Fourier Definition and 1000 Threads

Homework Statement $$u_{xx} + u_{yy} = 0 : x < 0, -\infty < y < \infty$$ Homework Equations We can use Fourier Transform, which is defined over some function ##f(x)## as ##F(f(x)) = 1/ 2\pi \int_{-\infty}^{\infty} f(x) \exp (i \omega x) dx##. The Attempt at a Solution Using the Fourier...

(First of all I never saw Hilbert spaces in a mathematical class, only used it in intro QM so far, so please don't assume I know that much when answering.) Let's consider the Hilbert space on the interval [a,b] and the operator ##\textbf{L} = \frac{d^{2}}{dx^{2}} ##. Then ##\textbf{L}## is...

If we have a standard function like x or x^2 defined between 0 and pi. Then why should we be interested in extending this function to give a Fourier series which resembles this function between 0 and pi? What is the whole purpose of this process? Does it have any real life application or is it...

Textbook says, Fourier transform expresses a function in time domain as a function in frequency domain. Basically, Fourier transform gives two different expressions in terms of t domain and f domain but they represent the same signal. It also says Hilbert transform is a different type of...

hi pf! My book presents a problem and has it boiled down to $$S(u) = -S(f(x)) \exp(- \omega y) / \omega$$ where ##S(u)## is the sine Fourier transform of the function ##u##. However, we cannot directly take the transform back since the singularity at ##\omega = 0##. Thus the book then takes...

Okay I have a question involving calculating the FFT of a signal from a sensor. I have simulated many different scenarios in MATLAB of various noise characteristics involving the signal. I want to take the FFT of a noisy signal. As long as my expected input signal has a higher amplitude than...

Suppose one has a simple aperture in one dimension across x direction (1D aperture). Illuminated by plane wave, this aperture will produce certain diffraction pattern which, at sufficiently large distance, is just the aperture's Fourier transform, and we place a detector to measure it. Now this...

Can someone explain the concept to me. Does it mean the the a's of n and b's of n are 90 degrees apart? I know the inner-product of the integral is 0 if the two are orthogonal.

Homework Statement Given x[n] with transform X(ejw), find the Fourier transform in terms of X(ejw). x1[n]=[0.9ncos(0.6*pi*n)] * x[n-2] Homework Equations time shift: x[n-k] -> e-jwkX(ejw) convolution: x[n] * h[n] -> X(w)H(w) freq. shift: x[n]ejwcn -> X(ew-wc) The Attempt at a Solution I...

Are there any resources which show Fourier series approximating a given waveform? I am looking for examples which have a real impact on students and provides motivation. I am trying to find something visual but it could be just audio based. Something to start the topic of Fourier series so that...

Hi PF! I was wondering if you could clarify something for me. Specifically, I am solving the heat equation ##u_t = u_{xx}## subject to ##| u(\pm \infty , t ) | < \infty##. Now this implies a solution of sines and cosines times an exponential. Since we have a linear PDE, we may superimpose each...

Just a question How does solving the nonlinear schrodinger equation using split step Fourier method makes us understand the four wave mixing process in optical fiber ? Any examples on how that happens Thank you

Homework Statement I am given f(t) = e^-|t| and I found that F(w) = ##\sqrt{\frac{2}{\pi}}\frac{1}{w^2 + 1}## The question says to use the nth derivative property of the Fourier transform to find the Fourier transform of sgn(t)f(t), and gives a hint: "take the derivative of e^-|t|" I also...

Homework Statement I have attached a screenshot of the question. I know how to use Fourier's theorem for one function but have no idea how to attempt it with a discontinuous function like this. I tried working out a0 by integrating both functions with the limits shown, adding them and...

If we have a simple periodic function (square wave) which can be easily written but the Fourier series is an infinite series of sines and cosines. Why bother with this format when we can quite easily deal with the given periodic function? What is the whole point of dealing this long calculation...

If we have a simple periodic function (square wave) which can be easily written but the Fourier series is an infinite series of sines and cosines. Why bother with this format when we can quite easily deal with the given periodic function? What is the whole point of dealing this long calculation?

I have recorded a micrograph of a 2-D array at a magnification of 43,000x on my DE-20 digital camera, which has a 6.4 μm pixel size and a frame size of 5120 × 3840 pixels. This magnification is correct at the position of the camera. I then compute the Fourier transform of the image. What is the...

I am working on a simple PDE problem on full Fourier series like this: Given this piecewise function, ##f(x) = \begin{cases} e^x, &-1 \leq x \leq 0 \\ mx + b, &0 \leq x \leq 1.\\ \end{cases}## Without computing any Fourier coefficients, find any values of ##m## and ##b##, if there is any...

If I write the basic scalar field as $$\phi(x)=\int\frac{d^3k}{(2\pi)^3}\frac1{\sqrt{2E}}\left(ae^{-ik\cdot x}+a^\dagger e^{ik\cdot x}\right),$$ this would seem to imply that the creation and annihilation operators carry mass dimension -3/2. That's the only way I can get the total field...

Homework Statement I have solved the following exercise, but I have obtained the half of the correct result! I can't understand where is the problem... ##f(x)=\begin {cases} 0, x \in[-\pi, 0]\\cos x, x \in[0, \pi]\end{cases}## 1) Find the Fourier Series (base: ##{\frac{1}{\sqrt{2 \pi}}...

Homework Statement Given f = a0 + sum(ancos(nx) + bnsin(nx)) and f' = a0' + sum(an'cos(nx) + bn'sin(nx)) The sums are over all positive integers up to n. show that a0' = 0, an' = nbn, bn' = -nan Then prove a similar formula for the coefficients of f(k) using induction. Homework EquationsThe...

I'm trying to solve Laplace equation using Fourier COSINE Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't think so). NOTE: U(..) is the Fourier Transform of u(..) This are the equations (Laplace...

i just wanted to get this cleared that a beam falling on a diffraction grating with a shape gives the Fourier images of the grating object which can be reobtained by placing a biconvex lens that would converge the rays and form a focussed Fourier image at its focal length and the image of the...

Homework Statement Assume ψ(x,0)=e^(-λ*absvalue(x)) for x ± infinity, find Φ(k) Homework Equations Φ(k)=1/√(2π)* ∫e(-λ*absvalue(x))e(-i*k*x)dx,-inf, inf[/B]The Attempt at a Solution , my thought was Convert the absolute value to ± x depending on what of the number line was being...

If a function f'(u) has Fourier coefficients anμ and bnμ, by integration one can make new coefficients Anμ ,Bnμ which include constants of integration. My question how can I verify that : Anμcos nτ + Bnμ sin nτ= -i/2 ((Bnμ) -Anμi) einτ- (Bnμ-iAnμ) e-inτ I assume this is the complex form of...

Are there any real life applications of Fourier series? Are there examples of Fourier series which have an impact on students learning this topic. I have found the normal suspects of examples in this field such as signal processing, electrical principles but there must be a vast range of...

Below is my walkthrough of a Fourier transform. My problem is that I want to do all the similar steps for a Fourier transform between position x and the wave vector k. That is working on a solution of the maxwell equations. The maxwell equations has many possible solutions for example: $$...

Homework Statement Homework EquationsThe Attempt at a Solution I tried to attempt the question but I am not sure how to start it, at least for part (i). My biggest question, I think, is how does the multiplication of a random complex number to a Fourier-Transformed signal (V(f)) have an...

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...

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 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...