Fourier series Definition and 706 Threads

Homework Statement Since I don't know how to insert equations into a message here, I've scanned both the problem and my attempt at a solution. Where I run into problems is how to find an. I'm not completely sure how to treat that integral and was hoping somebody could nudge me in the...

Hi all! I wanted to have a little clarification about this line in relation to Fourier Series: (It's about a periodic and symmetrical signal) "x(t) is a periodic signal. As cos nwt is an even function and sin nwt is an odd function. So, if x(t) is an even function of t, then x(t) cos nwt...

Homework Statement f(x) = sin(x) for 0\leqx<\pi. Extend f(x) as an even function . Obtain a cosine Fourier series for f. Homework Equations a_{0}/2 + \sum a_{n}cos(nx) The Attempt at a Solution So as far as I know, to extend sin(x) as an even function you have to make f(x)=-sin(x)...

Homework Statement This isn't really a homework question, but something I've been wanting to know out of curiosity in David Griffiths' Introduction to Electrodynamics. On pages 131 and 132, there is a Fourier series, V(x,y) = \frac{4V_0}{\pi}\sum_{n=1,3,5...}\frac{1}{n}e^{\frac{-n \pi...

Homework Statement An oscillator is driven by a triangular periodic force (if that makes sense), which has period \tau = 2. (a) Find the long-term motion x(t), assuming the following parameters: natural period \tau[naught] = 2 (that is, \omega[naught] = π), damping parameter ß = 0.1, and...

Homework Statement Find the Fourier series for: sin(a*pi*t). Consider what happens when a -> 1/L Homework Equations The Attempt at a Solution I keep getting zeros for a_o, a_n, and b_n. I though that atleast b_n should give me something, can anyone also confirm this?

Homework Statement i can find the Fourier coefficent of the signal but i could't understand how to find the frequencies which are present in the signal.can anyone help me?

Complex Fourier Series Coeffcients; what are they? what do they represent? Homework Statement I'm not sure if this is the right place for this but it seems appropriate. I just started an intro signals and systems course at my university at the beginning of this semester. We started Complex...

Homework Statement Homework Equations The Attempt at a Solution Ive numbered the solution steps, the ones that are giving me trouble are from 1 to 2 and from 3 to 4 From 1 to 2 i don't understand how there can be an exp(x) term taken out of the bracket and still be in the...

Homework Statement Express the function plotted in the figure below as a Fourier series. Homework Equations The Attempt at a Solution I have the fully worked out solution infront of me and I am ok with working out the a0, an and bn parts but what i want to know is why is the function...

can anyone give me a hint on deriving this identity: sum(((p^n))/n)*sin(n*Q)= atan(2*p*sin(q)/(1-p^2) n = 1 to infinity p and q are polar coordinates

Homework Statement Claim: If f(x) is a REAL-valued function on x E [-L,L], then the full Fourier series is exactly equivalent to the complex Fourier series. This is a claim stated in my textbook, but without any proof. I also searched some other textbooks, but still I have no luck of...

I was finally able to figure out how to find the sine series for cos(x), but only for [0,2pi]. A question i have though is what is the interval of validity? is it only [0,pi]? Ie if I actually had to sketch the graph of the sum of the series, on all of R, would I have cosine or just a periodic...

Homework Statement f(x) = 1 0<t<1 = -1 1<t<2 How can I simplify this given that function(on the attachment). What I mean is that how can I write the function in any other way? In addition, How can I know if the function can be written in other form? How can I...

I need to find the Fourier series for the function f; 0 if -\pi \prec x \leq -\frac{\pi}{2} 1+x if -\frac{\pi}{2} \prec x \prec \frac{\pi}{2} 0 if \frac{\pi}{2} \leq x \leq \pi I've never done a Fourier series computation before so I don't really know if any of what I'm...

Homework Statement The function g(x) is defined as follows: g(x) = \left\{ \begin{array}{rcl} {-\pi e^x} & \mbox{for} & -\pi < x < 0 \\ {\pi e^{ -x}} & \mbox{for} & 0 < x < \pi \end{array}\right. And the Fourier series for g(x) is as follows: \sum_{n=0}^\infty...

Homework Statement Using the Fourier trigonometrical series for f(x) = {x^2},{\rm{ }}0 \le x < 2\pi , prove that \sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}}}} = \frac{{{\pi ^2}}}{6} The Attempt at a Solution This is more of a "what am I doing wrong question". First, because I'm not...

Homework Statement I am given this function f(x) = \left\{\begin{array}{cccc} x^2 \ \mathrm{where} \ \frac{-\pi}{2} < x < \frac{\pi}{2} \\ \ \frac{1}{4}\pi^2 \ \mathrm{where} \ \frac{\pi}{2} < x < \frac{3\pi}{2} \end{array} Doesn't this mean that the function is periodic Fourier...

Does f(t)=1 have Fourier series expansion or not?

Is a Fourier series essentially the analogue to a Taylor series except expressing a function as trigs functions rather than as polynomials? Like the Taylor series, is it ok only for analytic functions, i.e. the remainder term goes to zero as n->infinity?

Homework Statement write the Fourier series of f(x,y)=Ke^(aix+biy) Homework Equations The Attempt at a Solution

Both (i d/dx) and (-i d/dx ) are Hermitian. For some reason (-i d/dx ) is chosen to be the momentum operator, and the consequences are that [x,p]=ih (and not -ih), and that e^{ipx} is an eigenvalue of momentum p (and not -p). Is there any fundamental reason why [x,p] can't be -ih, and...

Homework Statement The heat eq'n is Ut -4Uxx = 2t - xsin(x) Ux(0,t) = Ux(pi,t) = 0, U(x,0)=x^2+1 Homework Equations Using separation of variables, in obtaining the eigenvalues/eigenfunctions of X''=-lambdaX, it would appear that you would need to use a cosine series basis and expand the...

Let f(x):= x when 0<x<1. Find the Fourier series for f if: a) f is 1 periodic b) f is even and 2 periodic c) f is odd and 2 periodic. I am very lost and behind. I'm reading through my notes and book and hopefully will be able to to this soon, but can anyone give me a hint or just explain how...

Homework Statement From the Fourier series\frac{1}{2} - \frac{1}{4}cos(x) + \sum\frac{(-1)^{n}}{1-n^{2}}cos(nx)of (1/2)x*sinx on [-\pi,\pi], find the function whose Fourier Series on [-\pi,\pi] is \frac{3}{4}sin(x) - \sum\frac{(-1)^{n}}{n-n^{3}}sin(nx)} Both sums go from n=2 to n=infinity. The...

We were given in a previous question, s_{N}(x) = \frac{4}{\pi}\sum_{n=0}^{N-1}\frac{sin(2n+1)x}{2n+1} Homework Statement Show that s'_{N}(x) = \frac{2sin(2Nx)}{\pi sinx}, x \neq l\pi and s'_{N}(x) = \frac{4N}{\pi}(-1)^l, x = l\pi where l is any integer. The Attempt at a Solution...

Having some trouble with this, any help is appreciated Homework Statement Give the complex Fourier series for f(x) = 2 - x, -2<x<2 Homework Equations f(x) = \sum_{n=-\infty}^\infty C_ne^{\frac{i n \pi x}{l} C_n=\frac{1}{2l} \int_{-l}^l f(x)e^{\frac{-i n \pi x}{2}} dx The...

Homework Statement I have found the complex Fourier series corresponding to following function f(x) = x \cdot (\pi -x) defined on the interval (0,\pi) where I get that f(x) = \frac{\pi^2}{12} + \sum(\frac{-cos(n\pi)+1}{2n^2} + \frac{cos(n\cdot \pi)}{n^3\cdot \pi} \cdot i) Then I...

Plz find urgent ::Fourier Series problem? Plz find urgent ::Fourier Series problem? given function on (-L,L) If f(x) is even(cosine series) ,odd(sine series) ,either(Fourier series) what f is given on arbitrary interval(a,b) ?

Homework Statement Verify the formula x=2*(sin(x)-(1/2)sin(2x)+(1/3)sin(3x)-...), {x,-Pi,Pi}Homework Equations The Attempt at a Solution I guess, I am to show that the sum on the right converges to the function x. I began by rewriting the sum on the RHS as $\displaystyle\sum_{k=1}^k...

Homework Statement Find the Fourier series of the function \[f\in {{C}_{st}}\] that in the interval ]-pi, pi[ is given by: \[f\left( x \right)=\left\{ \begin{array}{*{35}{l}} 0for\,-\pi <x\le 0 \\ \cos \left( x \right)for\,0<x<\pi \\ \end{array} \right.\] and give the sum of...

I have to find the Fourier series of f(x)=sin(2x) but i always get all the coefficients (a0, an and bn) equal zero. is it right? Thank you

Hi I have been working on a example and have worked it out as this Homework Statement f(x) = \( \left( \left \begin{array}{ccc}0 & \mathrm{for} & -\pi < x \leq 0 \\ cos(x) & \mathrm{for} & 0 < x < \pi\end{array} where f is defined on the interval ]-\pi,\pi[. Find the...

I'm working on a P.D.E. homework problem and the one part of it my professor gave us a function and wants us to set it equal to a given series and find the variable. Specifically, find f(sub n) in x-x^{2}=\sum (from n=1 to \infty) f(sub n) sin(n\pix) I'm not sure exactly how to do this. I...

Homework Statement Show that provided that m and n are arbitrary integers, the two functions f(t) = sin nt and g(t) = cos mt are orthogonal over the interval (0,2\pi). Explain the significance of this result in Fourier series analysis. Hint: you may find the following trigonometric identity...

Well, the problem gave me a symmetric square wave f(x). f(x) = 1, when |x|<pi/2 and -1, pi/2 < |x|< pi I was able to solve for its Fourier series expansion given by: f(x) = (4/pi) * \Sigma (-1)n cos(2n+1)x / 2n+1 Now...

sir please give me a good way to learn about Fourier series, trigonometrical Fourier series. explain the term by examples and by 3-d figure and imagination

I have a function I want to model. It is periodic, but the period keeps decreasing. Basically, it'll be a periodic function "squished" for larger values of x. The typical Fourier series is... y = SUM{aSin(nx)} + SUM{bCos(nx)} I think I will attempt y = SUM{aSin(nx^2)} +...

Hey, I was wondering. Since the Fourier Series coefficients can just be represented in the form of a Fourier Transform, what is the point of ever finding the Fourier coefficients and not doing the transform?

Hello, I am attempting a past exam paper in preparation for an upcoming exam. The past exam papers do not come with answers and I'm a little unsure as to whether I'm doing all of the questions correctly and would like some feedback if I'm going wrong somewhere. Any help is greatly appreciated...

Homework Statement hello, so I've got a couple of problems I need someone to kind of check over because on one of them, I'm not sure if it's correct, and then the other looks to be incorrect. For problem 1, we have a 2\pi periodic function where f(x) = xsin(x) For the second problem, I...

Homework Statement f(x) = |cos x| if -\pi \leq x \leq\ \pi The Fourier series answer we should end up with is the following: 2/\pi - 4/\pi\sum\frac{-1^{k}}{(2k)^{2}-1}cos(2kx) where for the summation, k = 1 and goes to infinity. What I need to do is to actually go an solve for the...

Hello, everywhere I can see this a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(t) \cos(nt)\, dt b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(t) \sin(nt)\, dt etc... I can't find, how to derive this formulas. I'm really tired and a bit confused of this formulas, because I can't find possible way to derive...

Homework Statement Compute the Fourier series for the given function f on the specified interval f(x) = x^2 on the interval − 1 < x < 1 The Attempt at a Solution Just wondering if anyone can verify my answer? f(x)=1/3+\sum(4/(n^2*pi^2)*(-1)^n*cos(n*pi*x))

Homework Statement Hi I'm stuck with the following problem: Find the Fourier series of: f(t) = { -sint for -pi < t < 0 and sint for 0 < t < pi }The Attempt at a Solutionsince they are both even functions I expanded both using the cosine series and I get two integrals which are the same...

when modelling a function for a Fourier series, what determines which function i select from the forcing function for an and bn? eg in the following would 1-t^2 be used in the an or bn formula? http://users.on.net/~rohanlal/fourier2.jpg

Homework Statement Compute the Fourier sine series for the given function: f(x) = -1 0<x<1 Homework Equations http://mathworld.wolfram.com/FourierSineSeries.html The Attempt at a Solution I got: bn = 2 * integral( (-1) * sin(npix) dx) from 0 to 1 = 2*(-1)^n/(npi)...

Homework Statement Compute the Fourier cosine series for given function: f(x)=sinx 0<x<piHomework Equations for cosine series of f(x) on [0,T]... use this general equation: http://mathworld.wolfram.com/FourierCosineSeries.html The Attempt at a Solution so I get: a0 = (2/pi) *...

Could almost nothing really be something? Fourier series. Say we have a large box. Say we have some function defined in this box that is square integrable. Say this function is small except for some small region in the box. This function could be represented as an infinite Fourier series, an...

Homework Statement { 0 -pi < t < 0 E(t) = { sin(t) 0 < t < pi Find the Fourier series w = 1, T = 2pi, L = pi a0 = 1/(2L) integral(-L to L) f(t) dt an = 1/(L) integral(-L to L) f(t)cos(nwt)dt n = 1,2,3... bn = 1/(L) integral(-L to L)...