Transform Definition and 1000 Threads

Homework Statement I have to find the L-transform of ##f(x) = cos(\omega t + \phi)## Homework Equations . The Attempt at a Solution The straightforward approach is to write ##cos(\omega t + \phi)## as ##cos(\omega t)cos(\phi) - sin(\omega t)sin(\phi)## and it becomes: $$Lf(s) = \frac {s...

Homework Statement Calculate ##F(\frac 1 {1+x^4})##. Homework Equations ##\hat f (ξ) = \int_ℝ \frac 1 {1+x^4} e^{-2\pi i ξ x} dx## and Residue Theorem The Attempt at a Solution I know the function has to be real and even because ##\frac 1 {1+x^4}## is real and even, but I can't work out the...

Hi, I have to show that if ##f \in L^1(ℝ^n)## then: $$ ||\hat f||_{C^0(ℝ^n)} \le ||f||_{L^1(ℝ^n)}$$ Since ##|f(y)e^{-2 \pi i ξ ⋅y}| \le |f(y)|##, using the dominated convergence theorem, it is possible to show that ##\hat f \in C^0(ℝ^n)## but now I don't know how to go on. Thanks is advance.

It is often reported that the Fourier transform of a constant is δ(f) : that δ denotes the dirac delta function. ƒ{c} = δ(f) : c ∈ R & f => Fourier transform however i cannot prove this Here is my attempt:(assume integrals are limits to [-∞,∞]) ƒ{c} = ∫ce-2πftdt = c∫e-2πftdt = c∫ƒ{δ(f)}e-2πftdt...

Fourier Transform problem with f(t)=cos(at) for |t|<1 and same f(t)=0 for |t|>1. I have an answer with me as F(w)=[sin(w-a)/(w-a)]+[sin(w+a)/(w+a)]. But I can't show it.

Obviously in the title I mention the user that recently got banned, but the reason I do is because s/he was having some trouble accepting that a B field transforms into a mixture of E and B fields per the Lorentz transformation (and other assorted quackery), so it got me thinking about why this...

Homework Statement Solve the following partial differential equation , using Fourier Transform: Given the following: And a initial condition: Homework EquationsThe Attempt at a Solution First , i associate spectral variables to the x and t variables: ## k ## is the spectral variable...

Hello. Glad to meet you, everyone I am studying the [Mathematical Methods for Physicists; A Comprehensive Guide (7th ed.) - George B. Arfken, Hans J. Weber, Frank E. Harris] In Divergence of Vector Field, I do not understand that How to transform the equation in left side into that in right...

Homework Statement find the Fourier transform of the following function in two ways , once using direct computation , and second using the convolution therom . Homework Equations Acos(w0t)/(d2+t2) The Attempt at a Solution I tried first to solve directly . used Euler's identity and got...

Hello, If I have a homgeneous linear differential equation like this one (or any other eq): $$y''(x)-y'(x)=0$$ And they give me these Dirichlet boundary conditions: $$y(0)=y(1)=0$$ Can I transform them into a mixed boundary conditions?: $$y(0)=y'(1)=0$$ I tried solving the equation, derivating...

Hi All! I've been looking at this Fourier Transform integral and I've realized that I'm not sure how to integrate the exponential term to infinity. I would expect the result to be infinity but that wouldn't give me a very useful function. So I've taken it to be zero but I have no idea if you can...

Homework Statement The Fourier transfrom of the wave function is given by $$\Phi(p) = \frac{N}{(1+\frac{a_0^2p^2}{\hbar^2})^2}$$ where ##p:=|\vec{p}|## in 3 dimensions. Find N, choosing N to be a positive real number. Homework Equations $$\int d^3\vec{p}|\Phi(p)|^2=1$$ , over all p in the 3...

I'd like to see whether or not I understood correctly how massive particle states will transform under a homogeneous Lorentz transformation, in terms of the standard four-momentum ##k = (0,0,0,M)##. I suppose we can write $$U(\Lambda) \Psi \propto D^{(j)} (W(\Lambda)) \Psi$$ where ##U(\Lambda)##...

In the wikipedia page and on every book they proof the transformation by equaling the the equivalent resistance between any pair of terminals while disconnecting the other node.https://en.wikipedia.org/wiki/Y-%CE%94_transform Why this should make the two circuits equal? How can we apply...

After we proved Y-Δ transform using superposition theorem, The professor asked us what would happen if you have a Y circuit but a resistor is connected to the middle node. Can you do the transformation? She answered and said yes, and that resistor will be connected to nothing. How can that be...

Please, can anyone explain how formula (5) is obtained in J.J. Barton article ''Approximate translation of screened spherical waves" . Phys.Rew. A ,Vol.32,N2, 1985. ? https://doi.org/10.1103/PhysRevA.32.1019 The same formula are given in the book Pendry J.B. "Low enrgy electron diffraction. The...

So is there a proposed theoretical mechanism for transforming a particle into its own anti-particle? ##Electron \leftrightarrow Positron## ##Proton \leftrightarrow anti-Proton##

When you do a discrete Fourier transform (DFT) of a one-dimensional signal, I understand that the second half of the result is the complex conjugate of the first half. If you threw out the second half of the result, you're not actually losing any data and you would be able to recreate the entire...

Good afternoon, Not sure if this should be in the homework section or not but in any case... I'm having difficulty understanding the outputs from the Lorentz transform. Example problem. The Earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume...

Homework Statement I’m being asked to prove if and why (what instances in which) T<0 for the Laplace transform property of time shifting doesn’t hold. Homework Equations L{f(t-T)}=e^-aT* F(s) The Attempt at a Solution I know that for T<0 there are instances where the property cannot hold, but...

Hi all, just had a question about tensor/matrix notation with the inverse Lorentz transform. The topic was covered well here, but I’m still having trouble relating this with an equation in Schutz Intro to GR... So I can use the following to get an equation for the inverse...

Hi, outside the mathematical proof that shows that sines of different frequency are orthogonal... is there geometric interpretation/picture of this phenomena?

Homework Statement Given a continuous non-periodic function, its Fourier transform is defined as: $$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(x) e^{-ikx} dx$$ The problem is proving this is true by evaluating the...

A common use of the Fourier transform in physics is to transform between momentum-space and position-space. But what physical variables am I allowed to transform between? For instance can I use the Fourier transform to go from momentum space to frequency space or whatever?

Homework Statement The input signal of the circuit shown below is ##x(t)=2\sin (ω_ot + \pi/6)##. The switch in the circuit is controlled with a digital signal of the form ##s(t)=\sum_{k=-\infty}^{+\infty} (u(t+ε-kT_s) - u(t-ε-kT_s))##, ##\frac{2\pi}{T_s}=800\pi##, ##ε\to 0##, so that when the...

What is an orientation (i.e., set of Euler rotations) or shear transform collectively termed? It seems that these transforms, along with the scale transform are known as "linear" transforms, as described in the Venn diagram on page 2...

Hello! (Wave) I want to find $f(t)$ if its Laplace transform is $F(s)=\frac{1}{s(s^2+1)}$. We use the following formula, right? $$f(t)=\frac{1}{2 \pi i} \lim_{T \to +\infty} \int_{a-iT}^{a+iT} e^{st} F(s) ds$$ But how can we calculate the integral $\int_{a-iT}^{a+iT} e^{st}...

If a Laplace transform has a region of convergence starting at Re(s)=0, does the Laplace transform evaluated at the imaginary axis exist? I.e. say that the Laplace transform of 1 is 1/s. Does this Laplace transform exist at say s=i?

Only recently started to understand Euler angles and rotation matrices, and I am reasonably comfortable with the concepts already posted here. I am pretty sure I am missing something obvious, but I cannot figure out the way to solve this problem: A body in 3D space with a orientation defined by...

Hi particular solution only. As an example of what I am talking about, this method works for this DE: $$ 4y' + 2y = 10\cos(x) \\ \\ 10 \cos(x) = \Re( 10 e^{j(x)} ) = \Re(e^{j(x)} \cdot e^{j(0)} ) \rightarrow \text{complex number that captures the amplitude and phase of 10 cos x is} \\ 10...

The orientation of frequency components in the 2-D Fourier spectrum of an image reflect the orientation of the features they represent in the original image. In techniques such as nonlinear microscopy, they use this idea to determine the preferred (i.e. average) orientation of certain features...

Homework Statement I have derived the differential equations of a system. They are like the following: a\ddot{\theta} - b\ddot{x} + c \theta = 0 \\ d\ddot{\theta} + e\ddot{x} = F(t) where a,b,c,d,e are constants. I'm having trouble putting it into state space form, since I have the highest...

Homework Statement Y=(8s-4)/(s²-4) Homework EquationsThe Attempt at a Solution I rearranged the right side as: 8*(s/(s²-2²))-2*(2/(s²-2²)) Using the Laplace transform chart given in the class I was able to identify these as the transforms of hyperbolic sine and hyperbolic cosine making the...

$\displaystyle \begin{align*} F \left( \omega \right) &= \mathcal{F} \left\{ f \left( t \right) \right\} \\ &= \int_{-\infty}^{\infty}{ f\left( t \right) \mathrm{e}^{-\mathrm{j}\,\omega \, t}\,\mathrm{d}t } \\ &= \int_{-\infty}^{-2}{ 0\,\mathrm{d}t } + \int_{-2}^0{ \left( 1 + \frac{t}{2}...

Homework Statement Homework EquationsThe Attempt at a Solution 1. I got Y(s) = (15s +18)/(5s^2+s-2) 2. I got Y(s) = (7s - 7iw + 1)/((s+4)(5 - iw)) Was just wanting to make sure I solved these right. I would type it out but without formating, it will look messy.

Homework Statement I am supposed to compute the Fourier transform of f(x) = integral (e-a|x|) Homework Equations Fourier transformation: F(p) = 1/(2π) n/2 integral(f(x) e-ipx dx) from -infinity to +infinity The Attempt at a Solution My problem is, that I do not know how to handle that there...

Homework Statement Given the Fourier transformation pair ##f(t) \implies F(jw)## where ##f(t) = e^{-|t|}## and ##F(jw)=\frac{2}{w^2+1}## find and make a graph of the Fourier transform of the following functions: a) ##g(t)=\frac{2}{t^2+1}## b) ##h(t) = \frac{2}{t^2+1}\cos (w_ot)## Homework...

https://en.wikipedia.org/wiki/Discrete_Fourier_transform Why is the signal obtained from a DFT periodic? The time signal x[n] is finite and the number of sinusoids being correlated with it is finite, yet its said the frequency spectrum obtained after the DFT is periodic. I've also read the...

Homework Statement Solve ##\frac{dy}{dt} -y = 1, y(0) = 0## using the laplace transform Homework EquationsThe Attempt at a Solution ##\mathcal{L}\big\{\frac{dy}{dt}\big\} - \mathcal{L}\big\{y\big\} = \mathcal{L}\big\{1\big\}## ##sY(s) - y(0) - \frac{1}{s^2} = \frac{1}{s}## ##Y(s)...

I would like to express that when I am viewing the repetitive Fourier transform on Internet I encounter that for instance twice Fourier transform may lead the same value at the end of first Fourier transform. When does repetitive( twice or third... consecutively)fourier transform be same with...

Homework Statement I am reading the book of Gerry and Knight "Introductory Quantum Optics" (2004). In page 60, Chapter 3.7, there is two equation referring Fourier Transformation in the complex plane as follows: $$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$...

Hi, I'm struggling with a conceptual problem involving the Fourier transform of distributions. This could possibly have gone in Physics but I suspect what I'm not understanding is mathematical. The inverse Fourier transform of a Cauchy distribution, or Lorentian function, is an exponentially...

Hi everybody. There has been a thread about this on physics forums, where the Fourier transform X(w) of x(t) volts (with time units in seconds) could be considered as volt second, or volt per Hz. So when we see tables of Fourier transform pairs, we might see Fourier transform plots associated...

1. The problem statement, all variables, and given/known data Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) =...

Homework Statement I've written a program that calculates the discrete Fourier transform of a set of data in FORTRAN 90. To test it, I need to "generate a perfect sine wave of given period, calculate the DFT and write both data and DFT out to file. Plot the result- does it look like what you...

I have two functions ##\phi(t)=\cos(\omega t)## and ##f(t)=u(t)−u(t−k)## with ##f(t)=f(t+T)##, ##u(t)## is the unit step function. The problem is to find Laplace transform of ##\phi(t) \cdot f(t)##. I have tried convolution in frequency domain, but unable to solve it because of gamma functions...

Homework Statement Homework Equations Scaling property and property of dual. I got the answer. The Attempt at a Solution I got the answer using scaling property and using property of dual. x1(t)---> X2(W)----(another Fourier transform)--->2(3.14) x1(-w) But I think the final answer should be...

Hi, let's take this ode: y''(t) = f(t),y(0)=0, y'(0)=0. using the FT it becomes: -w^2 Y(w) = F(w) Y(w)=( -1/w^2 )F(w) so i can say that -1/w^2 is the Fourier transorm of the green's function(let's call it G(w)). then y(t) = g(t) * f(t) where g(t) = F^-1 (G(w)) (inverse Fourier transorm) how can...

I'm doing a research project over the summer, and need some help understanding how to construct an inverse Fourier transform (I have v. little prior experience with them). 1. Homework Statement I know the explicit form of ##q(x)##, where $$ q(x) = \frac{M}{2 \pi} \int _{- \infty}^{\infty} dz...

Homework Statement Homework Equations if x(t) --> X(W) then x(-t) --> X(-W) and x(t+a)-->ejwX(W) The Attempt at a Solution I'm getting right answer for 1st part. For second part book says right answer is C. Where am I wrong?[/B]