Fourier series Definition and 750 Threads

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!

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...

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...

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...

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...

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...

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...

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...

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...

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} \} =...

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...

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...

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))...

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...

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...

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)...

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...

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...

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...

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...

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...

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...

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...

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?

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...

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...

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 :(

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

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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+...

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...

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...

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...

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...

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...

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...

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