What is Transform: Definition and 1000 Discussions
In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable
t
{\displaystyle t}
(often time) to a function of a complex variable
s
{\displaystyle s}
(complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.For suitable functions f, the Laplace transform is the integral
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?
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...
I came across this problem and solved it using different approach. I get a slightly different answers.
here's how it goes,
1st approach
$\mathscr{L}[te^{2t}\cos(3t)]$
first I get the laplace of something that's familiar to me which is $\mathscr{L}[t\cos(3t)]$
using this...
Heya folks,
I'm currently pondering how to decide whether a function has an inverse Laplace transform or not. In particular I am considering the function e^(-is), which I am pretty sure does not have an inverse Laplace transform. My reasoning is that when calculating the inverse by the Bromwich...
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...
Hi,
My aim is to get a series of images in 2D space that run over different timestamps and put them through a 3D Fourier Transform. So my 3D FT has 2 spatial axes and one temporal axis. However I have never done anything like this before, and I have a very basic knowledge of Python.
So...
Firstly, if this is an inappropriate forum for this thread, feel free to move it. This is a calculus-y equation related to differential equations, but I don't believe it's strictly a differential equation.
The question I'm asking is which functions...
Homework Statement
OK, we're given to practice Fourier transforms. We are given
f(x) = \int^{+\infty}_{-\infty} g(k) e^{ikx}dk
and told to get a Fourier transform of the following, and find g(k):
f(x) = e^{-ax^2} and f(x) = e^{-ax^2-bx}
Homework Equations
The Attempt at a Solution
For...
Homework Statement
Hi.
I need help to resolve the inverse laplace transform of {1/((x^2)+1)^2}2. The attempt at a solution
I have tried to do:
{(1/((x^2)+1) * (1/((x^2)+1)}
then, convolution, sen x
But, isn't working
Thanks for your help :)
Hello. I'm wondering if anyone has a table of transforms showing the result of an Abel transform on a Gaussian distribution. I have been unable to find the solution to this. Many thanks for any help. I'm reconstructing an an image from a picture that fits a Gaussian very well, hence I'm hoping...
I have this expression:
f(\tau) = 4 \pi \int \omega ^2 P_2[\cos (\omega \tau)] P(\omega) \, \mathrm{d}w \quad [1] where P_2 is a second order Legendre polynomial, and P(\omega) is some distribution function.
Now I am told that, given a data set of f(\tau), I can solve for P(\omega) by either...
A first stage of the determination.
We have a body of length L = AB, which moves along the x-axis with velocity v,
say that coming to us.
A -------- B v <---
|
| h - vertical distance
| ./ - a light converges to us with the speed eq. c
| /
O ---------> x
At some time t = 0, we see the point A...
Hi All,
I'm playing around with an arduino, and have build a PID controller that controls the temperature of a light bulb, measured with an NTC. All is working fine
I'm looking to get a bit more theoretical on the subject and have modeled the system in a simulink like environment. I want...
Basically I am trying to lorentz transform the magnetic field along θ of a bunch particles which have a gaussian distribution to the radial electric field. However the magnetic field in θ is dependent on the longitiudinal distribution.
Now initially i thought we would just use the standard LT...
Hi! I'm trying to show that the differential from equation
$$D \star F = 0$$ transform homogeneously under the adjoint action ##F \mapsto gFg^{-1}## of the lie group ##G##, where ##D## denotes the covariant exterior derivative ##D\alpha = d \alpha + A \wedge \alpha## for some lie algebra valued...
When I sample a certain digital signal with increasing sampling frequency, the fast Fourier transform of the sampled signal becomes finer and finer. (the image follows) Previously I thought higher sampling frequency makes the sampled signal more similar to the original one, so the Fourier...
before I go to bed(it's 11:30pm in my place), here is the last problem that I need help with
find the inverse Laplace Transform
$\frac{4s-2}{s^2-6s+18}$
the denominator is a non-factorable quadratic. I don't know what to do.
thanks!
find the inverse Laplace of the ff:
1. $\frac{n\pi L}{L^2s^2+n^2 \pi^{2}}$
2. $\frac{18s-12}{9s^2-1}$
for the 2nd prob
I did partial fractions
$\frac{18s-12}{9s^2-1}=\frac{9}{3s+1}-\frac{3}{3s-1}$
$\mathscr{L}^{-1}\{\frac{18s-12}{9s^2-1}\} =...
$\mathscr{L}\{\sin^{2}4t\}$
$\mathscr{L}\{\sin(3t-\frac{1}{2}\}$for the 2nd prob here's what I have tried
$\mathscr{L}\{\sin(3t)\cos(0.5)-\cos(3t)\sin(0.5)}$
$\cos(0.5)\mathscr{L}\{\sin(3t)\}-\sin(0.5) \mathscr{L}\{\cos(3t)\}$
$\frac{3\cos(0.5)-s\sin(0.5)}{s^2+9}$ ---> is this correct?
for...
please help me solve this problem
$\mathscr{L}\{e^{3a-2bt}\}$
here's my attempt
$\mathscr{L}\{e^{3a}\cdot e^{-2bt}\}$ from here I couldn't continue
I looked up my table of transform but nothing matches the problem above. I'm not sure if the first shift formula would work here. please help...
To calculate the DOS of a material, the electronic structure typically needs to be calculated first. This requires lots of expertise and the accuracy is questionable.
I'm interested in seeing if there's some shortcut to get some general properties of the DOS:
If I could arbitrarily deform...
As it can be read here, http://en.wikipedia.org/wiki/Laplace_transform#Relation_to_power_series
the Laplace transform is a continuous analog of a power series in which the discrete parameter n is replaced by the continuous parameter t, and x is replaced by exp(-s).
Therefore, computing a...
Hi All,
I have a problem I've been thinking about for a while, but I haven't come up with a really satisfactory solution:
I want to do a discrete Fourier transform on data that has been sampled at 2 different sampling frequencies. I've attached a picture of what my data will look like...
Homework Statement
Find the FT of the following signal
The function is: f(t) = t(\frac{sen(t)}{t\pi})^2
Homework Equations
Fourier transform: F(\omega)= \int_{-\infty}^\infty f(t)e^{-jt\omega}
My attempt began with this Fourier transform, and that's my goal:
F[tf(t)]=...
I was wondering if anyone could help me with this integral. I've heard of contour integration but I'm unsure of how it would be used for this integral.
1. Hi! I am new at this forum, and english is not my native language,
so, I hope I can make myself clear. A teacher send us a list of activities,
but he did not give us the theory about it (the theoretical class). So, I have
read a few things on the internet and I have solved some exercises. I...
Hello,
I am searching for the Laplace transform of this function
u_a(y)\frac{\partial c(t)}{\partial t}
where u_a(y) is the Heaviside step function (a>0).
Can anyone help me?
Thanks in advance! Paolo
Here we will use the following transforms: $\displaystyle \begin{align*} \mathcal{F}^{-1} \left\{ \frac{n!}{ \left( a + \mathrm{i}\,\omega \right) ^{n+1} } \right\} = t^n\,\mathrm{e}^{-a\,t}\,\mathrm{H}(t) \end{align*}$ and $\displaystyle \begin{align*} \mathcal{F}^{-1} \left\{...
Homework Statement
\int_{-\infty}^{\infty} |k|^{2\lambda} e^{ikx} dkHomework Equations
The Attempt at a Solution
As you can guess, this is the inverse Fourier transform of |k|^{2\lambda}. I've tried splitting it from -infinity to 0 and 0 to infinity. I've tried noting that |k| is even, cos is...
For two-body decay, in the center of mass frame, final particle distribution is,
$$
W^*(\cos\theta^*,\phi^*) = \frac{1}{4\pi}(1+\alpha\cos\theta^*)
$$
We have the normalization relation , ##\int W^*(\cos\theta^*,\phi^*)d\cos\theta^* d\phi^*=1##.
And we also know that in CM frame ##p^*##...
I have a third order derivative of a variable, say U, which is a function of both space and time.
du/dx * du/dx * du/dt or (d^3(U)/(dt*dx^2))
The Fourier transform of du/dx is simply ik*F(u) where F(u) is the Fourier transform of u. The Fourier transform of d^2(u)/(dx^2) is simply...
We have a wave ψ(x,z,t). At t = 0 we can assume the wave to have the solution (and shape)
ψ = Q*exp[-i(kx)]
where k = wavenumber, i = complex number
The property for a Fourier transform of a time shift (t-τ) is
FT[f(t-τ)] = f(ω)*exp[-i(ωτ)]
Now, assume ψ(x,z,t) is shifted in time...