What is Fourier series: Definition and 750 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. 2

    Prove Lim N→∞ of Rudin Fourier Series 8.19

    Rudin 8.19 f is a continuous function on R, f(x+2Pi)=f(x), and a/pi is irrational. Prove that lim N goes to infinity (Sum n=1,...,N f(x+na)) =(1/2pi) * \int f(t)dt from -pi to pi for every x. Hint: do it first for f(x)=exp(ikx) THANKS!
  2. F

    Constructing a function - Fourier Series?

    Homework Statement Construct a function that is infinitely differentiable, f(x) in [0,1] for all x, and f(x)=1 for -1<x<1, f(x)=0 for |x|>2. Homework Equations None. The Attempt at a Solution I thought of doing it using a Fourier series for a square wave, in the way that f(x)=1...
  3. jegues

    Simple Fourier Series Question

    Homework Statement See figure attached for problem statement as well as my attempt. Homework Equations The Attempt at a Solution I'm stuck as to how to rewrite this sin term. I know that at n = 1 it will be positive, n = 3 it will be negative, n = 5 it will be positive and so...
  4. P

    What is the Fourier Series for f(x) = (pi - |x|)^2 on [-pi,pi]?

    Homework Statement For those of you who own baby rudin this is problem #14 in Chapter 8. For those of you don't I am given that f(x) = (pi - |x|)^2 on [-pi,pi], I need to show that f(x) = (pi^2)/3 + ∑(4/n^2)cos(nx) The series above is an infinite series from n=1 to infinity. I know I...
  5. K

    Find the Fourier series of a 'broken' function which is periodic in 2L

    Homework Statement Sketch the function: f(x) = \begin{Bmatrix} \frac{-x}{a} & 0 \leq x \leq a \\ \frac{x-L}{L-a} & a \leq x \leq L where f(x) is an odd function and is periodic in 2L. And a is a constant less than L/2 Find the Fourier series for the function f(x). Homework...
  6. A

    Complex Fourier Series of f: Evaluating Parseval's Relation

    Define f(t)=e^{-t} ont he interval [-\pi,\pi),and extend f to 2\pi-periodic.Find the complex Fourier series of f.Then, apply Parseval's relation to f to evaluate \sum^{\infty}_0 \frac{1}{1+k^{2}} For the first part when I calculate c_k \frac{1}{2\pi}\ \int^\pi_{-\pi} e^{-ikt-t}dt...I get...
  7. J

    Understanding the proof of the parseval theorem in fourier series

    Homework Statement I want to prove this http://img84.imageshack.us/img84/918/asdfo.png The Attempt at a Solution here is part of the proof [PLAIN]http://img824.imageshack.us/img824/4513/parseval.png i can't understand the red part, can someone help me? thanks edit: i...
  8. E

    Fourier Series - Summation to Integral

    Homework Statement My question involves the mid-point in deriving some of the equations to solve Laplace's equation in rectangular coordinates. The question may no make sense as it isn't problem specific. I can provide boundary values if necessary - just let me know. Homework Equations...
  9. R

    Fourier series functions homework

    Homework Statement Which of the following functions, when extended as 2pi periodic function, are equal to their Fourier series? a. F(x) = 2x, pi<x<-pi b. f(x)= 3 abs value (x) Homework Equations none The Attempt at a Solution After i graphed the functions into periodic...
  10. mnb96

    Fourier series of a train of gaussians

    Hello, I am following a proof in a book, in which the author makes use of the fact (without proving it) that the Fourier serie of a train of Gaussians is given by the following relationship: \mathcal{F} \{ \sum_{n=-\infty}^{+\infty}\frac{1}{\tau}e^\frac{-\pi(x-n)^2}{\tau^2} \} =...
  11. Z

    Fourier Series, Sin(t), but skips every other period Hard

    Homework Statement f(t)= sin(t), so as expected the period goes from 0 to 2pi. But 2pi to 4 pi is function is ZERO, and then starts up again from 4pi to 6pi, then zero from 6pi to 8pi,etc So basically sin(t) that skips every other period. Homework Equations The Attempt at a...
  12. S

    So the statement sin(n+1)pi/2 = cos(n)pi/2 is true by this property.

    Question - Find the Fourier series for f(x) = |cos(x)| in the interval (-π, π). Right, I attempted the question and the integration that followed. I'm having trouble in the integration itself.. Firstly I found that f(x) is an even function, so the sine term(bn) of the Fourier expansion would be...
  13. D

    Fourier series for particular solution when n=1

    hi, i already got the Fourier series for f(x) = x where -pi/2 =< x =< pi/2 which is f(x) = sigma, n=1 to infinity ( (-1)^n+1*sin (2nx) / n ) in order to find particular solution for y'' + 4y = f(x) i have to equate with with y(x)_p = A0 + sigma, n=1 to infinity (An*cos(2nx) + Bn*sin(2nx))...
  14. T

    Discovering the Fourier Series of a 2pi Periodic Function | Homework Help

    Homework Statement A 2pi peroidic function f is defined in the interval (-pi, pi) by f = t. Sketch the graph of the function and show that it's Fourier series is given by \frac{\pi^{2}}{3} + 4\sum^{\infty}_{n=1}\frac{(-1)^{n} \cos(nt)}{n^{2}} Homework Equations The Attempt at a...
  15. J

    Understanding Fourier Series, Transform and DFT

    Hello All, I am little confused with the Fourier family of transforms. I would really appreciate it if someone could have a look at them My understanding is as follows: Fourier Series: Only used for Peiodic continuous signals. Fourier Transform: Can be used for periodic or...
  16. C

    Discovering the Fourier Series of a General Expression: A Comprehensive Guide

    How can I represent a general "expression" as a Fourier series? For example, I want to find the Fourier series of sum: \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} ... (infinite). using the value of the Fourier series at x = 0 (because this will give the value of the infinite sum)...
  17. R

    Fourier Trigonometric Series for Symmetric Function: Homework Check

    Homework Statement Determine the first three terms in the Fourier trigonometric series for the function shown below. Homework Equations Look at attached jpeg. This tex tool is terrible. After making correction to the code, the preview still shows previous preview posts. The Attempt...
  18. E

    Fourier Series - Half Range Expansions

    1. Homework Statement [/b] Find the half range expansion of the given function. I tried to show f(x) using LaTeX but couldn't figure out how to stack three rows, so I have attached an image instead. Homework Equations Fourier series expansions for a0, an, bn - I can post these once I...
  19. B

    Fourier series/ periodic function

    Homework Statement Suppose f is a periodic function of period 2pi and that g is a horizontal shift of f, say g(x) = f(x + a). Show that f and g have the same energy. Homework Equations n/a The Attempt at a Solution i can see that if f(x) is shifted by 'a' that it does not make the...
  20. L

    Coefficient ck of the fourier series

    1. find the complex Fourier series and the real Fourier series f(t) = 0 , -pi <= t < 0, f(t)= t, 0<=t< pi Homework Equations 3. to find the coefficient ck i got 2 different answers one is if the index k is even and another if the index is odd, how am i supposed to represent this...
  21. L

    Finding a Fourier Series: What to Do and How?

    1. We are supposed to find a Fourier series knowing only f(t)= acos(kt)+bcos(kt) and some values of Fourier coefficients... please see #2 on this link http://www.math.ubc.ca/~oyilmaz/courses/m267/hmk3.pdf 2. I am using ck=1/2pi \intf(t)e-iktdt 3. are we supposed to convert f(t) into expnential...
  22. N

    How is this possible? (simple fourier series)

    Hello, Given a well-behaved function f:[0,1] \to \mathbb C with f(0)=0, is it then totally equivalent to write it as a sum of e^{2 \pi i n x} (n \in \mathbb Z) or as a sum of \sin{\pi n x} (n = 1,2,3,...) -- this last one by defining f:[-1,1] as an uneven function and then applying the first...
  23. B

    How Can Fourier Series be Used to Expand a Continuous Function of Period 2L?

    please help on this question Any continuous function of period 2L can be expanded as a Fourier series f(x)=a0/2+∑(from n=1 to∞) (ancos(n pi x/L)+bnsin(n pi x/L)) Using ∫(from -L to +L) sin(m pi x/L)sin(n pi x/L)dx=L kronecker delta m n Show that Bn=1/L∫(from -L to+L) sin(n pi...
  24. V

    One fundamental property of Fourier Series

    Suppose the functions f(t) and g(t) are periodic with periods P and Q, respectively. If the ratio P/Q of their periods is a rational number, show that the sum f(t)+g(t) is a period function. How to prove this?
  25. B

    Fourier series homework problem

    Not really a homework question; I typed this sum into Wolframalpha and it gave a nice, compact expression, but I couldn't figure out where to begin finding it. Is there a way to find it using just calc II-level knowledge of infinite sums...
  26. N

    Heat equation with a Fourier Series on an infinitely long rod

    Homework Statement The heat equation for an infinitely long rod is shown as: \alpha^2 \frac{\partial^2}{\partial x^2}u(x,t) = \frac{\partial}{\partial t}u(x,t) u(0,t) = u(L,t) = 0,\ \forall \ t > 0 u(x,0) = sin(\pi x) \ \forall \ 1 < x...
  27. T

    Harmonic oscillator and fourier series

    Hello, Attached are two problems I can not solve, thanks for the help. The Attempt at a Solution For the first question, I understand that I need insert A1coswt+A2sinwt into the homogenous equation , but don't know what's then .. But I'm pretty much lost on both of em :(
  28. C

    MATLAB Transfer fourier series into matlab code.

    Hey guys, i need to transfer Fourier series equation into MATLAB code. Anyone can help me in transfering?
  29. F

    Diff eq Fourier Series Stuck halfway through,

    Homework Statement Find the Fourier series of f on the given interval: EDIT: For the result of an, it should be multiplied by \frac{2}{1+n^{2}}. That has been corrected.
  30. J

    Solving Complex Fourier Series Problem - Help Needed!

    Hello Folks, I have a problem understanding a step of the complex Fourier series; it’s a step which involves simple addition and subtraction of exponentials (regrettably not simple for me). I have attached a picture of the step I am having a problem with would really appreciate if someone...
  31. M

    Fourier series and parseval's theorem

    A square wave has amplitude 3 and period 5. calculate its power? Using Fourier series for this square wave and Parseval’s theorem, calculate the power in a signal obtained by cutting out frequencies above 1 Hz in the square wave? i am able to obtain the Fourier series for the square wave...
  32. T

    Fourier series - correspondence between complex and real coefficients.

    Hello, I know how to get the full Fourier series with complex coefficients and with real coefficients, and I know the relationship between An, Bn and Cn. However, I don't know why the relationship between them is what is it. Can someone either explain to me where the relationship comes from...
  33. M

    Fourier Series based on 2 limits for x

    i have an exam in these kind of questions in a few days so i was pracitsing a a few problems but I can't do them! Any help would be appreciated. Calculate the Fourier series for f(x) when f(x) = 0, on -pi <= x <= 0, and f(x) = coshx, on 0 <= x <= pi. and show that SUM (from n=1 to...
  34. G

    Calculating Fourier Series for an Odd Function

    Homework Statement f(t) is given as: from 0 to 0.2s, f(t) = 5 from 0.2s to 0.6s, f(t) = 0 from 0.6s to 0.8s, f(t) = 5, etcHomework Equations for an odd function a0 = 2/p * integral(from -p/2 to p/2) of f(t) dt bn = 4/p * integral(from 0 to p/2) of f(t)*sin(2*pi*n*t/p) dtThe Attempt at a...
  35. B

    Finding the Fourier Series of f(x)

    Homework Statement Trying to find the Fourier series for the function f(x) = 0 for -pi<x<0 and f(x) = sinx for 0<x<pi The Attempt at a Solution im having a little trouble working it out.. are any of the sets of coefficients = 0? Im getting two non-zero integrals for the...
  36. J

    Solution of Heat Equation Through Fourier Series.

    OK, so I was trying to solve the Heat Equation with Inhomogeneous boundary conditions for a rod through Fourier Series when I got stuck at the solution for the coefficient c_n, the part where I'm stuck is highlighted in red. The following is just a step-by-step solution of how I got to c_n...
  37. T

    Orthogonality, Fourier series and Kronecker delta

    Homework Statement Show that the orthogonality relation for the "cosine basis functions" used in the Fourier series is 1/L\intcos[(n*pi*x)/L)]cos[(m*pi*x)/L)]dx = {Sin([n-m]*pi)}/[(n-m)*pi] + {Sin([n+m]*pi)}/[(n+m)*pi] By considering the different integer n and m, show that the right...
  38. S

    Deriving equations from fourier series representations

    Say you have the coefficients a_k of a Fourier series representation of some function x(t). You can easily then give x(t) as $$x(t) = \sum_{k = -\infty}^{\infty} a_k e^{i k \omega_0 t}$$ But this doesn't do much good in telling you what the actual function looks like. For example, if we have...
  39. E

    Alternate expressions of Fourier series formula

    Homework Statement Show that the Fourier series formula F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt)) can be expressed as F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}c_{n}cos(nwt-\phi_{n}). Relate the coefficients c_{n} to a_{n} and b_{n}. Homework Equations We...
  40. E

    Fourier series representation for F(t) = 0 or sin(wt) [depending on range]

    Homework Statement Obtain the Fourier series representing the function F(t)=0 if -2\pi/w<t<0 or F(t)=sin(wt) if 0<t<2\pi/w. Homework Equations We have, of course, the standard equations for the coefficients of a Fourier expansion...
  41. G

    How Can I Calculate the Fourier Series and Use it to Evaluate Infinite Series?

    http://img69.imageshack.us/img69/6758/123123123nx.jpg http://img819.imageshack.us/img819/5390/fsdfsdfsdf.jpg To calculate the Fourier series, I used the formulae above, and I got: [PLAIN][PLAIN]http://img831.imageshack.us/img831/2008/xcvxcvxcv.jpg and i substituted the...
  42. P

    Finding Fourier Series Coefficients

    Homework Statement Find the Fourier series coefficients X_k of the periodic signal: x(t) = 5cos(6w_0t+pi/2) (digital or discrete spectrum) Homework Equations The Attempt at a Solution I am really confused with all of this and don't...
  43. R

    Fourier Series Expansion of coshx

    Homework Statement Expand the function into Fourier series f(x) = coshx, |x|\leq \pi Homework Equations Fourier series will be C_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}(\frac{e^{x}+e^{-x}}{2}})e^{-inx}}dx \frac{1}{4\pi}\int_{-\pi}^{\pi}({e^{x}e^{-inx})dx+...
  44. S

    How Do I Solve for the Fourier Series of sin(6t) with f(t)=f(t+2pi)?

    Homework Statement f(t)=sin(|6t|), −pi<t<pi with f(t) = f(t+2pi) Homework Equations Show that the Fourier series for f(t) can be written as (24/pi) time the sum, from n=0 to infinity, of ( 1/( 36 - (2k+1)^2 ) )cos(2k+1)t. The Attempt at a Solution I have an answer of a0 being 0...
  45. O

    Fourier Series from Eigenfunctions

    Homework Statement "Using the eigenfunctions for the Hamiltonian of an infinite square-well potential defined over[-1,1] in the standard, dimensionless setting, construct Fourier series representation of the following functions..." the functions are e^(-100x^2), e^(-5x^2), e^(-x^2) It also...
  46. C

    Showing fourier series of sin^2(x)=(1/2)-(cos(2x)/2)

    Homework Statement f(x)=sin^2(x) Homework Equations The Attempt at a Solution solving for a(0)= i did (1/2Pi)*int(sin^2(x),x,-Pi..Pi)=1/2 b(n)=0 because sin^2(x) is an even function...
  47. T

    Complex Fourier Series for f(t)=2sin(πt) with Periodicity

    Homework Statement Find the complex Fourier series of the periodic function f(t)=2sin(πt) 0 < t < 1 and f(t+1) = f(t) for all t. (π is pi)Homework Equations http://upload.wikimedia.org/math/9/d/7/9d7f73fbcba87cbff485e66646aa541d.png...
  48. B

    Sum of infinite Fourier series

    Homework Statement Show that \sum_{r=0}^\infty\frac{1}{(2r+1)^2}=\frac{\pi^2}{8}Homework Equations The equation of the function is F(t)&=&\dfrac{\pi}{4}-\dfrac{2}{\pi}\left(\cos t+\dfrac{\cos3t}{3^{2}}+\dfrac{\cos5t}{5^{2}}+\cdots\right)-\left(\sin...
  49. L

    Hilbert Spaces for PDEs: How are they used?

    Hi, When I solve the diffusion equation for a spherically symmetric geometry in spherical coordinates I obtain the following general solution (after application of the boundary conditions). T(r,t) = \sum_{n=1}^{\infty}\, \frac{A_n}{r}\sin(\lambda_nr)\exp(-\alpha\lambda_n^2t) So to...
  50. R

    Complex form of Fourier series

    Though I can compute the coefficients of Trigonometric form of Fourier series, how can I compute the coefficients of complex form of Fourier series.
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