Fourier Definition and 1000 Threads

Homework Statement Homework Equations The Attempt at a Solution I did Fourier transform directly to the eigenvalue equation and got Psi(p)=a*Psi(0)/(p^2/2m-E) But the rest, I don't even know where to start. Any opinion guys?

http://ms.mcmaster.ca/courses/20102011/term4/math2zz3/Lecture1.pdfOn pg 10, the example says f(x)=/=0 while R.H.S is zero. It is an equations started from the assumption in pg 9; f(x)=c0f(x)0+c1f(x)1…, then how do we get inequality? if the system is complete and orthogonal, then...

I am not quite clear on the use of Fourier series to solve the Schrodinger equation. Can you point me to a source of some simple one dimensional examples?

I'm trying to solve this exercise but I have some problems, because I haven't seen an exercise of this type before. "f(x)= \pi -x in [0, \pi] Let's consider the even extension of f(x) in [-\pi, \pi] and write the Fourier Series using this set ( \frac{1}{\sqrt{2 \pi}}, \frac{1}{\sqrt {\pi}}...

Homework Statement Sketch the waveform defined below and explain how you would obtain its Fourier series: f(wt) = 0 for 0 ≤wt ≤pi/2 (w=omega) f(wt) = Vsin(wt) for pi/2 ≤wt ≤pi f(wt) = 0 for pi ≤wt ≤3pi/2 f(wt) = Vsin(wt) for 3pi/2 ≤wt ≤2pi Develop the analysis as far as you are...

Hey guys, long story short. I am completing my Math minor this semester and need to decide on whether Topology or Fourier Analysis. I am an undergraduate physics major and neither one of those classes is required for my B.S. in physics. So what do you guys think, Topology or Fourier Analysis?

Hi Folks, The Fourier Cosine Transform of cos(x) for 0<x<a and 0 everywhere else is given as F(\omega)=\displaystyle\frac{1}{\sqrt{2 \pi}}[\frac{\sin a (1-\omega)}{1-\omega}+\frac{\sin a (1+\omega)}{1+\omega}] I can plot this and we get a continuous amlitude spectrum of F(\omega) against...

when a function doesn't satisfy dirichlet condition, why do we not care and go ahead finding the Fourier transform anyway? What is the use? Eg: unit impulse, dirac delta function, etc. don't statisfy the dirichlet conditions but its like dirichlet conditions arent really conditions?

Homework Statement [/B] f(x)=\left\{\begin{array}{cc}0,&\mbox{ if } 0< x < 2\\1, & \mbox{ if } 2<x<4\end{array}\right. Show that the Cosine Fourier Series of f(x) for the range [0,4] is given by: A + B\sum^{\infty}_{n=0}\frac{(-1)^n}{(2m+1)}cos(\frac{(2m +1) \pi x}{2}) Homework Equations...

Homework Statement Noting that J_0(k) is an even function of k, use the result of part (a) to obtain the Fourier transform of the Bessel function J_0(x). Homework Equations In (a) I am asked to show that the Fourier transform of f(x)=\dfrac{1}{\sqrt{1-x^{2}}} is...

Homework Statement The question is to get Fourier sine series of e^-x =f(x) on 0<x<1 Homework Equations Bn = 2/L ∫ (e^-x) * sin(nπx/L) over the limits 1 to 0, where L = 1 f(x) = summation of Bn*sin(nπx/L) The Attempt at a Solution So I integrated ∫ by part integration so I took u =...

Homework Statement So the question is how does 4/π*(sin(πx))+4/3π *(sin(3πx))+4/5π *(sin(5πx)) = 1 for values of 0<x<1 Homework Equations No relevant equation needed just don't understand which values of x to take. The Attempt at a Solution I am not sure which value of x to start with, it...

Why is the summation for the discrete Fourier series from 0 to N-1 (where N is the fundamental period of the signal) wheras it goes from minus infiniti to infiniti for continuous Fourier series...Thank you

Hi To properly understand introductory quantum mechanics, I want to understand what the Fourier transform actually gives me mathematically. What book do you recommend? I found one book, but it doesn't get to Fourier transformations until after seven long chapters. Is that what I have to expect...

So I have been away from education for a little while now and I'm going through some refresher stuff - in particular I have been playing around with FFTs. If i take (with MATLAB notation): time = 0:0.01:10 y = fft(sin(2*pi*f*time)) with f = 5 then the maximum amplitude of the fft output is...

Homework Statement This comes up in the context of Poisson's equation Solve for ##\mathbf{x} \in \mathbb{R}^n ## $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$ Homework Equations $$\int_0^\pi \sin\theta e^{ikr \cos\theta}\mathop{dk} = \int_{-1}^1 e^{ikr \cos\theta}\mathop{d\cos \theta }$$...

Hello everyone, I'm in need for the best books that you know out there for PDE (Partial Differential Equations) and everything related to Fourier (series, transform, etc.). Any help would be much appreciated. Thank you and happy holidays!

Homework Statement this is something i noticed doing homework rather than homework itself. I plot fft output from different frequency signals, i am not sure why power changes with increasing frequency? Homework Equations if i take (with MATLAB notation): time = 0:0.01:10 y =...

Homework Statement I am having trouble understanding this: I have a Dirac Delta function $$ \delta (t_1-t_2) $$ but I want to prove that in the frequency domain (Fourier Space), it is: $$\delta(\omega_1+\omega_2) $$ Would anyone have any ideas how to go about solving this problem? I know...

Hi, I've been reading a paper on renormalisation theory as applied to a simple one-particle Coulombic system with a short-range potential. In the process of renormalisation, the authors introduce an ultraviolet cutoff into the Coulomb potential through its Fourier transform: ## \frac{1}{r}...

Hello everyone, I know that the integral of an odd function over a symmetric interval is 0, but there's something that's bothering my mind about it. Consider, for example, the following isosceles trapezoidal wave in the interval [0,L]: When expressed in Fourier series, the coefficient...

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...

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 Evaluate following series: \sum_{n=1}^\infty \frac{1}{(4n^2-9)^2} by finding the Fourier series for the 2\pi-periodic function f(x) = \begin{cases} sin(3x/2) & 0<x<\pi \\ 0 & otherwise \end{cases} Homework Equations a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}...

I am carrying out FFT analysis to compare two waves. One looks very much like a sine wave the other has an extra dip occurring at half the frequency of the main wave. I have been thinking around how I might expect this to show up in the FFT analysis. At first i was expecting to see a smaller...

Homework Statement Find the following Fourier series in trigonometric form. Homework Equations $$y(t)=a_0+\sum\limits_{n=1}^{\infty} a_n cos(n\omega_{0}t)+b_n sin(n\omega_{0}t)$$ The Attempt at a Solution The graph above is represented by the function: $$ x(t) = \left\{ \begin{array}{ll}...

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...

Homework Statement Homework Equations The Attempt at a Solution I don't really understand why my solution is wrong as I think I have substituted everything in correctly.. Is it okay if anyone can help me take a look at my solution? Thank you. :) My solution: (Only bn) My...

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.

(NOTE: Maybe this post belongs in the Number Theory Forum? Apologies if it is wrongly located!) I am reading Julian Havil's book, "The Irrationals: The Story of the Numbers You Can't Count On" In Chapter 4: Irrationals, Old and New, Havil gives a proof of the irrationality of e which was...

Hi, friends! Let ##f:[a,b]\to\mathbb{C}## be an http://librarum.org/book/10022/173 periodic function and let its derivative be Lebesgue square-integrable ##f'\in L^2[a,b]##. I have read a proof (p. 413 here) by Kolmogorov and Fomin of the fact that its Fourier series uniformly converges to a...

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...

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

Homework Statement Hi, so I am doing some past exam papers and there was this question; Homework EquationsThe Attempt at a Solution a0 and an both are equal to zero, this leaves only bn. Since you can only use the sine series for an odd function, and cos(t) is even, does this mean i have to...

If you take the Fourier series of a function $f(x)$ where $0 < x < \pi$, then would $a_{0}$, $a_{n}$, and $b_{n}$ be defined as, $a_{0} = \displaystyle\frac{1}{\pi}\int_{0}^{\pi}f(x)dx$ $a_{n} = \displaystyle\frac{2}{\pi}\int_{0}^{\pi}f(x)\cos(nx)dx$ $b_{n} =...

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 was working on a problem where I had been given a differential equation to be solved using separation of variables. Two coordinates: a time coordinate and a single spatial coordinate (1-D problem). Homework Equations The domain for the spatial part was [0, L]. Given...

Homework Statement Find Fourier series of f(x) = Acos(\pix/L) I know how to do this, I just don't know the value of L. If it's equal to \lambda/2, then I know the solution. But the question does not specify the value of L. L is just the length of the entire wave that I'm working with, right? If...

Hi everybody! I'm studying the Fourier integral operators but I can't resolve a pass. I'm considering the following operator: $$Au(x)=\frac{1}{{(2\pi h)}^{n'}}\int_{\mathbb{R}_y^m\times\mathbb{R}_\theta^{n'}} e^{i\Psi(x,y,\theta)/h}a(x,y,\theta,h)u(y)\, dy\, d\theta$$ where $$Au\in C^0...

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...

Ok so this isn't a homework question per se, but I'm currently writing a report on Fourier Analysis but a bit stuck as to what the results can actually help with. I realized that I don't grasp how a Fourier Transform can be used. In the experiment we recorded the signal created by a remote...

Suppose we have some function f(x) with period L. My book states that if it is even around the point x=L/4, it satisfies f(L/4-x)=-f(x-L/4), whilst if it is odd it satisfies f(L/4-x)=f(x-L/4). Then we define s=x-L/4 so we have for the function to be odd or even about L/4 that f(s)=±f(-s)...

Homework Statement The problem is finding the Fourier series of f(t) = e^(-t) from [0,2] where T=2 and without using complex solution. [/B]Homework Equations f(t) = a0/2 + ∑ (anCos(nωt) +bnsin(nωt) NOT using f(t) = ∑dne^(inωt)The Attempt at a Solution I tried once but got completely wrong...

Homework Statement Define ##f : [−π, π) → \mathbb R ## by ##f(x)## = ##−1## if ##− π ≤ x < 0##, ##1## if ##0 ≤ x < π.## Show that the Fourier series of f is given by ##\frac{4}{π} \sum_{n=0}^\infty \frac{1}{(2k+1)} . sin(2k+1)x##Homework Equations The Fourier series for ##f## on the interval...

Hi there, I am reading a material on the application of Fourier transformation in physics. One application is to transform the position-dependent function to k-dependent function, i.e.## F(k) = FFT[f(x)]## We know that the in physics, the wavenumber could be written in momentum as...

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...