frac{2s+7}{s^{2}+2s+7} + \frac{e^{-s}}{s(s^{2}+2s+7)} +...

Hi all, I'm a complete novice when it comes to describing images in frequency space and i understand that it is a way of representing images as being composed of a series of sinusoids. So a horizontal striped pattern with a single spatial frequency would have a magnitude image in frequency...

Hi! I am new here, thought to join as I am trying to learn Relativity, in this case Special Relativity. I have solved a bunch of problems already but ... The Lorentz Transform formulation I am dealing with is a 4x4 matrix. I understand the invariance of the spacetime interval and have...

is it also possible to transform any these kinds summation to any product notation: 1. infinite - convergent 2. infinite - divergent 3. finite (but preserves the "description" of the sequence) For example, I could describe the number 6, from the summation of i from i=0 until 3. Could I...

Hey guys! if anyone can help me I guess it is you! :) I'm trying to find the Fourier Series demonstration to continuous and periodic functions. I don't understand why people keep using X(jw) and X[e^jw] and even sometimes X(w) and X(f) If anyone can help me I'm really not understanding that...

Homework Statement Find the inverse Fourier transform of F(jω) = cos(4ω + pi/3)Homework Equations δ(t) <--> 1 δ(t - to) <--> exp(-j*ωo*t) cos(x) = 1/2 (exp(jx) + exp(-jx))The Attempt at a Solution So first I turned the given equation into its complex form using Euler's Formula. F(jω) = 1/2...

can you use Fourier transform to find a moving average on a data set? so, you do a Fourier transform on your one dimensional data set. next remove high order harmonics from FT result. do reverse Fourier transform on new FT result. And, vola! smoothed out data set.

Also just working on another question, especially stuck with the last part. It's basically definitions. This is what I've got so far, correct me if I'm wrong. a) k components, k components. b) R^n to R^n c) R^rank(T) d)R^nullity(T) e) Completely unsure (need help with this)

Please i need someone who can help me with these, by introduction, giving examples and explaining how i can tackle questions on it..thanks

I have posted a question on here before regarding the generation of a number sequence. I followed up that question with an answer. However, as I have developed my code more I need to use an equation instead of a lookup table. Note: I'm using MATLAB Given a matrix A of size r x c where r >=...

Homework Statement Find the following integral: Homework Equations \int \frac{e^{x}}{\sqrt{(1+e^{2x})(1-e^{4x})}}dx The Attempt at a Solution I changed the integral to: \int \frac{e^{x}}{(1+e^{2x})\sqrt{(1-e^{2x})}}dx The let u=e^x The integral becomes: \int...

Hi, I was wondering what would the Fourier transform of a signal like below give: s(t) = sin(2πt*10) ; t in [0s,5s] = sin(2πt*20) ; t in [5s,10s] I certainly did not expect it to give me 2 sharp peaks at frequencies 10Hz and 20Hz - because I understand that the addition of...

Function f(t)=\frac{t-i}{t+i} for t\in \mathbb{R} maps real ax into complex circle. Show that for any hermitian operator H operator U:=(H-iI)(H+iI)^{-1} is unitary (where H+iI is reversible) If I understand correctly U is unitary when U=U^{T} right? So I tried to show that U is unitary like...

Homework Statement (6-t)heaviside(t-2) This is just one term of the real problem I'm working, but it will serve to help me figure this out. Homework Equations The Attempt at a Solution http://www.wolframalpha.com/input/?i=laplace+transform+%7B%286-t%29heaviside%28t-2%29%7D...