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. matqkks

    MHB Why Use Half Range Fourier Series for Functions Like x and x^2?

    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...
  2. A

    Fourier Series and orthogonality

    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.
  3. C

    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...
  4. D

    Summing series through Fourier series

    Hi, I was trying to sum the Series S(a)=1+exp(-a^2)+exp(-4a^2)+exp(-9a^2)+... According to the notes where I found it it could be done through Fourier Series. I managed to find a relation between S(a) and S(pi/a), and it works, but I can't find S(a) alone. Can anybody help me find a way to do...
  5. matqkks

    MHB Uncovering the Hidden Significance of Fourier Series in Physics and Engineering

    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...
  6. matqkks

    Uncovering the Hidden Power of Fourier Series

    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?
  7. A

    Converge pointwise with full Fourier series

    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...
  8. B

    Exercise proofreading about Fourier Series

    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}}...
  9. ElijahRockers

    Fourier series coefficients: proof by induction

    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...
  10. matqkks

    MHB Real-Life Applications of Fourier Series

    Why are Fourier series important? 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...
  11. matqkks

    Why are Fourier series important?

    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...
  12. K

    Fourier series and orthogonality, completeness

    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...
  13. C

    Not quite clear in application of fourier series

    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?
  14. B

    Fourier series and even extension of function

    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}}...
  15. M

    Fourier series and sketch the waveform

    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...
  16. O

    Fourier Series of a step function

    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...
  17. J

    Fourier series understanding problem

    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...
  18. Ahmad Kishki

    Discrete Fourier Series question

    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
  19. J

    Heat equation problem so confusing

    Homework Statement The problem is f(x) = sin2πx - (1/πsquare)*sinπx and its given Bn sin (nπx) = f(x) Question is find Bn. Homework Equations Bn = 2/L ∫ (sin2πx - (1/πsquare)*sinπx) * sin(nπx/L) where L is 1 The Attempt at a Solution I did [/B] ∫ sin2πx * sin (nπx) - (1/πsquare)*sin...
  20. M

    Even and Odd functions - Fourier Series

    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...
  21. T

    Fourier Series for f(x) = sin(3x/2) and Evaluating Series for (1/(4n^2-9))^2

    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}...
  22. I

    Find the following fourier series in trigonometric form

    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}...
  23. G

    Fourier Series (Half-range expansion)

    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...
  24. S

    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} =...
  25. M

    Fourier Series For Function Not Centred at Zero

    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...
  26. L

    Find fourier series of wave function

    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...
  27. P

    Quarter period symmetry in Fourier series

    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)...
  28. A

    Fourier Series without complex

    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...
  29. C

    Fourier Series for f on the Interval [-π, π) | Homework Statement

    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...
  30. G

    Fourier Series (Clarification of Concept)

    Hi everyone. I ran into a problem while attempting my Fourier Series tutorial. I don't really understand the "L" in the general formula for a Fourier Series (integration form). I shall post my question and doubts as images. Thank you for any assistance rendered. <I am solving Q3 in the image.>
  31. S

    How to Calculate Error in Fourier Series and its Approximation of Angles?

    Does anyone know how to calculate the error between a function and its Fourier series representation as a function of the partial sums of the series? So far I haven't been able to find anything in the literature that talks about this. I'm also interested in looking at how well a Fourier series...
  32. E

    Complex form of Fourier series

    Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ $$a_0=\frac{2}{b-a}\int_{a}^{b}f(t)dt,$$ $$a_n=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi t}{b-a}dt,$$...
  33. P

    Solving cos ax/sin pi*x: Fourier Series Approach

    I' m trying to solve something as apparently simple like this cos ax/sin pi*x which appears solved in https://archive.org/details/TheoryOfTheFunctionsOfAComplexVariable in the page 157, exercise 9. second part. I'm trying by Fourier series, but by the moment I can't achieve it. Thanks.
  34. ranju

    Why do Fourier series require specific limits for integration?

    Homework Statement The major problem I am facing while solving for Fourier series is about the limits to be taken while integrating..! In the general equation of Fourier series the upper & lower limits are t1 & t1+T respectively..while solving for even functions we take t1 =-T/2..! Why is it...
  35. E

    Can a discontinuous function have a uniformly convergent Fourier series?

    Let's say I have Fourier series of some function, f(t), f(t)=\frac{a0}{2}+\sum_{n=1}^{\infty}(an\cos{\frac{2n\pi t}{b-a}}+bn\sin{\frac{2n\pi t}{b-a}}), where a and b are lower and upper boundary of function, a0=\frac{2}{b-a}\int_{a}^{b}f(t)dt, an=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi...
  36. E

    MATLAB How can I optimize my MATLAB code for faster Fourier series plot?

    Hi! Here is my m-file for Fourier series plot: clear clc syms n a0=input('Enter coefficient a0: '); an=input('Enter coefficient an: '); bn=input('Enter coefficient bn: '); a=input('Enter lower boundary: '); b=input('Enter upper boundary: '); t=linspace(a,b,10000); sum=0; for n=1:10 %%n could...
  37. E

    Representing Signals with Fourier Series in Multisim

    Is it possible to represent some signal in terms of Fourier series in Multisim? For example, Fourier series of sawtooth voltage with period T=2pi is $$\sum_{n=1}^{\infty }\frac{2}{n}(-1)^{n+1}sin{(nt)}=2sin{(t)}-sin{(2t)}+\frac{2}{3}sin{(3t)}-\frac{1}{2}sin{(4t)}+...$$. These terms on right side...
  38. K

    Why did Fourier choose sinusoids as the basis functions in Fourier series?

    Fourier said that any periodic signal can be represented as sum of harmonics i.e., containing frequencies which are integral multiples of fundamental frequncies. Why did he chose the basis functions i.e., the functions which are added to make the original signal to be sinusoidal? I know...
  39. J

    MATLAB Verifying Fourier Series In MATLAB

    HI please help me this could someone verify it for me please find attachement clc; clear all; k=0; s=0; N=inf; for i=1:N s=s+(1/(k^2+1)); k=k+1; end syms x n a0=1/pi*int(cosh(x),-pi,pi); an=1/pi*int(cosh(x)*cos(n*x),-pi,pi); bn=1/pi*int(cosh(x)*sin(n*x),-pi,pi); fs=0...
  40. B

    How to Determine the Correct Fourier Series for a Given Waveform?

    Homework Statement Sketch the waveform and develop its Fourier series. f(\omega t)= \begin{cases} 0 & if & 0 \leq \omega t \leq \frac{π}{2} \\ V*sin(\omega t) & if & \frac{π}{2} \leq \omega t \leq π\\ 0 & if & π \leq \omega t \leq \frac{3π}{2} \\ V*sin(\omega t) & if & \frac{3π}{2} \leq...
  41. Chacabucogod

    Fourier Series Convergence Criterion

    I'm currently reading Tolstov's "Fourier Series" and in page 58 he talks about a criterion for the convergence of a Fourier series. Tolstov States: " If for every continuous function F(x) on [a,b] and any number ε>0 there exists a linear combination σ_n(x)=γ_0ψ_0+γ_1ψ_1+...+γ_nψ_n for which...
  42. G

    What is the correct Fourier series for f(x) = 2x-1 on the interval 0<x<1?

    Homework Statement Hello guys, I have to solve one basic problem, but I got the result twice smaller that it should be. So, I am thinking that I must have missed something basic. The problem is f\left(x\right) = 2x-1 for ##0<x<1##. I have to find the Fourier coefficients. I have found A_n...
  43. S

    Discovering the Type of Waveform from a Fourier Series | Homework Help

    Homework Statement what type of waveform would this make ? Homework Equations V(t)=2/π(sin(ωt)+1/2sin(2ωt)+1/3sin(3ωt)+1/4sin(4ωt)+...) 5sin(ωt)+5sin(2ωt)+5sin(3ωt)+5sin(4ωt)... The Attempt at a Solution
  44. M

    Help with Triangle Wave using complex exponential Fourier Series

    I'm participating in research this summer and it's has to do with the Fourier Series. My professor wanted to give me practice problems before I actually started on the research. He gave me a square wave and I solved that one without many problems, but this triangle wave is another story. I've...
  45. B

    Clepsydra shape using Fourier series

    Our Fluid Mechanics professor gave us a challenge: to find the shape of a vessel with a hole at the bottom such that the water level in the vessel will change at a constant rate (i.e. if z is the height of the water in the tank dz/dt=constant). I presented a solution assuming that the vessel...
  46. S

    Fourier series for a random function

    Hello! My problem consists of : there is a representation of an uneven surface in terms of Fourier series with random coefficients: The random coefficients are under several conditions: W - function is undefined. Maybe you've confronted with such kind of expressions. The...
  47. M

    MHB Calculating the coefficients with the Fourier series

    Hey! :o I have to solve the following initial and boundary value problem: $$u_t=u_{xx}, 0<x<L, t>0 (1)$$ $$u(0,t)=u_x(L,t)=0, t>0$$ $$u(x,0)=x, 0<x<L$$ I did the following: Using the method separation of variables, the solution is of the form: $u(x,t)=X(x)T(t)$ Replacing this at $(1)$, we...
  48. U

    Use fourier series to find sum of infinite series

    Homework Statement Find the value of An and given that f(x) = 1 for 0 < x < L/2, find the sum of the infinite series. Homework Equations The Attempt at a Solution The basis is chosen to be ##c_n = \sqrt{\frac{2}{L}}cos (\frac{n\pi }{L}x)## for cosine, and ##s_n = \sqrt{\frac{2}{L}}sin...
  49. S

    Validity of Fourier Series Expansion for Non-Periodic Functions

    Homework Statement Given ∑^{∞}_{n=1} n An sin(\frac{n\pi x}{L}) = \frac{λL}{\pi c} σ(x-\frac{L}{2}) + A sin(\frac{\pi x}{2}), where L, λ, c, σ and A are known constants, find An. Homework Equations Fourier half-range sine expansion. The Attempt at a Solution I understand I...
  50. D

    Complex analysis fourier series

    Hello, Homework Statement Develop in Fourier series 1/cos(z) and cotan(z) for Im(z)>0 Homework Equations The Attempt at a Solution I really don't know how to do this, i was looking at my notes and we just saw Fourier transform and there is no example for complex functions. I...
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