Transform Definition and 1000 Threads

I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?

Would someone be able to explain like I am five years old, what is the precise relationship between Fourier series and Fourier transform? Could someone maybe offer a concrete example that clearly illustrates the relationship between the two? I found an old thread that discusses this, but I...

The definition of Fourier transform (F.T.) that I am using is given as: $$f(\vec{x},t)=\frac{1}{\sqrt{2\pi}}\int e^{-i\omega t}\tilde{f}(\vec{x},\omega)\,\mathrm{d}\omega$$ I want to show that: $$\frac{1}{c\sqrt{2\pi}}\int e^{-i\omega t}\omega^2...

I am interested in modeling a battery charging/discharging. I am starting off with a simple model using a voltage source in series with a parallel RC branch which is in series with a resistor. I will be measuring the open circuit voltage between the last series resistor and the bottom of the...

Given ##f(\vec{x})##, where the Fourier transform ##\mathcal{F}(f(\vec{x}))= \hat{f}(\vec{k})##. Given ##\vec{x}=[x_1,x_2,x_3]## and ##\vec{k}=[k_1,k_2,k_3]##, is the following true? \begin{equation} \begin{split} \mathcal{F}(f(x_1))&= \hat{f}(k_1) \\ \mathcal{F}(f(x_2))&= \hat{f}(k_2) \\...

Hi I would like to transform the S-parameter responce, collected from a Vector Network Analyzer (VNA), in time domain by using the Inverse Fast Fourier Transform (IFFT) . I use MATLAB IFFT function to do this and the response looks correct, the problem is that I do not manage to the time scaling...

I have not studied the Fourier transform (FT) in great detail, but came across a problem in electrodynamics in which I assume it is needed. The problem goes as follows: Evaluate ##\chi (t)## for the model function...

It's a silly example, but hopefully it will help me to understand the maths. Two guys ##A## and ##B## are initially at the same spacetime event ##O##, and then ##B## receives an impulse along the ##x##-direction giving him an initial coordinate velocity ##\dot{x}_B = v_0## as measured by ##A##...

The standard Clarke Transform is ## i_{alpha} = i_a; -> 1 i_{beta} = \frac {(i_a + 2i_b)} {\sqrt3} ->2 ## I am trying to derive it, but missing a factor. Basically converting the 3 phase currents ## I_a, I_b, I_c ## into the 2 axis ##I_{\alpha}, I_{\beta} ## resolving along the x-axis...

In August, "Quantum Information Processing" published an article describing a full FFT in the quantum domain - a so-called QFFT, not to be confused with the simpler QFT. According to the publication:

How to solve the transforms below \[ \mathscr{L}^{-1} \frac{a(s+2 \lambda)+b}{(s+ \lambda)^2- \omega^2} \]

I am attempting to be able to do this problem with the help of Mathematica and Fourier transform. My professor gave us instructions for Fourier Transformation and Inverse Fourier, but I don't believe that my output in Mathematica is correct.

I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts. It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...

I've included the problem statement and a bit about the function but my main issue is with the equation after "then" and the one with the red asterisk. I don't understand why the Laplace transform for a u(t)*e^(-t/4) isn't (1/s)*(1/(s+1/4)). The book I am reading says it's(1/(s+1/4)).

Image below. Is the Lorentz transform just switching between a stationary frame and a moving frame? I forgot to write Alice's frame but I assume that is obvious.

Hi, I was hoping to gain more insight into these window questions when looking at frequency spectra questions. I don't really know what makes windows better than one another. My attempt: In the question, we have f(t) = cos(\omega_0 t) and therefore its F.T is F(\omega ) = \pi \left(...

Hello, I am unfamiliar with Maxwell's equations' Fourier transform. Are there any materials talking about it?

Homework Statement:: Why is the heaviside function in the inverse laplace transform of 1? Relevant Equations:: N/A This is a small segment of a larger problem I've been working on, and in my book it gives the transform of 1 as 1/s and vice versa. But as I've looked online for help in figuring...

I read in one book about the deduction of Lorentz transform. It writes: ' $$ \begin{aligned} t^\prime & = \xi t + \zeta x (1) \\ x^\prime & = \gamma x + \delta t (2) \\ y^\prime & = y (3) \\ z^\prime & = z (4) \end{aligned} $$ from (2), it gives: $$ \begin{aligned} {dx \over dt} = -{ \delta...

From the sketch, I know that this function is an even function. So, I simplify the Fourier transform in the limit of the integration (but still in exponential form). Then, I try to find the exponential FOurier transform. Here what I get: $$g(a)=\frac{2}{2\pi} \int_{0}^{\infty} e^{-x} e^{-iax}...

Suppose I'm solving $$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?

I have the following probability density function (in Maple notation): f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi] Now I want to transform x so that 0 -> (3/2) * Pi and 3 * Pi -> (15/2) * Pi and the new function is still a probability density function. How should I...

Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform. So here's my attempt to this problem so far...

Hi, In class I have learned how to find the Fourier transform of rectangular pulses. However, how do I solve a problem when I should sketch the Fourier transform of a pulse that isn't exactly rectangular. For instance "Sketch the Fourier transform of the following 2 pulses" Thanks in advance...

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}})...

I found this video on youtube which is trying to explain Fourier transform using the center of mass concept At 15:20 the expression of the x coordinate is given in the video. I believe it is wrong, and it should be: ##\frac{{\int g(t)e^{(-2 \pi ift)}.g(t).2 \pi f.dt}} { \int g(t).2 \pi...

Was just practicing some problems on the Fundamentals of Electric Circuits, and came across this question. I understand I will have to transform to the s domain circuit, which looks something like this: Then doing nodal analysis, I will get the following for the first segement (10/s-V1)/1 =...

Summary:: Discrete Fourier transform exam question Hi there, I'm not really sure how to do this question at all. Any help would be appreciated.

I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m + 1)...

I struggle to find an appropriate inverse Laplace transform of the following $$F(p)= 2^n a^n \frac{p^{n-1}}{(p+a)^{2n}}, \quad a>0.$$ WolframAlpha gives as an answer $$f(t)= 2^n a^n t^n \frac{_1F_1 (2n;n+1;-at)}{\Gamma(n+1)}, \quad (_1F_1 - \text{confluent hypergeometric function})$$ which...

Hi, A rectangular pulse having unit height and lasts from -T/2 to T/2. "T" is pulse width. Let's assume T=2π. The following is Fourier transform of the above mentioned pulse. F(ω)=2sin{(ωT)/2}/ω ; since T=2π ; therefore F(ω)=2sin(ωπ)/ω Magnitude of F(ω)=|F(ω)|=√[{2sin(ωπ)/ω}^2]=|2sin(ωπ)/ω|...

Hi, I was trying to find Fourier transform of two rectangular pulses as shown below. The inverted rectangular pulse has unit height, -1, and lasts from -π to 0. The other rectangular pulse has unit height, 1, and lasts from 0 to π. I was making use of Laplace transform and its time shifting...

Hi again, The previous problem was done using y′′(t)+2y′(t)+10y(t)=10 with with intial condition y(0⁻)=0. In the following case, I'm using an initial condition and setting the right hand side equal to zero. Find y(t) for the following differential equation with intial condition y(0⁻)=4...

Hi, This thread is an extension of this discussion where @DrClaude helped me. I thought that it'd be better to separate this question. I couldn't find any other way to post my work other than as images so if any of the embedded images are not clear, just click on them. It'd make them clearer...

Hi, I 'm trying to find the Laplace transform of the following expression. I used the following conversion formulas. I think "1" is equivalent to unit step function who Laplace transform is 1/s. I ended up with the following final Laplace transform. Is my final result correct? Thank you...

Hello i have an assignment. From given circuit i need to find s domain and inverse them back to t domain. can you help me by explain this circuit?

I used partial fraction method first as: 1/s(s^2+w^2)=A/s+Bs+C/(s^2+w^2) I found A=1/w^2 B=-1 C=0 1/s(s^2+w^2)=1/sw^2- s/s^2 +w^2 Taking invers laplace i get 1/w2 - coswt But the ans is not correct kindly help.

Since this is of the form $\displaystyle \frac{f\left( t \right)}{t} $ we should use $\displaystyle \mathcal{L}\,\left\{ \frac{f\left( t \right) }{t} \right\} = \int_s^{\infty}{F\left( u \right) \,\mathrm{d}u } $. Here $\displaystyle f\left( t \right) = \cosh{\left( 4\,t \right) } - 1 $ and so...

Shouldn't the equation be x' = x + (v')(t) instead of x' = x + (v)(t)?

Upon taking the Laplace Transform of the equation we have $\displaystyle \begin{align*} s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) + 4\,Y\left( s \right) &= -\frac{8\,\mathrm{e}^{-6\,s}}{s} \\ s^2 \,Y\left( s \right) - 2\,s - 0 + 4\,Y\left( s \right) &=...

In a numerical Fourier transform, we find the frequency that maximizes the value of the Fourier transform. However, let us consider an analytical Fourier transform, of ##\sin\Omega t##. It's Fourier transform is given by $$-i\pi\delta(\Omega-\omega)+i\pi\delta(\omega+\Omega)$$ Normally, to find...

$\displaystyle \begin{align*} \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \sinh{ \left( 11\,t \right) } \right\} &= \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \cdot \frac{1}{2} \left( \mathrm{e}^{11\,t} - \mathrm{e}^{-11\,t} \right) \right\} \\ &= \frac{5}{2} \,\mathcal{L} \left\{...

So I have completed (a) as this (original on the left): I have then went onto (b) and I have equated T(s)=Z(s) as follows: and due to hence Does this look correct to you smarter people? Thanks in advance! All replies are welcome :)

I would like to know the equation of Fourier transform when the data has lack. like this sine wave.

In my QFT homework I was asked to prove that $$\int d^3x \int \frac{d^3k}{(2\pi)^3} e^{i\mathbf{k} \cdot (\mathbf{x} - \mathbf{y})} k_j f(\mathbf{x}) = i \frac{df}{dx_j}(\mathbf{y}) $$ Using ##\frac{\partial e^{i\mathbf{k} \cdot (\mathbf{x} - \mathbf{y})}}{\partial x^j} = i k_j e^{i\mathbf{k}...

Problem: Find a (limited?) solution to the diff eq. At the end of the solution, when you transform $$\frac{-1}{s+1} + \frac{2}{s-3}$$ why doesn't it become $$-e^{-t} + 2e^{3t} $$, t>0 ?

Hi, I've been looking all over the net for good examples but I've only found some intro but no examples being solved. If you know of good resources (both theories and problems) please let me know! a) Calculate Fourier and inverse Fourier transform of f(t). b) Calculate the limit. My...

So the task is to solve the following integral with laplace transform. Since t>0 we can multiply both sides with heaviside stepfunction (lets call it \theta(t)). What I am unsure about is what happens with the integral part and how do we inpret the resulting expression? What will it result...

Hello, everyone. :) I'm trying to do a certain problem regarding Fourier transforms (but one that's supposedly easy, because of just using tables, rather than fully computing stuff), and I know how to do it, but I don't know why it works. Here's the problem statement.: "Compute the Fourier...