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. Z

    Fourier series of this scale and shift of triangle wave

    This function seems to be ##tr\left (t+\frac{\pi}{2}\right )-\frac{\pi}{2}## where ##tr(t)## is the triangle wave function. $$tr(t)=|t|\ \ \ \ \ -\frac{\pi}{2}<t<\frac{\pi}{2}\tag{2}$$ ##tr## has Fourier series $$tr(t)=\frac{\pi}{2}-\frac{4}{\pi}\sum\limits_{n=1}^\infty...
  2. Z

    How to express this Fourier coefficient without trig function?

    Here is a little table I made with the values of ##b_n## for ##n=1,2,3,4,5,6##. Is there a way to write a formula for ##b_n## not involving a trigonometric function?
  3. Z

    A few questions to make sure I understand Fourier series

    Here is a plot We see that this function has minimal period of ##\pi##. Initially, the material was presented with functions that had a period of ##2\pi## and were explicitly defined on the interval ##[-\pi,\pi]## and defined everywhere else by ##f(x+n2\pi)=f(x)##. Then it was shown that...
  4. Z

    Why only odd cosine terms in this Fourier series?

    This is part of a problem in a problem set in MIT OCW's 18.03 "Differential Equations" course. This problem uses a nice online applet made for playing with Fourier coefficients. I was able to solve everything which mainly involved finding the coefficients both by tinkering with the UI and...
  5. saadhusayn

    I I want to expand a Gaussian wavepacket in terms of sines

    From this paper, I am trying to compute the coefficients in the expansion of the Gaussian wavepacket $$\phi(x) = \frac{1}{(2\pi\sigma^2)^{\frac{1}{4}}}\exp \Big(-\frac{(x-x_{0})^{2}}{4\sigma^{2}} + ik_{0}(x-x_{0})\Big) $$ where $$\sigma << 1$$and $$k_{0} >> \frac{1}{\sigma}$$ in terms of the...
  6. Q

    I Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines

    Hello everyone, I've been delving deep into the realm of periodic functions and their properties. One of the fundamental concepts I've come across is the use of sines and cosines as an orthogonal basis for representing any functions. This is evident in Fourier series expansions, where any...
  7. G

    Showing that a certain summation is equal to a Dirac delta?

    I'm studying Quantum Field Theory for the Gifted Amateur and feel like my math background for it is a bit shaky. This was my attempt at a derivation of the above. I know it's not rigorous, but is it at least conceptually right? I'll only show it for bosons since it's pretty much the same for...
  8. Skaiserollz89

    Relationship between Fourier coefficients and power spectral density

    Here, ##\Phi(f_{x_n},f_{y_m})=|\mathscr{F(\phi(x,y))}|^2 ## is the Power Spectral Density of ##\phi(x,y)## and ##\mathscr{F}## is the Fourier transform operator. Parseval's Theorem relates the phase ##\phi(x,y)## to the power spectral density ##\Phi(f_{x_n},f_{y_m})## by...
  9. Euge

    POTW Fourier Series on the Unit Interval

    Evaluate the Fourier series $$\frac{1}{\pi^2}\sum_{k = 1}^\infty \frac{\cos 2\pi kx}{k^2}$$ for ##0 \le x \le 1##.
  10. P

    V Space With Norm $||*||$ - Fourier Series

    Hi, a question regarding something I could not really understand The question is: Let V be a space with Norm $||*||$ Prove if $v_n$ converges to vector $v$. and if $v_n$ converges to vector $w$ so $v=w$ and show it by defintion. The question is simple, the thing I dont understand, what...
  11. S

    Comp Sci Plot periodic function with Fourier coefficients

    I have plotted the function for ##T=15## and ##\tau=T/30## below with the following code in Python: import numpy as np import matplotlib.pyplot as plt def p(t,T,tau): n=np.floor(t/T) t=t-n*T if t<(2*np.pi*tau): p=np.sin(t/tau) else: p=0 return p...
  12. R

    Fourier series, periodic function for a string free at each end

    From the statement above, since the ring is massless, there's no force acting vertically on the rings. Thus, the slope is null. ##\frac{\partial y(0,0)}{\partial x} = \frac{\partial y(L,0)}{\partial x} = 0## ##\frac{\partial y(0,0)}{\partial x} = A\frac{2 \pi}{L}cos(\frac{2 \pi 0}{L}) =...
  13. S

    MHB Fourier Series involving Hyperbolic Functions

    Hello everyone first time here. don't know if it's the correct group... Am having some issues wiz my maths homework that going to count as a final assessment. Really Really need help. The function (f), with a period of 2π is : f(x) = cosh(x-2π) if x [π;3π].. I had to do a graph as the first...
  14. A

    The value of a Fourier series at a jump point (discontinuity)

    Greetings according to the function we can see that there is a jump at x=e and I know that the value of the function at x=e should be the average between the value of f(x) at this points my problem is the following the limit of f(x) at x=e is -infinity and f(e)=1 how can we deal with such...
  15. docnet

    I The precise relationship between Fourier series and Fourier transform

    Would someone be able to explain like I am five years old, what is the precise relationship between Fourier series and Fourier transform? Could someone maybe offer a concrete example that clearly illustrates the relationship between the two? I found an old thread that discusses this, but I...
  16. A

    Problem with the sum of a Fourier series

    Good day I really don't understand how they got this result? for me the sum of the Fourier serie of of f is equal to f(2)=log(3) any help would be highly appreciated! thanks in advance!
  17. Tony Hau

    Finding the Fourier Series of a step function

    The answer in the textbook writes: $$ f(x) = \frac{1}{4} +\frac{1}{\pi}(\frac{\cos(x)}{1}-\frac{\cos(3x)}{3}+\frac{\cos(5x)}{5} \dots) + \frac{1}{\pi}(\frac{\sin(x)}{1}-\frac{2\sin(2x)}{2}+\frac{\sin(3x)}{3} + \frac{\sin(5x)}{5}\dots)$$ I am ok with the two trigonometric series in the answer...
  18. no_drama_llama_77

    I Fourier Series and Cepheid Variables

    If given a set of data points for the magnitude of a cepheid variable at a certain time (JD), how can we use Fourier series to find the period of the cepheid variable? I'm trying to do a math investigation (IB math investigation) on finding the period of the cepheid variable M31_V1 from data...
  19. rannasquaer

    MHB Dirac Delta and Fourier Series

    A beam of length L with fixed ends, has a concentrated force P applied in the center exactly in L / 2. In the differential equation: \[ \frac{d^4y(x)}{dx^4}=\frac{1}{\text{EI}}q(x) \] In which \[ q(x)= P \delta(x-\frac{L}{2}) \] P represents an infinitely concentrated charge distribution...
  20. M

    How to 'shift' Fourier series to match the initial condition of this PDE?

    Hi, Question: If we have an initial condition, valid for -L \leq x \leq L : f(x) = \frac{40x}{L} how can I utilise a know Fourier series to get to the solution without doing the integration (I know the integral isn't tricky, but still this method might help out in other situations)? We are...
  21. agnimusayoti

    Fourier series for trigonometric absolute value function

    First, I try to define the function in the figure above: ##V(t)=100\left[sin(120\{pi}\right]##. Then, I use the fact that absolute value function is an even function, so only Fourier series only contain cosine terms. In other words, ##b_n = 0## Next, I want to determine Fourier coefficient...
  22. Adesh

    Analysis Books for learning Fourier series expansion

    Hello Everyone! I want to learn about Fourier series (not Fourier transform), that is approximating a continuous periodic function with something like this ##a_0 \sum_{n=1}^{\infty} (a_n \cos nt + b_n \sin nt)##. I tried some videos and lecture notes that I could find with a google search but...
  23. M

    Fourier series and the shifting property of Fourier transform

    Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform. So here's my attempt to this problem so far...
  24. lottotlyl

    Engineering Why Multiply by Exponential Terms in Fourier Series Calculations?

    i tired using complex identity equation for sin(pi*k/3) but it doesn't work out
  25. Jason-Li

    Comp Sci Fourier analysis & determination of Fourier Series

    ANY AND ALL HELP IS GREATLY APPRECIATED :smile: I have found old posts for this question however after reading through them several times I am having a hard time knowing where to start. I am happy with the sketch that the function is correctly drawn and is neither odd nor even. It's title is...
  26. Raihan amin

    I A claim regarding Fourier Series

    This is written on Greiner's Classical Mechanics when solving a Tautochrone problem. Firstly,I don’t understand why we didn’t use the term ##m=0## and Sencondly, how the integrand helps us to fulfill the Dirichlet conditions. That means,how do we know that the period is 1?Thanks
  27. L

    Green's functions (Fourier Series)

    In order to obtain equation (3), I think I have to do the Fourier transform in the x direction: \begin{equation} \tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx \end{equation} So I have: \begin{equation} -k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...
  28. rude man

    Compact summary of Fourier series equations

    Greg has kindly allowed me to post these equations which I compiled many years ago. Somehow I like them better than anything I've ever run across so maybe someone else will find them useful also. Actually, I have given some thought to the Fourier series and how they tie in with sampled-data...
  29. Phys pilot

    I Fourier series coefficients in a not centered interval

    Hello, so for a Fourier series in the interval [-L,L] with L=L and T=2L the coefficients are given by $$a_0=\frac{1}{L}\int_{-L}^Lf(t)dt$$ $$a_n=\frac{1}{L}\int_{-L}^Lf(t)\cos{\frac{n\pi t}{L}}dt$$ $$b_n=\frac{1}{L}\int_{-L}^Lf(t)\sin{\frac{n\pi t}{L}}dt$$ But if we have an interval like [0,L]...
  30. PainterGuy

    I What does it mean when an integral is evaluated over a single limit?

    Hi, A function which could be represented using Fourier series should be periodic and bounded. I'd say that the function should also integrate to zero over its period ignoring the DC component. For many functions area from -π to 0 cancels out the area from 0 to π. For example, Fourier series...
  31. E

    Fourier series for a series of functions

    ## ## Well I start with equation 1): ## e^{b\theta }=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in\theta } ## If ## \theta =0 ## ##e^{b(0)}=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in(0) }## ##1=\frac{sinh(b\pi )}{\pi...
  32. K

    Why Does the Fourier Series of |sin(x)| Treat n=1 Differently?

    Homework Statement Hello, i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1...
  33. merlyn

    Fourier series equation derivation

    Hi all. Could someone work out for me how equation 21 in attachment left side becomes right side. Please show in detail if you could. It's for exponential Fourier series. Drforbin thank you
  34. J

    B Why the Fourier series doesn't work to solve any differential equation?

    I know this may sound as a stupid question but I would like to clarify this. An arbitrary function f can be expressed in the Fourier base of sines and cosines. My question is, Can this method be used to solve any differential equation? You plug into the unkown function the infinite series and...
  35. N

    Fourier Series: How to interpret the function?

    This is a rather simple question, but am I understanding the following correctly? 1. Homework Statement The Attempt at a Solution This isn't really the problem, but I have a feeling my problem the assignments, is me misunderstanding the function description. I don't see how this 2 pi...
  36. D

    Finding the fourier spectrum of a function

    Homework Statement Find the Fourier spectrum ##C_k## of the following function and draw it's graph: Homework Equations 3. The Attempt at a Solution [/B] I know that the complex Fourier coefficient of a rectangular impulse ##U## on an interval ##[-\frac{\tau}{2}, \frac{\tau}{2}]## is ##C_k =...
  37. T

    A ELI5: Fourier Series Explained

    Working on some microwave stuff, read about this but can't understand the explanations online.
  38. C

    Arbitrary Circulation Calculation with Fourier Series

    Homework Statement Homework Equations The Attempt at a Solution I am stuck trying to figure out why there are three different alphas and why in the equation we are supposed to use has a theta and what that means. If I can set up the Fourier series I can properley I know how to solve it for...
  39. DoobleD

    I Fourier series of Dirac comb, complex VS real approaches

    Hello, I tried to compute the Fourier series coefficients for the Dirac comb function. I did it using both the "complex" formula and the "real" formula for the Fourier series, and I got : - complex formula : Cn = 1/T - real formula : a0 = 1/T, an = 2/T, bn = 0 This seems to be valid since it...
  40. L

    Find the Fourier Series of the function

    Homework Statement Find the Fourier series of the function ##f## given by ##f(x) = 1##, ##|x| \geq \frac{\pi}{2}## and ##f(x) = 0##, ##|x| \leq \frac{\pi}{2}## over the interval ##[-\pi, \pi]##. Homework Equations From my lecture notes, the Fourier series is ##f(t) = \frac{a_0}{2}*1 +...
  41. A

    I Complex Fourier Series: Even/Odd Half Range Expansion

    Does the complex form of Fourier series assume even or odd half range expansion?
  42. Duke Le

    I [Signal and system] Function with fourier series a[k] = 1

    We have: Period T = 4, so fundamental frequency w0 = pi/2. This question seems sooo easy. But when I use the integral: x(t) = Σa[k] * exp(i*k*pi/2*t). I get 1 + sum(cos(k*pi/2*t)), which does not converge. Where did I went wrong ? Thanks a lot for your help.
  43. S

    Is My Fourier Series Expansion of a Sawtooth Wave Correct?

    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...
  44. ElPimiento

    Coefficients for an exponential Fourier Series

    I'm kinda just hoping someone can look over my work and tell me if I'm solving the problem correctly. Since my final answer is very messy, I don't trust it. 1. Homework Statement We're asked to find the Fourier series for the following function: $$ f(\theta)=e^{−\alpha \lvert \theta \rvert}}...
  45. J6204

    What formula should be used to find the Fourier series of an even function?

    Homework Statement In the following problem I am trying to extend the function $$f(x) = x $$ defined on the interval $$(0,\pi)$$ into the interval $$(-\pi,0)$$ as a even function. Then I need to find the Fourier series of this function.Homework EquationsThe Attempt at a Solution So I believe I...
  46. J6204

    Calculating the Fourier integral representation of f(x)

    Homework Statement Considering the function $$f(x) = e^{-x}, x>0$$ and $$f(-x) = f(x)$$. I am trying to find the Fourier integral representation of f(x). Homework Equations $$f(x) = \int_0^\infty \left( A(\alpha)\cos\alpha x +B(\alpha) \sin\alpha x\right) d\alpha$$ $$A(\alpha) =...
  47. J6204

    Extending function to determine Fourier series

    In the following question I need to find the Fourier cosine series of the triangular wave formed by extending the function f(x) as a periodic function of period 2 $$f(x) = \begin{cases} 1+x,& -1\leq x \leq 0\\ 1-x, & 0\leq x \leq 1\\\end{cases}$$ I just have a few questions then I will be able...