Transform Definition and 1000 Threads

Homework Statement Given that ##x_\mu x^\mu = y_\mu y^\mu## under a Lorentz transform (##x^\mu \rightarrow y^\mu##, ##x_\mu \rightarrow y_\mu##), and that ##x^\mu \rightarrow y^\mu = \Lambda^\mu{}_\nu x^\nu##, show that ##x_\mu \rightarrow y_\mu = \Lambda_\mu{}^\nu x_\nu##. Homework Equations...

I'm recently new to the field of 2D Fourier Transform Infrared Spectroscopy and am learning its applications. I would like to know its applications in biology. Specifically, is there anything in the 400 nm to 1000 nm range that is important in protein structure, protein dynamics or biology in...

Homework Statement I do not know how to find f(t) with the given Ampliture 40 and a=pi Homework EquationsThe Attempt at a Solution I have the solution above. my set up was 1/2y''+y'+5=f(t) 1/2S^2* Y(s) + Y(s)+5=f(t)

Homework Statement I'm given a transfer function C(s)=10R(s)/(s+4) And I have to find c(t) for r(t)=6u(t) The Attempt at a Solution First I did this problem by taking inverse laplace of the transfer function, and inserting the value of r(t) in it. Next I did the same problem by first...

Alright guys. First off, this is my first post (happy to be here!) and I'm hoping this is the correct section of the forum. I'm an engineering student, currently working towards finishing my master's thesis. Short introduction. I am trying to simulate an ocean wave environment, as a...

I'm asked to transform y(t) = x(t)*x(t) (where * is the convolution product) and x(t)= sinc(t)cos(2π10t) ( sinc(t)= sin(πt)/(πt) ).The attempt at a solution Clearly everything is simple if you know X(f), because y(t)=InverseFourier{ X(f)2 }. The problem is that I can't find X(f). By the way...

Hi, friends! In order to find an orthogonal basis of eigenvectors of the Fourier transform operator ##F : L_2(\mathbb{R})\to L_2(\mathbb{R}),f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x## for Euclidean separable space ##L_2(\mathbb{R})##, so that ##F## would be represented by...

Homework Statement Solve the integral y(t) + \int_0^t (t-u)y(u) \, du = 3sin(2t) Homework EquationsThe Attempt at a Solution Rewrite the equation: y(t) = 3sin(2t) - \int_0^t (t-u)y(u) \, du I assume the integral to be the convolution: f(t) * y(t) = t * y(t) as f(t-u) = f(t) = t...

Homework Statement Solve the IVP : dy/dt + y = f(t) y(0) = -5 where f(t) = -1, 0 <= t < 7 -5, t >= 7 y(t) for 0 <= t < 7 = ? y(t) for t >= 7 = ? Homework EquationsThe Attempt at a Solution So I have never seen a problem of this type, excuse my silly mistakes if I'm interpreting this question...

Homework Statement Determine the Laplace transform of the given function: f(t) = sin(t) for 0 <= t < \pi and f(t) = 0 for \pi <= t Homework EquationsThe Attempt at a Solution Ok, I've been having some trouble figuring out how I should write the above branched function (sorry for the...

Homework Statement A damped harmonic oscillator is driven by a force of the form f(t)=h(t) t^2 Exp(-t), where h(t) is a Heaviside step function. The Oscillator satisfies the equation x''+2x'+4x=f(t). Use pencil-and-paper methods involving Fourier transforms and inverse transforms to find the...

Homework Statement Using a theorem (state which theorem you are using and give the formula), Calculate the Fourier Transform of 1. rect(x)triangle(x) 2.cos(pi*x)sinc(x) 3.rect(x)exp(-pi*x^2) 4.sinc(x)sin(pi*x) 5. exp(-pi*x^2)cos(pi*x) Homework Equations not sure what theorem to use for the...

Inverse Laplace transform \mathcal{L}^{-1}[F(p)]=\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}F(s)e^{st}dp=f(t) Question if we integrate along a straight line in complex plane where axis are Re(p), Im(p), why we integrate from c-i \ínfty to c+\infty? So my question is, because Im(p) are also...

Homework Statement Find the inverse Fourier transform of X(ejw = 1/(1-ae-jw)2 using the convolution theorem. Homework EquationsThe Attempt at a Solution I tried finding the partial fraction coefficients but without success.

Homework Statement Find L[x(t)], where $$ x(t) = tu(t) + 3e^{-1}u(-t) $$ Also determine the region of convergenceHomework EquationsLaplace properties, Laplace table: L[te-at = 1/(s+a)2 L[u(t)] = 1/s L[t] = 1/s2 The Attempt at a Solution I don't really know what to do with this as my table...

Hello, everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field: \tilde{p}=-\frac{i}{{{k}^{2}}}\mathbf{F}\centerdot \mathbf{k} where i is the imaginary unit...

Homework Statement Division by s Equals integration by t: For this problem use the following property (see relevant equations) to find the inverse transform of the given function: F(s) = \frac{1}{s(s-1)} Homework Equations L^{-1}(\frac{F(s)}{s}) = \int_{0}^{t} f(\tau)\,d \tau The Attempt...

Homework Statement If L[x(t)] = (s + 4)/(s2 + 1), find L[tx(t)] Homework Equations Laplace transform: F(s) = 0∫ f(t)e-stdtLaplace table The Attempt at a Solution Clearly it's not just asking for a Laplace transform. Not sure what it's specifically asking to be honest. t multiplied by...

Homework Statement x(t) = cos(3πt) h(t) = e-2tu(t) Find y(t) = x(t) * h(t) (ie convolution) Homework Equations Y(s) = X(s)H(s) and then take inverse laplace tranform of Y(s) The Attempt at a Solution L(x(t)) = \frac{s}{s^2+9π^2} L(h(t)) = \frac{1}{s+2} I then try to find the partial...

Homework Statement A 4-pound weight stretches a spring 2 feet. The weight is released from rest 15 inches above the equilibrium position, and the resulting motion takes place in a medium offering a damping force numerically equal to 7/8 times the instantaneous velocity. Use the Laplace...

Homework Statement x(t) = cos(3πt) h(t) = e-2tu(t) Find y(t) = x(t) * h(t) (ie convolution) Homework Equations Y(s) = X(s)H(s) and then take inverse laplace tranform of Y(s) The Attempt at a Solution L(x(t)) = [πδ(ω - 3π) + πδ(ω + 3π)] L(h(t)) = \frac{1}{s+2} Laplace Transform inverse ...

Any books on Fourier transforms and other transformations? (Thanks, for any help)

Hello, I hope somebody can help me with this. 1. Homework Statement I am supposed to show that if there is a function \phi(x,t) which is real, satisfies a linear wave equation and which satisfies \phi(x,0)=0 for x<0 then the Fourier Transform \tilde{\phi}(k) of \phi(x,0) is in the lower...

Can anyone point me to some material on applying the Fourier transform to the case of an analytic function of one complex variable? I've tried to generalize it myself, but I want to see if I'm overlooking some important things. I've started by writing the analytic function with u + iv where u...

Homework Statement I keep getting the wrong answer, and wolphram seems to back me up. Here's the system of equations ##(-10+s)X(s)-7Y(s)=\frac{10}{s}## ##X(s)+(-2+s)Y(s)=0## Homework EquationsThe Attempt at a Solution Using Cramer's rule I've got...

Find the LT and specify ROC of: x(t) = e-at, 0 ≤ t ≤ T = 0, elsewhere where a > 0 Attempt: X(s) = - 1/(s+a)*e-(s+a) integrated from 0 to T => -1/(s+a)[e-(s+a) + 1] Converges to X(s) = 1/(s+a) , a ⊂ R, if Re{s} > -a for 0≤t≤T Elsewhere ROC is empty (LT doesn't exist). Is this...

we have a wavefunction \psi (x) the question asks for \psi (p) and says to use this to calculate the expectation value of momentum. The problem is the expectation value of momentum is integrated over dx so after transforming how do you get the integral to be over dp? thanks for any help with...

Mod note: Please don't tinker with SIZE tags. Things are perfectly readable without them. Homework Statement Find x(t) = L-1[(4e-4s - 3)/(s2 + 6s + 25)]. Homework Equations L(x(t)) = ∞∫∞x(t)e-stdt L-1(x(s)) = (1/2π)(σ - ∞j)∫(σ + ∞j)[x(s)est]ds, "But you want to avoid this integral." Laplace...

Hi Folks, I need to evaluate the following function f(t)=A[1+B \cos(\omega_1 t+ \phi)] \cos(\omega_2 t+ \phi) to find f(\omega) using the Fourier transform. Ie, the Fourier transform I use is f(\omega)=\displaystyle \frac{1}{\sqrt {2 \pi}} \int^{\infty}_{-\infty} f(t) (\cos \omega t+ j \sin...

<< Moderator Note -- thread moved to the Homework Help forums >>[/color] I'm stuck on a problem, and I'm in serious need of help. I) Problem: Find the solution to f (t) = 2 \int^t_0 f'(u) sin 3 (t-u) \ du + 2 cos (3t) . Also find f (0) .II) Solution, so far: F(s) = 2 (s F(s) - f(0)) *...

I've learned that a vector in coordinate system can be expressed as follows: A = axAx+ayAy+azAz. ai, i = x, y, z, are the base vectors. The transformation matrix from cylindrical coordinates to cartesian coordiantes is: Ax cosΦ -sinΦ 0 Ar Ay = sinΦ cosΦ...

Hello everyone, I have spend whole day but still not figure out an inverse Laplace transform. Hope someone can help me. The question is in the attachment. I'm trying to extract u^2/4D^2 out the bracket to match the standard inverse table, but it seems difficult to deal with the square root...

Homework Statement we have to solve the given circuit using laplace transform for i(t) for t>0 when switch k opened at t=0.. now all I wnated to make clear is the final circuit diagram of this in s-domain..[/B]Homework EquationsThe Attempt at a Solution as I solved I considered the circuit to...

Hi everyone, do you know how to calculate the Fourier transform for the infinitely deep circular well (confined system)? The radial wave function is given by R=N_m J_m (k r). k=\alpha_{mn}/R. R is the radius of the circular well. R(k R)=0. Thanks. Another question is that The k in J_{m}(k r)...

Hi, I have a simple harmonic oscillation problem whose Green function is given by $$\Bigl[\frac {d^2}{dt^2}+ \omega_{0}^{2}\Bigl] G(t, t') = \delta(t-t')$$ Now I found out the Fourier transform of $G(t, t')$ to be $$G(\omega)= \frac{1}{2\pi} \frac{1}{\omega_{0}^{2}-\omega^2}$$ which has poles...

Hello, Let's suppose we are given a function f:\mathbb{R}\rightarrow \mathbb{R}, and we assume its Fourier transform F=\mathcal{F}(f) exists and has compact support. What sufficient condition could we impose on f, in order to be sure that F is also bounded?

Whais is the effect of a multiplication by (-1)^n in the DTFT ?? In other words, what's is this transform : x(n)* (-1)^n ??

Hi, I am totally a non-math guy. I had to attend a training (on automobile noise signals) that had a session discussed about Fourier Transform (FT). Let me pl. write down what I understand: "The noise signal observed at any point in the transmission line can be formed using a sum of many sine...

Homework Statement Silly question, but I can't seem to figure out why, in e.g. Peskin and Schroeder or Ryder's QFT, the Fourier transform of the (quantized) real scalar field \phi(x) is written as \phi (x) = \int \frac{d^3k}{(2\pi)^3 2k_0} \left( a(k)e^{-ik \cdot x} + a^{\dagger}(k)e^{ik...

Homework Statement Wondering if I did this correctly.. Find the laplace transform: $$z(t)=e^{-6t}sin(\omega_{1}t)+e^{4t}cos(\omega_{2}t)$$ for ##t\geq 0## Homework Equations The Attempt at a Solution For the first part, I assume I can do this, but I'm not too sure. This is my main question...

please check my work here $\mathscr{L}[2\sin(bt)\sinh(bt)]$ I know that $\sinh(bt) = \frac{e^{bt}-e^{-bt}}{2}$$\mathscr{L}[2\sin(bt)\left(\frac{e^{bt}-e^{-bt}}{2}\right)]$...

Homework Statement A traveler in a rocket of length 2d sets up a coordinate system S' with origin O' anchored at the exact middle of the rocket and the x' axis along the rocket’s length. At t' = 0 she ignites a flashbulb at O'. (a) Write down the coordinates t'_F, x'_F, and t'_B, x'_B for...

I have been given this y(t)=\frac{sin(200πt)}{πt} All I want is to find, is how the rectangular pulse will look like if I take the transformation of the above. That "200" kind of confusing me, because it isn't a simple sinc(t)=\frac{sin(πt)}{πt} I need somehow to find the height of the...

Homework Statement Decide the inverse laplace transform of the problem below: F(s)= \frac{4s-5}{s^2-4s+8} You're allowed to use s shifting. Homework Equations The Attempt at a Solution By looking at the denominator, I see that it might be factorized easily, so I try that...

Homework Statement Find the inverse Laplace transform of the expression: F(S) = \frac{3s+5}{s^2 +9} Homework Equations The Attempt at a Solution From general Laplace transforms, I see a pattern with laplace transforming sin(t) and cos(t) because: L{sin(t)+cos(t)} =...

Hi, I'm new here, I was just wondering if anyone could help clarify a subject I'm having difficulties teaching myself... In thermo we perform a "Legendre transform" on the internal energy with respect to entropy. The stated purpose of this is so that we don't have to work in the entropy...

I took an introduction to ODEs course this past spring semester. It always bothered me where this thing came from. I did a little bit of research and found a video of a professor explaining how it is the continuous analog of an infinite sum. He did a little bit of a derivation using that...

Hi! I am taking a second look on Fourier transforms. While I am specifically asking about the shape of the Fourier transform, I'd appreciate if you guys could also proof-read the question below as well, as I've written down allot of assumptions that I've gained, which might be wrong. OK...

1. why do we need to use shifted unit step function in defining second shifting theorem? 2. why don't we instead calculate laplace transform of a time shifted function just by replacing t by t-a? 3. everywhere in the books as well as internet i see second shifting theorem defined for...

Hi bros, so I feel like I am very close, but cannot find out how to go further. Q.1 Compute the DTFT of the following signals, either directly or using its properties (below a is a fixed constant |a| < 1): for $x_n = a^n \cos(\lambda_0 n)u_n$ where $\lambda_0 \in (0, \pi)$ and $u_n$ is the...