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. H

    How to find the sum of two series using the Weierstrass M-test?

    Hi Given the function f(t) = t^2, were t \in ]- \pi, pi[, and is continious find the Fourier series for f(t). L = 2 \pi. Then A_0 = \frac{1}{2 \pi} \int \limit_{-\pi} ^{\pi} t^2 dt = \frac{\pi ^2}{3} A_n = \int \limit_{-\pi} ^{\pi} t^2 \cdot cos(\matrm{n} \pi \mathrm{t}) dt A_n = \int...
  2. G

    Fourier Series Help: Find Steady State Solution of Diff Eq

    Can anyone help me out with this? Find the steady state periodic solution of the following differential equation. x''+10x= F(t), where F(t) is the even function of period 4 such that F(t)=3 if 0<t<1 , F(t)=-3 if 1<t<2. Im basically just having a problem findind the general Fourier...
  3. G

    What is the general Fourier series for an even function with a period of 4?

    Can anyone help me out with this? Find the steady state periodic solution of the following differential equation. x''+10x= F(t), where F(t) is the even function of period 4 such that F(t)=3 if 0<t<1 , F(t)=-3 if 1<t<2. Im basically just having a problem findind the general Fourier...
  4. F

    Fourier Series - Will someone walk me through this

    The book for the class that I'm currently taking is "Introduction to Applied Mathematics" by Gilbert Strang. Things have been good with this class until this chapter. If you have used this book, you will understand what I mean when I say it is different. Things have been ok, because I've been...
  5. F

    Laplaces Equation with Fourier Series

    Ok, I'm going to bed. But I have to ask ANOTHER question about my homework... so I can get up early and work on it. Q: Around the unit circle suppose u is a square wave: u_0 = \left\{\begin{array}{c} +1 \,\,\,\, on\,the\,upper\,semicircle \,\,\,\, 0<\theta < \pi \\ -1 \,\,\,\...
  6. F

    How Does the Fourier Series of a Square Wave Lead to the Leibniz Formula for Pi?

    Here is the question: At x= \frac{\pi}{2} the square wave equals 1. From the Fourier series at this point find the alternating sum that equals \pi . \pi = 4(1 - \frac{1}{3}+\frac{1}{5}-\frac{1}{7} + \ldots I do not understand what the question is asking. I'm not knowledgeable enough...
  7. G

    Find the fourier series representation of x^2

    ok, i wasn't sure if i ought put this in math or phys, we're going over it my phys class, but its just math... whatever.. So i had to find the Fourier series representation of x^2 in the intervals (-pi, pi) and (0, 2pi). i haven't even started the (0, 2pi) one, cause i can't get the first...
  8. M

    MATLAB MATLAB Fourier Series: Evaluate & Tabulate 1st 8 Terms

    hey guys, I've got to do some Fourier series work using matlab, but I have no idea what to do. Ive found the coefficients by hand, but now I need to use MATLAB to evaluate and tabulate the first 8 terms. I then have to evaluate the series at 1000 points over a certain range. Does...
  9. benorin

    Does the Fourier Series of a Continuous Function Converge Uniformly?

    So I'm working this HW problem, namely Suppose f is a continuous function on \mathbb{R}, with period 1. Prove that \lim_{N\rightarrow\infty} \frac{1}{N}\sum_{n=1}^{N} f(\alpha n) = \int_{0}^{1} f(t) dt for every real irrational number \alpha. The above is for context. The hint says...
  10. B

    Fourier Series for |x| and Finding g(x): Help Needed

    Hi, can someone help me out with the following question? Q. Show that the Fourier series for the function y(x) = |x| in the range -pi <= x < pi is y\left( x \right) = \frac{\pi }{2} - \frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\cos \left( {2m + 1} \right)x}}{{\left( {2m + 1}...
  11. M

    Differential Equation and Fourier Series

    I have this problem. I would appreciate it if anyone can help me get started. Question: Consider the differential equation: \frac{d^2 y(x)}{dx^2} + y(x) = f(x) \ \ ; \ \ 0 \leq x \leq L \\ The boundard conditions for y(x) are: y(0) = y(L) = 0 \\ Here f(x) is assumed to be a known function...
  12. F

    Understanding Fourier Series: Solving Problems and Exploring Applications

    http://www4.okfoto.co.kr/S_storage4/314500/A05120618494148_t.jpg http://www4.okfoto.co.kr/S_storage4/314500/A05120618494175_t.jpg It's solution is...
  13. M

    Complex Fourier Series and Phase Spectra

    Please check my solution and I need help on understanding the second part of the question. Q:Obtain the complex form of the Fourier series of the sawtooth function. f(t) = \frac{2t}{T} \ \ \ 0 < t < 2T\\ So if the period is 2l = 2T then l = T \\ c_n = \frac{1}{2l} \int_{-l}^{l} f(x) e^{in\pi...
  14. siddharth

    Evaluation of Numerical series by Fourier series

    I have some problems which says show that (i) \sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90} and (ii) \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} = \frac{\pi^2}{12} And another one which says, show that for 0<x<\pi sin x + \frac{sin 3x}{3} + \frac{sin 5x}{5} + ... = \frac{\pi}{4} The...
  15. S

    Finding Fourier Series f(x)=1 & Integrating for g(x)=x on 0<=x<=pi

    Find the Fourier series for f(x)=1 on the interval 0 <=x <= pi in temrs of phi = sin nx. By integrating thi series find a convergent series for hte function g(x) =x oin this interval assuming that the set {sin nx} is complete i can find the Fourier series for f(x) =1. But i would like to know...
  16. S

    Combining Fourier Series for Step Function: $\phi_{n} = \sin(nx)$

    Find the FOurier Series in terms of \phi_{n} = \sin(nx) of the step function f(x) = 0 for 0 \leq x \leq \frac{1}{2} \pi [/tex] f(x) =1 for [itex] \frac{1}{2} \pi < x \leq \pi now i have no problem finding the series for each branch. But how would i combine them? for the 0 to 1/2 pi...
  17. S

    Prove Fourier Series Limit: Integral of Ln(x) Sin(nx)

    Show that \lim_{n \rightarrow \infty} \int_{0}^{\pi} Ln(x) Sin(nx) dx i was told to use this identity given that int f^2 \rho dx is finite then c_{n}^2 \int_{a}^{b} \phi_{n}^2 \rho dx = \frac{(\int_{a}^{b} f \phi_{n} \rho dx)^2}{\int_{a}^{b} \phi_{n}^2 \rho dx} \rightarrow 0 and n...
  18. S

    Finding the Fourier Series of a Step Function

    Find the Fourier Series in terms of \phi_{n} = \sin{nx} of the step function f(x) = 0 for 0 \leq x \leq \frac{1}{2} \pi = 1 for \frac{1}{2} \pi < x \leq \pi Solution Fourier Series is \Sigma c_{n} \phi_{n} and for hte interval for x between 0 and 1/2 pi c_{n} =...
  19. S

    Can a fourier series of a function just be a constant?

    say if you get a result for a0 but 0 for an and bn(using my book's notation where the Fourier series is a0+an*cos(nx)+bn*sin(nx)
  20. M

    Solving Fourier Series Prob: Need Help With Integral Parts

    I've got parts of this problem but I'm stuck on some of the integration. See attached. Thanks!
  21. S

    Fourier series / Kpler's equation

    GIVE ME A HINT! Fourier series / Kepler's equation By expanding e \sin\psi in a Fourier series in \omega t, show that Kepler's equation has the formal solution \psi = \omega t + \sum_{n=1}^{\infty}{\frac{2}{n}J_{n}(ne)\sin{\omega t}} where J_{n} is the Bessel function of order n. For small...
  22. T

    Fourier Series of simple function.

    f(x) = cos(x) x from [-PI, 0] f(x) = -cos(x) x from ] 0, PI] I'm not sure how to deal with getting a Fourier series for this function. (don't bother explaining theory, I know that, just can't apply it in this case) Could anyone help me out?
  23. G

    Fourier series of a lineer function

    Hello, My QP homework involves (not is) Fourier expansion. i think I'm done with the physics part and for the answer, i need to expand a function to Fourier series and solve it. So far well, but I couldn't solve that simple function: f(x) = x (in -1,1 interval) I've found various...
  24. D

    Need Help with Fourier Series: sinh t & 1+ltl

    I don't solve Fourier series of 1) sinh t :-1<t<1. 2) 1+ltl : -p<t<p anyone please suggest to me. Thank you very much
  25. P

    Need help to find application of the Fourier series and Fourier Transforms

    Hi, i got a task in school, in which I shall find as many application of the Fourier series And Fourier Transforms as possible. Any suggestion? Kindly Paul-Martin
  26. V

    Solving a Differential Equation with a Fourier Series

    I am not sure I am doing this correctly, so here it is. Problem: Find the Fourier Series f(s)\,=\,\left\{\begin{array}{ccc}x^2&-\pi\,<\,x\,<\,0\\0 &0\,<x\,<\,\pi \\}\end{array}\right Answer(supposedly): a_0\,=\,\frac{\pi^3}{3} a_n\,=\,-\frac{2}{n^2}...
  27. N

    Fourier Series Question: Express q(t) as a Fourier Expansion

    I got this question out of a book, but I can't get the book's answer. Since I can't draw, I'll just describe the graph given. Express q(t) as a Fourier series expansion. The charge q(t) on the plates of a capacitor at time t is shown as a saw-tooth wave with period 2\pi and its peak is at t =...
  28. G

    Tackling Fourier Series: Need Help With Examples

    I've been trying pretty consistently to work out the Fourier series for a number of functions, but continually fail to find the correct series. My book is terrible in that it only has one poorly explained example of how to do a Fourier series. I was wondering if anyone has any links to worked...
  29. M

    How to Find B_n in a Fourier Series Problem on the Interval -L < x < L?

    Hi, I need help on the following problem on Fourier series: Let phi(x)=1 for 0<x<pi. Expand 1 = \sum\limits_{n = 0}^\infty B_n cos[(n+ \frac{1}{2})x] a) Find B_n. b) Let -2pi < x < 2pi. For which such x does this series converge? For each such x, what is the sum of the series? c) Apply...
  30. Oxymoron

    Prove only using Fourier Series

    Prove only using Fourier Series! By considering the Fourier Series of f(x)=x^2 prove that \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}
  31. T

    What is the Correct Fourier Series for e^x?

    Hi all, I've been having little problems getting Fourier series of e^x. I have given f(x) = e^{x}, x \in [-\pi, \pi) Then a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi} e^{x}\ dx = \frac{2\sinh \pi}{\pi} a_{n} = \frac{1}{\pi}\int_{-\pi}^{\pi} e^{x}\cos (nx)\ dx =...
  32. J

    Understanding Fourier Series: Conceptual Explanation and Calculation Tips

    I'm having a hard time grasping exactly what a Fourier series is. I know the book definition and that it represents any periodic function as an infinite series. I also know that the a0 term is the average value of the function over one period. I can calculate the terms and everything but I...
  33. G

    What is the process for determining the Fourier series of (sin(x))^2?

    Hey guys i was working on an algorithm for one of my CS classes that included working out the Fourier series for the function f(x) = (sin(x))^2. it's been a few years since I've done anything like this, so I did some googling to refresh my memory of how to determine the Fourier coefficients...
  34. S

    Fourier Series / Fourier Transform Question

    Hello there, Im sure someone on this forum must know how to go about this. It is part of an exam question. Firstly I must draw a sketch of this pulse: v=0 when |t| > a v=V0( 1 + t/a ) when -a < t <= 0 v=V0( 1 - t/a ) when 0 < t < a v represents amplitude, V0 represents peak...
  35. S

    Finding Fourier Series for u(x): Challenges with Integration

    I have tried to find the Fourier series for a function u(x): u(x)=\sin((1+3\cos(t))t) The function is odd, hence the Fourier coefficients a_n equal zero and the b_ns are given as b_n=\frac{4}{T}\int_0^{T/2}u(t)\sin(n\omega t)\,\text{d}t where T=2\pi and \omega=2\pi/T=1...
  36. M

    Fourier Series Problem - Representing an Even Function

    In quantum mechanics, a free particle in an infinite potential well has the wave function (ie. overlap <x/phi>). Its eigenfunctions take the form: (2/a)^1/2 * sin(n*pi*x/a), n is ofcourse an integer. My question is that do all eigenfunctions form a basis? And if so how can you represent an...
  37. D

    Half vs Full Range Fourier Series: Odd & Even Functions

    I'm a little confused about the difference between the half range Fourier series and the full range Fourier series. What is the difference between the two in an odd function like f(x)=x and an even function like f(x)=x^2 ? Maybe an example to clear things up. Thank you.
  38. D

    How Do Fourier Series Model the Motion of a Struck String?

    Can someone give me some hints on this problem please? A string (length L) clamped at both ends and initially at rest, the boundary conditions for the wave function y(x,t) are: y(x,0)=y(0,t)=y(L,t)=dy/dt(x,0)=0 A note is obtained by striking the string with a hammer at some point a...
  39. S

    Do Fourier Series Remain Unique When Functions Are Shifted?

    For a given function with a certain finite period, is there only one set of Fourier series coefficients a_n and b_n? The reason I ask is, I was doing a problem where it asked for the coefficients for a certain odd function, and then it asked for the coefficients for that same function shifted...
  40. C

    Solving Fourier Series: A Simple Guide

    What is the best and simplest way to solve Fourier series problems?
  41. Z

    Understanding Fourier Series Basics

    Hi, could someone give me an explanation of Fourier Series' please or a link that would give someone who has no idea about them a working grasp of what they are.
  42. M

    The difference between Fourier Series, Fourier Transform and Laplace Transform

    Mathematically, these are three distinct, although related beasts. Laplace transform (function f(x) defined from 0 to inf) integral of f(x)e-xt, defined for t>=0. Fourier transform (function f(x) defined from -inf to inf) integral of f(x)e-itx defined for all real t. Complex Fourier series...
  43. B

    Checking Fourier Series Quickly - Best Way?

    What is the best way to (quickly) check if a calculated Fourier series is the correct one?
  44. B

    Fourier series problem (again)

    Can anyone help me with this? Show that \sum_{n=1}^{\infty}(-1)^{n+1}\frac{\cos{nx}}{n^2} = \frac{\pi^2-3x^2}{12} \quad , \quad x \in [-\pi,\pi]. I have tried writing the right-side expression as a Fourier series, but it leads nowhere. What should I do?
  45. B

    Find the Fourier series for the function f(x) = sin(4x)

    I have to find the Fourier series for the function f(x) = sin(4x), but no matter what I find _all_ the Fourier coefficients to be zero; i.e. (2\pi)^{-1}\int_{-\pi}^{\pi}sin(4x)e^{-inx}dx = 0 for all n. I can't see the point in finding the Fourier series for sin(4x) anyway, since the...
  46. R

    Quantum Physics 2nd Year Exam Help: Understanding Fourier Series

    I have a quantum physics 2nd year undergraduate exam in a few weeks, I'm a complete beginner to Fourier series, can anyone help explain how to answer this question please? Thanks, rob. The question is http://mobilecrazy.net/fourier.jpg
  47. B

    Finding Fourier Series for f(x)=2x+e^x-e^-x (-1< x >1)

    I wondered if someone could help me to find the Fourier series for this function please. I believe it's an odd function. f(x) = 2x+e^x-e^-x (-1< x > 1) This is my first post, so I'm going to try this LayTex typing too! Here goes! f(x)=2x+e^x-e^-^x (-1< x >1) Thanks
  48. A

    How Do You Calculate Fourier Series Coefficients for f(t) = sin(pi*t)?

    I am having trouble finding the An and Bn coefficients for the Fourier series f(t) = sin (pi*t) from 0<t<1, period 1 Please help! Thank you!
  49. C

    Solving the Wave Eqt Using Fourier Series-

    We've been looking at the Forurier series in our lectures which has been fine and it's okay for heat transfer, but we covered the wave eqt today and I've been left baffled. Using the boundary conditions of he diaplacement at each end of the string to be 0, we got the general eqt y(x,t) =...
  50. R

    Proving the Fourier Series for a Real, Odd Function

    The Fourier Series! Hi guys, I'm having a bit a trouble helping my daughter with this question on the Fourier series approximation: The Fourier series for a real, odd function, f(t) can be written as: f(t) = [SUM to infinity, n=1, of]: b[subscipt n] sin(nwt) where f(t=T)=f(t) and...
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