# What is Fourier expansion: Definition and 31 Discussions

In mathematics, a Fourier series () is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

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1. ### A How to get a converging solution for a second order PDE?

I have been struggling with a problem for a long time. I need to solve the second order partial differential equation $$\frac{1}{G_{zx}}\frac{\partial ^2\phi (x,y)}{\partial^2 y}+\frac{1}{G_{zy}}\frac{\partial ^2\phi (x,y)}{\partial^2 x}=-2 \theta$$ where ##G_{zy}##, ##G_{zx}##, ##\theta##...
2. ### Fourier Transform Proof

Homework Statement Given a continuous non-periodic function, its Fourier transform is defined as: $$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(x) e^{-ikx} dx$$ The problem is proving this is true by evaluating the...
3. ### Fourier Series Expansion

Homework Statement There is a sawtooth function with u(t)=t-π. Find the Fourier Series expansion in the form of a0 + ∑αkcos(kt) + βksin(kt) Homework Equations a0 = ... αk = ... βk = ... The Attempt at a Solution After solving for a0, ak, and bk, I found that a0=0, ak=0, and bk=-2/k...
4. ### I Amplitudes of Fourier expansion of a vector as the generalized coordinates

When discussing about generalized coordinates, Goldstein says the following: "All sorts of quantities may be impressed to serve as generalized coordinates. Thus, the amplitudes in a Fourier expansion of vector(rj) may be used as generalized coordinates, or we may find it convenient to employ...
5. ### How do you find the fourier expansion coefficients?

Homework Statement I need to expand this piecewise function f(x) = h for a<x<L and f(x) = 0 for 0<x<a. I am told that this is a square wave so ao and an in the expansion are 0 (odd function). Therefore I only need to worry about bn. The limits on the integral are from a to L, but what about the...
6. ### Need Help with Fourier Coefficients for 1-t Function?

Homework Statement [/B] I am looking for help with part (d) of this question 2. Homework Equations The Attempt at a Solution I have attempted going through the integral taking L = 4 and t0 = -2. I was able to solve for a0 but I keep having the integrate by parts on this one. I've tried it...
7. ### Fourier transform of periodic potential in crystal lattice

Homework Statement Homework Equations I'm not sure. The Attempt at a Solution I started on (i) -- this is where I've gotten so far. I am asked to compute the Fourier transform of a periodic potential, ##V(x)=\beta \cos(\frac{2\pi x}{a})## such that...
8. ### Fourier Series for Periodic Functions - Self Study Problem

Self Study 1. Homework Statement Consider a periodic function f (x), with periodicity 2π, Homework Equations ##A_{0} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)dx## ##A_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)cos\frac{2\pi rx}{L}dx## ##B_{n} =...
9. ### Series expansion for 2D dipole displaced from the origin

I learn that we can expand the electric potential in an infinite series of rho and cos(n*phi) when solving the Laplace equation in polar coordinates. The problem I want to consider is the expansion for the potential due to a 2D line dipole (two infinitely-long line charge separated by a small...
10. ### Quantum Books on waves with Fourier Transforms

There are many waves and oscillations books out there that also include Fourier analysis but very few give the subject a thorough treatment, they just pass it in a few pages. If anybody has any sources(particularly books) that have Fourier analysis and particularly Fourier Transforms, I would...
11. ### Fourier Series Homework (Discontinuous Function)

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...
12. ### Did I set this Fourier series up correctly?

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} =...
13. ### Understanding Lissajous Curves for Fourier Synthesis

i have a question about Fourier synthesis and how this relates to lissajous curves. I have two sets of test data; spatial data in the x and y directions. When I plot the x data against the y I get an ellipse. ( it represents a wing moving in a circular motion) I am trying to recreate this...
14. ### Can Sound Waves Create Customized Noise Patterns in a Room?

I learned that with Fourier expansions any function can be approximated by an infinite sum of sine and cosine waves. Is it possible to use this fact to create an arbitrary distribution of sound and silence in a given room. Using a simple example, is it possible to make it so there is noise in...
15. ### How can a Fourier expansion contain all the same info as original f'n?

We know that a function f(x) over an interval [a, b] can be written as an infinite weighted sum over some set of basis functions for that interval, e.g. sines and cosines: f(x) = \alpha_0 + \sum_{k=1}^\infty \alpha_k\cos kx + \beta_k\sin kx. Hence, I could provide you either with the function...
16. ### Fourier expansion of boolean functions

Any boolean function on n variables can be thought of as a function f : \mathbb{Z}_2^n \rightarrow \mathbb{Z}_2 which can be written as f(x) = \sum_{s \in \mathbb{Z}_2^n} \hat{f}(s) \prod_{i : x_i = 1} (-1)^{x_i} where \hat{f}(s) = \mathbb{E}_t \left[ f(t) \prod_{i : s_i = 1}...
17. ### Square root of volume in fourier expansion of the vector potential

Hi. I just wondered why we use a 1/\sqrt{V} in the Fourier expansion of the vector potential. A regular 3 dimensional Fourier expansion is just f(\vec r) = \sum_{\vec k} c_\vec{k} e^{i \vec k \cdot \vec r} but as the solution to the equation (\frac{\partial ^2}{\partial t^2} -...
18. ### Maths help on Fourier Expansion

Dear all, Iam just wondering whether the term 1/a can be expanded using Fourier expansion. If it does, please let me know how to to do this. Thank for any kind help.
19. ### How Fourier Expansion indicates the Amplitude w.r.t a certain frequency?

Homework Statement I want to know how could I extract the amplitude(of the sinusoid component) of a random continuous wave w.r.t a certain frequency response? The teacher said the Fourier Expansion can do that but I'm really confused by the limits and integrals. Homework Equations...
20. ### Fourier expansion between two different intervals

Homework Statement f(x) = x+1 for -1,x<0 x-1 for 0<x<1 0 for x=0 expand it in an appropriate cosine or sine series Homework Equations f(x) = a0/2 + \sum [ancos (n\pix/p) + bn sin (n\pix/p) a0 = 1/p \intf(x).dx an = 1/p \int f(x)cos...
21. ### Need help to prove expressions using Fourier Expansion

Dear all, Please help me to prove the Fourier expansion for three different cases as follows. I need some help to show that L.H.S = R.H.S Case1: when m not equal to 0 and n not equal to 0. dU^0/dz (1- x/a) - dU^a/dz (x/a) = -(2/m^2 Pi^2)(1- Cos[m*Pi]) [dUn^0/dz - dUn^a/dz] Case2...
22. ### Finding summation when given a fourier expansion of a function

Homework Statement I have a function f(x) = x^2/4 for |x|<π I have the Fourier series of this function which is and I need to prove that The Attempt at a Solution I tried to use dirichlet for x = 0 but I get -pi^2/3

Homework Statement To prove: e^{ax} = \frac{sinh(\pi a)}{\pi}\sum_{n=-\infty}^{\infty} \frac{(-1)^n}{a^2+n^2}(a cos(nx) + n sin(nx)) For [itex] -\pi < x < \pi \hbox{ and a is real and not equal 0} [/tex] Homework Equations Given: \int^{\pi}_{\pi} e^{ax} e^{-jnx}dx =...
24. ### Integral of exponentiated Fourier expansion

Hello everyone, I need to solve an integral, opfully extracting Fourier coefficients, and I don't have a clue. It is just $\int^{2\pi}_0 d\sigma \text{exp}{A(\sigma)}$ where $A(\sigma) = \sum_n a_n e^{-i n \sigma }$ I tried to work with complex coordinates...
25. ### Finding Fourier Expansion of f(t): Part b & c

Find the Fourier expansion of one period of f(t)=1+t absolute value of t<1 I found this to be 1+2/pi Sigma(0 to infinity) ((-1^(n+1))/n)sinnpit by just the standard methods of the a0 an and bn formuals, which I know is correct Now the parts I am having problems with is part b and c...
26. ### Damped Harmonic Oscillator Fourier Expansion

Homework Statement A damped harmonic oscillator originally at rest and in its equilibrium position is subject to a periodic driving force over one period by F(t)=-\tau^2+4t^2 for -\tau/2<t<\tau/2 where \tau =n\pi/\omega a.) Obtain the Fourier expansion of the function in the integral...
27. ### Deriving a Monster: Fourier Expansion Help Needed

I'm supposed to derive this monster! \frac{1}{2} + \frac{2}{\pi} \sum^{\infty}_{k = 1}\frac{1}{2k-1}sin(2k-1)x = \left\{^{0 \ for \ -\pi < x < 0}_{1 \ for \ 0<x<\pi} I don't even know where to start right now. And no examples to work from. Can anyone get me started? the Chapter is on...
28. ### Problems about Fourier expansion?

Homework Statement just wondering if anyone knew a website that has solved problems about Fourier expansion?? all i can find are notes and discussions about it.. a gazallion thanks everyone Homework Equations The Attempt at a Solution
29. ### Fourier expansion (i cant understand a thing darn)

Homework Statement what is the Fourier expansion of the periodic function whose definition in one period is f(t)= {^{0}_{1} ^{-\pi \leq t < 0}_{sint 0\leq t \leq \pi} uh sorry about the small font i don't know how to make it bigger about the question, as much as i would...
30. ### Fourier Expansion in 3D: Expansion of V(r)

I'm used to use \tilde{f} (x) = a_n|e_n> where |e_n> = e^{2 \pi inx / L} and a_n = \frac{1}{L}<e_n|f> for my Fourier expansions. How do I expand a function in 3 dimensions, for example V(\vec{r}) = \frac{e^{-\lambda r}}{r} ?
31. ### Pointwise convergence of fourier expansion

I am reading in Wald "General Relativity" page 394, that the expansion of a scalar field over an orthonormal basis with creation and annihilation operators does not converge pointwise. What does this mean for a quantum field and what are the consequences of this?