Fourier Definition and 1000 Threads

what is the difference? It seems like in T, you choose the RHS first, but in f, you choose the LHS first. Is this the only difference? Because the Fourier coefficients of f is derived in a standard way, right? As in, couldn't I derive the coefficients for T in the same way as I did for f? In...

Hi. Light travelling in dispersive media is normally treated by being broken up into its harmonic constituents by Fourier analysis and those then travel at frequency-dependent, but constant speed. However, from a mathematical point of view, there should be infinitely many other bases of the...

Proof: Let ## \epsilon=0 ##. Then the unperturbed equation is ## \ddot{x}+x=0 ## and the general solution is ## x(t)=A\sin\omega t+B\cos\omega t ## where ## \omega=1 ## is the angular frequency with the constants ## A ## and ## B ##. With the initial condition ## x(0)=0 ##, we have that ##...

Hi, I'm not sure if I have Fourier transformed the expression correctly For the Fourier transformation, I used the following formula ##\int_{-\infty}^{\infty} f(t) e^{i \omega t}dt## $$\frac{4 \pi d_0^2}{\hbar}\int_{-\infty}^{\infty} \sin(\omega_0(t-t')) e^{i \omega t}dt$$ $$\frac{4 \pi...

Proof: Let ## f(x) ## be a function of the real variable ## x ## such that the integral ## \int_{-\pi}^{\pi}f(x)dx ## exists and if the Fourier coefficients ## (a_{n}, b_{n}) ## are defined by ## a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx dx, n=0, 1, ..., ## and ##...

This is my attempt at a solution. I have used Eulers formula to rewrite the sine function and then used the Fourier transform of complex exponentials. My solution is not correct and I don't understand if I have approached this problem correctly. Please help. $$ \mathcal{F}\{\sin (4t-4) \} =...

Dear all, I am having some difficulties in generating some heat conduction curves. My problem is: I have an object at a temperature (Th) of 900 K placed on top of a surface with a temperature (Ta) of 300 K (see Figure). The thermal conductivity (K; W M-1 K-1) is 1.5 whilst the thermal...

To compute the Fourier transform of the ##t-V## model for the case where ##t = 0##, we start by expressing the Hamiltonian in momentum space. Given that the hopping term ##t## vanishes, we only need to consider the potential term: $$\hat{H} = V \sum_{\langle i, j \rangle} \hat{n}_i \hat{n}_j$$...

OBS: Ignore factors of ## (2 \pi) ##, interpret any differential ##dx,dp## as ##d^4x,d^4p##, ##\int = \int \int = \int ... \int##. I am using ##x,y,z## instead of ##x_i##. Honestly, i am a little confused how to show this "triangle-star duality". Look, the propagators in positions space gives...

This is technically a Fourier transform of a quantum function, but the problem I'm having is solely mathematical. Conducting this integral is relatively straightforward. We can pull the square roots out since they are constants, rewrite the bounds of the integral to be from ##-a## to ##a##...

As in title: Plugging in the definition is straight forward, I am too lazy to type, I will just quote the book Fetter 1971: Up to here everything is very straight forward, in particular, since we are working on free electron gas, ##E=\hbar \omega## However, I have no idea how to arrive...

Hey all, I just wanted to double check my logic behind getting the Fourier Transform of the following Hamiltonian: $$H(x) = \frac{ie\hbar}{mc}A(x)\cdot\nabla_{x}$$ where $$A(x) = \sqrt{\frac{2\pi\hbar c^2}{\omega L^3}}\left(a_{p}\epsilon_{p} e^{i(p\cdot x)} + a_{p}^{\dagger}\epsilon_{p}...

Evaluate the Fourier series $$\frac{1}{\pi^2}\sum_{k = 1}^\infty \frac{\cos 2\pi kx}{k^2}$$ for ##0 \le x \le 1##.

In my system I am trying to represent two lenses. L1 with focal length f1=910mm and the other lens, L2 with focal length f2=40mm. These lenses are space such that there is a distance of f1+f2 between the lenses. I have a unit amplitude plane wave incident on L1. My goal is to find the...

This is on a building in Korea, and the F0 made me think of the Fourier number, but the rest of the formula is unfamiliar to me. Does anyone recognize it?

I was wondering if the following is true and if not, why? $$ \begin{split} \hat{f}_1(\vec{k}) \hat{f}_2(\vec{k}) &= \hat{f}_1(\vec{k}) \int_{\mathbb{R}^n} f_2(\vec{x}) e^{-2 \pi i \vec{k} \cdot \vec{x}} d\vec{x} \\ &= \int_{\mathbb{R}^n} \hat{f}_1(\vec{k}) f_2(\vec{x})...

First, ##\tilde{f}(\omega)=\int_{-\infty}^{\infty}e^{a|t|}\cos(bt)e^{-i\omega t} \mathrm{d}t## We can get rid of the absolute value by splitting the integral up ##\int_{-\infty}^{0}e^{at}\cos(bt)e^{-i\omega t} \mathrm{d}t+ \int_{0}^{\infty}e^{-at}\cos(bt)e^{-i\omega t} \mathrm{d}t## Using...

First of all i tried to follow the textbook. Here they start of by modelling the atom as an harmonic oscilator: Then they find the solution as: They neglect the second term as omega_0 >> gamma which also makes good sense so they end up with: So far so good. After this they state the...

My question is; is showing the highlighted step necessary? given the fact that ##\sin (nπ)=0##? My question is in general i.e when solving such questions do i have to bother with showing the highlighted part... secondly, Can i have ##f(x)## in place of ##x^2##? Generally, on problems to do...

>>> from numpy import exp, pi >>> exp(1j*pi) (-1+1.2246467991473532e-16j) The fact that the imaginary part of this is not zero is wrecking a fourier collocation scheme for a nonlinear PDE with periodic boundary conditions: the coefficient corresponding to the Nyquist frequency, which should be...

Hi, a question regarding something I could not really understand The question is: Let V be a space with Norm $||*||$ Prove if $v_n$ converges to vector $v$. and if $v_n$ converges to vector $w$ so $v=w$ and show it by defintion. The question is simple, the thing I dont understand, what...

I am unsure if ##h(x,t)## really is a wave packet, but it looks like one, hence the title. Anyway, so I'd like to determine ##\hat{h}(k,t=0)##. My attempt so far is recognizing that, without the real part in the integral, i.e. ##g(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} a(k)e^{i(kx-\omega...

I have plotted the function for ##T=15## and ##\tau=T/30## below with the following code in Python: import numpy as np import matplotlib.pyplot as plt def p(t,T,tau): n=np.floor(t/T) t=t-n*T if t<(2*np.pi*tau): p=np.sin(t/tau) else: p=0 return p...

I would try to determine whether ##p(t)## is even or odd. This would be so much easier if the values of ##\tau## and ##T## would be specified, but maybe it's possible to do without it, which I'd prefer. If for example ##\tau=1/2## and ##T=2\pi##, then ##p(t)=\sin{(2t)}## for ##0\leq t <\pi ##...

The 2D Fourier transform is given by: \hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy In terms of polar co-ordinates: \hat{f}(\rho,\phi)=\int_{0}^{\infty}\int_{-\pi}^{\pi}rf(r,\theta)e^{-ir\rho\cos(\theta-\phi)}drd\theta For Fourier transforms in cartesian co-ordinates, relating the...

I am unsure about what is being asked for in the question. At first I thought the question asks one to calculate the inverse Fourier transform and then to analyze its depends on ##t##, however, the "estimate" makes me think otherwise.

Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. I tried each part of the problem, but I'm only left in terms of limits as x -> infinity of sin or cos...

So, ##\hat{p}(\omega)=\int_{-\infty}^{\infty} p(t)e^{-i\omega t}\mathrm{d}t=A\int_{0}^{\infty}e^{-t(\gamma+i(\omega+\omega_0))}=A\left[-\frac{e^{-t(\gamma+i(\omega+\omega_0))}}{\gamma+i(\omega+\omega_0)}\right]_0^\infty,## provided ##\gamma+ i(\omega+\omega_0)\neq 0## for the last equality. Now...

I was studying a Michelson interferometer for infrared absorption in Fourier transform and I've found these two images (taken from https://pages.mtu.edu/~scarn/teaching/GE4250/ftir_lecture_slides.pdf ) representing an infrared monochromatic beam of light going into the interferometer and the...

My notes say that the Resolution of the Aperture(in the Electric field of the wave) is the Fourier transformation of the aperture. Then gives us the equation of the aperture: and says that for the circular aperture in particular also: My attempt at solving this: We know that the Fourier...

Hello! I am reading about Fourier Transform MW spectroscopy in a FB cavity, which seems to be quite an old technique and I want to make sure I got it right. As far as I understand, this is very similar to normal NRM, i.e. one applies a MW ##\pi/2## pulse which puts the molecules in a linear...

Here is the question: Here is my answer

What is the Fourier transform of a beat? For example, I want to calculate the Fourier transform of the function ##f(t)=\cos((\omega_p+\omega_v) t)+\cos((\omega_p-\omega_v)t),## where ##$\omega_p+\omega_v=\Omega,\space\omega_p-\omega_v=\omega## and ##\Omega\simeq\omega.## I think it is equal to...

It has been 35 years since I did the math for Fourier analysis, and I have forgotten what the subtleties are. Please be kind. So this is not a how do I calculate a DFT (though that may be my next question) but rather how do I use it, and interpret the results. All the online and software I find...

Hello, First of all, I checked several other threads mentioning duality, but could not find a satisfying answer, and I don't want to revive years old posts on the subject; if this is bad practice, please notify me (my apologies if that is the case). For the past few days, I have had a lot of...

Doing the Fourier transform for the function above I'm getting a result, but since I can't get the function f(t) with the inverse Fourier transform, I'm wondering where I made a mistake. ##F(w) = \frac{1}{\sqrt{2 \pi}} \int_{0}^{\infty} te^{-t(a + iw)} dt## By integrating by part, where G = -a...

I think that is with the Fourier transform.

I wanted to do an investigation about how the same musical chord can have the same pitch but sound different on different musical instruments. Like how chord C major would sound higher played in the electric guitar than a C major played on piano. How should I approach this investigation?

My Professor has started on the Fourier Transforms Topic in the Introductory Mathematical Physics class and gave us a small homework to try our concepts on. I have attached a clear & legible snippet of my solution. I request someone to please have a look at it & determine if my solution is...

Hello everybody! I have a question concerning the Fourier transformation: So far I have experimentially measured the magnetic field of a quadrupole but as the hall effect sensor had a fixed orientation I did two series, one for the x, one for y component of the magnetic field, I have 50 values...

I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...

From the statement above, since the ring is massless, there's no force acting vertically on the rings. Thus, the slope is null. ##\frac{\partial y(0,0)}{\partial x} = \frac{\partial y(L,0)}{\partial x} = 0## ##\frac{\partial y(0,0)}{\partial x} = A\frac{2 \pi}{L}cos(\frac{2 \pi 0}{L}) =...

I was assigned an experiment of Fouriers optics where I have to use different Filters. One of them was the dark spot and the tiny hole. As of my understanding, for tiny hole, we cut off all high-frequency light related to diffraction and refraction, thus using only the low freuency part of the...

I have tried to Fourier transform in ##x## and get the result in the transformed coordinates, please check my result: $$ \tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky} $$ However, I'm having some problems with the inverse transform: $$ \frac{1}{2\pi}\int_{-\infty}^\infty...

Homework Statement:: . Relevant Equations:: . I am having a hard time thinking about Fourier transform, because there are so many conventions that i think i got more confused each time i think about it. See an example, "Find the Fourier transform of $$V(t) = Ve^{iwt} \text{ if } nT \leq t...

Hello again. Having some issues on Fourier transform. Can someone please tell me how to proceed? Need to solve this then use some software to check my answer but how to solve for a and b. Plzz help

I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where \begin{equation} \begin{split} \hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3...

I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time. The time-domain form is, s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}} Here, \mu , \Omega and \lambda are constants. A quick...