Fourier series Definition and 706 Threads

1) Compute Fourier series of $f(x)=\left\{\begin{array}{cl}\sin x,&\sin x\ge0,\\0,&\sin x<0.\end{array}\right.$ 2) Consider the $2\pi-$periodic function $f(x)=\left\{\begin{array}{cl}1,&\text{ if }\phantom{-}0\le x<\pi,\\0,&\text{ if }-\pi\le x<0.\end{array}\right.$ Use Parseval's identity to...

1) Find the Fourier series of the $2\pi-$periodic function defined by $f(x)=2x,\,-\pi\le x<\pi.$ 2) Use the Fourier series of $f(x)=\cos \alpha x,$ with $0\ne\alpha\in\mathbb R$ to show that $\displaystyle\cot \alpha \pi = \frac{1}{\pi }\left( {\frac{1}{\alpha } - \sum\limits_{n = 1}^\infty...

Homework Statement Find a formal solution of the heat equation u_t=u_xx subject to the following: u(0,t)=0 u_x(∏,t)=0 u(x,0)=f(x) for 0≤x≤∏ and t≥0 Homework Equations u(x,t)=X(x)T(t)The Attempt at a Solution I first began with a separation of variables. T'(t)=λT(t) T(t) =...

Homework Statement Hi. I want to find the Fourier series of sin^2 x + sin ^3x and sin θ = [e^iθ - e-iθ]/2i Homework Equations The Attempt at a Solution So if i use sin x = [e^ix - e-ix]/2i I will get for the first term: [e^2ix + e^-2ix -2]/-4 I can do the same for...

Homework Statement Given f(x) = 2x+1 periodic from -pi<x<pi. Find the Fourier series from (-pi to pi) as well as the sine series from [0, pi] and cosine series from (0,pi)Homework Equations The Attempt at a Solution So since f(x) is periodic it is odd, so then a0 and ak = 0. for bk: bk =...

Homework Statement I'm trying to find the Fourier series of x/3. Homework Equations The Attempt at a Solution So i believe that it is an odd function so a0, ak = 0. To find bk: bk = 1/3pi int (x sin kx dx) from -pi to pi. From integration by parts = 1/3pi ([x sin kx from...

Homework Statement p(t)={ -1 from -1/220 to -1/330 0 from -1/330 to -1/660 1 from -1/660 to 1/660 0 from 1/660 to 1/330; -1 from 1/330 to 1/220 } p(t) represents the period of the excess air pressure of a sound wave. Find the harmonics...

Homework Statement Please take into account I'm not very good at maths so I would just like to make sure that what I am doing so far is correct Obtain the Fourier transform for: f(x) = x(2∏-x) 0<x<2∏ f(x)=f(x+2∏) Homework Equations f(x) = 1/2 a0 + Ʃ {an cosnx + bn sinnx} = ao/2...

I need to write the Fourier series in the Trigonometric form up to 9th harmonic Fourier Series for Square Wave with odd symmetry Peak value of 0.3 volts Average value of zero fundamental frequency of 1Hz Is this right? 4*0.3V/(pi*N)*sin(Nt)? please respond I really need help and am...

Write the Fourier series in the Trigonometric form up to 9th harmonic Fourier Series for Square Wave with odd symmetry Peak value of 0.3 volts Average value of zero fundamental frequency of 1Hz Is this right? 4*0.3V/(pi*N)*sin(Nt)?

Suppose, I have a non-periodic signal for which amplitude spectrum is to be obtained . For this why should Fourier integral be used instead of Fourier series expansion . I want to know when to use Fourier inerals and when to use Fourier series . Please let me know all the details .

i have been trying to solve a pde problem for 3 days but i couldn't even find the answer,now i feel i m about to have a mental disease,anyone can help me ?the question is u(x,0) = x u(x,2) = 0 u(0,y) = 0 d u(1,y) / dx = 0 [ d^2 u / dx^2 ] + [ d^2 u / dy^2 ] = 0 i will really be...

Fourier series of complex numbers with diffrent limits of integration? Dear all, i don't know how to simplify a COMPLEX NUMBER Fourier series with LIMITS OF INTEGRATION that are not complementary. I MEAN limits LIKE this X to -X being easy to solve and SIMPLIFY but Not X to -Y or...

Can someone please help me with a Fourier series equation for a even square wave shown below: F(t) = 0 when -2ms <t < -1ms k when -1ms <t < 1ms 0 when 1ms <t < 2ms T=4 Im after finding the first 10 harmonic components. To be honest struggling with...

Homework Statement Find the Fourier series expansion for f(x)=x^{3}, a periodic function on -\pi<x<\pi Use this to compute \zeta(6)=\sum\frac{1}{n^{6}}Homework Equations Parsevals Theorom, Real Fourier series The Attempt at a Solution I got the Fourier series to be...

Dear all, I am wondering why the Fourier Series converges at a finite discontinuity of a periodic function at 1/2*(y1+y2) at the point f(x1), where x1 is the point at which the discontinuity occurs and y1 is the limiting value for the function when we approach x=x1 from one side and y2 is the...

Homework Statement Homework Equations Usual equations for calculating Fourier series coefficients The Attempt at a Solution Well essentially I don't know what to let f(x) equal to for calculating the coefficients a0, an and bn. Should I use 1 + x/pi or 1 - x/pi? And what about...

Homework Statement f(x)= -cos(x) when -π<x<0 cos(x) when 0<x<π Decide if f is an even, odd function or either. Find the Fourier series of f. Homework Equations odd function: f(x)=f(-x) even function: -f(x)=f(-x) or f(x)=-f(-x) The Attempt at a Solution substitute -x into...

Homework Statement Homework Equations Equations to compute Fourier series given in question The Attempt at a Solution Ok so I've just done this problem and I'm having some trouble computing the Fourier series of x^{2} between -∏ < ∏ So first I calculate a_{0} and that turns...

Homework Statement http://imageshack.us/photo/my-images/824/50177563.png/ I need to find the Fourier coefficients and estimate the series for certain values of n. (4, 20 and 100) Homework Equations http://imageshack.us/photo/my-images/839/32591148.png/ The Attempt at a Solution...

Homework Statement f(t) defined by f(t) = |t| for (-pi,pi) and f(t+2pi)=f(t) the graph is just ^^^ where w=2pi/T = 1 Homework Equations Periodic function using Trigonometric from Even Function f(t) = (1/2)anot + (the sum from n=1 to inf) (an)*COS(nwt), where an = 4/T Integrated from 0 to...

Hello, I recently learned about the Fourier Series and how it can be used decompose a periodic signal into a sum of sinusoids. I can calculate all the coefficients by hand, but I wanted Mathematica to do that for me. I attempted to write a code, and it does give the desired output. I...

Homework Statement f(t)= -1 if -∏ < t ≤ 0 1 if 0 < t ≤ ∏ f(t+2∏) = f(t) question asks to compute first 3 non-zero terms in Fourier series expansion of f(t) Homework Equations The Attempt at a Solution since this is an odd function i used the Fourier sine series...

Homework Statement Show that: \sum_{n=1}^{\infty}\frac{1}{n^4} = \frac{π^4}{90} Hint: Use Parseval's theorem Homework Equations Parseval's theorem: \frac{1}{\pi}\int_{-\pi}^{\pi} |f(x)|^2dx = \frac{a_0^2}{2}+\sum_{n=1}^{\infty}(a_n^2+b_n^2) The Attempt at a Solution I've been trying to solve...

Hey there, First time user of this forum. I have a question regarding an integral I've been stuck on for the past few days. I would really appreciate any eye opener into this problem! How do I find the Fourier coefficients for f(x)=xcos(x), x [-∏,∏] So when calculating the...

Homework Statement I have a periodic function which has a period 2, which is defined as f(x) = ( 1.7(x + 1) , -1 ≤ x < 0 With f(x+2) = f(x) for all x ( 1.7(x - 1), 0 < x ≤ 1 I have to determine a0The Attempt at a Solution I have started this as a0 = 1/T ∫ top limit = T/2...

If so, could it be used to integrate sin(x)/ln(x)

Homework Statement [PLAIN]http://pokit.org/get/d9a4a8542c2acdfedbb1038292f6817b.jpg Find the coefficients of this function Ck Homework Equations The Attempt at a Solution \begin{align}\Large C_{k}=\frac{1}{T}\cdot \int_{-T/2}^{T/2} f(t)\cdot e^{\frac{-jk2\pi t}{T}}dt=\frac{1}{T}\cdot...

Hi I'm trying to understand what we mean when we say that the Fourier transform is used to transform a signal from the time domain to the frequency domain and what we actually have in the frequency domain. In Fourier series we are actually using a different representation of the signal in terms...

Homework Statement Studying for an exam, this is a question from an earlier exam: Calculate: sum k from 1 to infinity, 1/k^2 I have from previous question: f(t)=pi+sum n from 1 to infinity: 2*(-1)^n/n*sin(nt) f(t)=pi-t, -pi<=t<pi So i need to rearrange this to calc the sum...

hey guys need some basic help in Fourier series.. these are a couple of problems i got stuck at... 1. f(x) = 50 -8<x<-2 = 0 -2<= x<= 4 2. f(x) =Sin x 0<x<= pi = 0 pi <= x<= 2pi 3. f(x) = 0 0<x<1 = 1 1<= x<= 2 please...

Homework Statement Find the minimum number required (value of n) for the average deviation of the Fourier Series to fall below 2% Homework Equations Use the Uniform Convergence of Fourier Series. Where Sm is the partial sum of the Fourier Series. C is constant. Here C is ∏^2 So...

Homework Statement Given each of the functions f below, describe the set of points at which the Fourier series converges to f. b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x) Homework Equations Theorem: If f(x) is absolutely integrable, then its Fourier series converges to f...

Homework Statement Hello. I need help with orthogonality of the Fourier series coefficients. I know you can use the dirac delta function, (or the kronecker function) in the orthogonality relationship. I want to try and see the derivation using complex form rather than sines and cosines...

Triangular and sawtooth waveforms. I solved for the coeffiecients for the square wave, and was wondering if you do the same for triangular and sawtooth. I keep getting large/messy integrals that do not seem to simplify. The thing that is messing me up is that the waveform is linear with a...

Homework Statement Consider the function: f(x) = {0 if 0<x<L/2 x-L/2 if L/2<x<L} Define a periodic extension, obtain the complex Fourier series, and show that Ʃ1/(2m+1)^2 = pi^2/8... Homework Equations complex Fourier series The Attempt at a Solution I defined it as an even function by...

Hey guys, I'm having trouble with a problem assigned for homework in an EE course on Fourier series. We have yet to have a lecture on Fourier series when the homework is due Thursday, and because of the long weekend break we don't have class Tuesday. With little knowledge on Fourier series, from...

Is it possible to deduce a function from its Fourier series ?

Homework Statement Find the Fourier series of f(x) = sin^2(x) Homework Equations bn = because f(x) is even ao = (1/(2*∏))*∫(f(x)) (from 0 to 2*∏) an = (1/(∏))*∫(f(x)*cos(x)) (from 0 to 2*∏) The Attempt at a Solution ao = (1/(2*∏))*∫(f(x)) (from 0 to 2*∏) = ao = 1/2 an =...

If you are given part of a period of a Function, what rules would you apply to draw out the full function, so that it converges as quickly as possible as a Fourier series? thanks

Hi, I have a general question regarding the computation of Fourier Series coefficients for real and odd inputs. In this case, the following should be true: ∫x(t)*e^(-j*k*w*t)dt = ∫x(t)*sin(k*w*t)dt However, every time I compute my coefficients this way, I get the inverse sign of what it...

Okay, so I didn't really understand the professor when he talked about the speed of convergence of Fourier series. The question is what kind of functions converge faster than what kind of other functions using Fourier series representation. My guess from what I have absorbed is that functions...

I'm trying to get Fourier series for some function however I get the same nonsense series each time: f(x)=\begin{cases} 0, & \mbox{if x $\in [-\pi,0]$}\\ x, & \mbox{if x $\in (0,\pi)$} \end{cases} \pi a_n=\int_{-\pi}^\pi f(x)cos(nx)dx=\int_{-\pi}^0 0 \cdot cos(nx)dx+\int_0^\pi...

I'm trying to find Fourier series for the following function: f(x) = \begin{cases}1, & \mbox{if x $\in (-\frac{\pi}{2}+2\pi n,\frac{\pi}{2}+2\pi n)$ } \\ -1, & \mbox{if x $\in [\frac{\pi}{2}+2\pi n,\frac{3\pi}{2} + 2\pi n]$} \end{cases} This is how I calculated a_n and b_n: [Please See...

Homework Statement Let f be the 2pi periodic function defined by f(x)=e^{cos(x^{2})} for 0 < x < 2pi. What is the value of the Fourier series at x=4pi Homework Equations The Attempt at a Solution I don't even know where to start. All help is much appreciated!

Homework Statement A function f(x) is given as follows f(x) = 0, , -pi <= x <= pi/2 f(x) = x -pi/2 , pi/2 < x <= pi determine if it's Fourier series (given below) F(x)=\pi/16 + (1/\pi)\sum=[ (1/n^{2})(cos(n\pi) - cos(n\pi/2))cos(nx) -...

Homework Statement f(x) = 5, -pi <= x <= 0 f(x) = 3, 0 < x <= pi f(x) is the function of interest Find the x-points where F(x) fails to converge to f(x) Homework Equations F(x) = f(x) if f is continuous at x\in(-L,L) F(x) = 0.5[ f(x-) + f(x+) ] if f is...

Hello, I am trying to formalize the logical steps to prove that the Fourier Series of a function with period\rightarrow \infty leads to the Fourier transform. So let's have the Fourier series: f(x)=\sum_{n=-\infty}^{+\infty}c_n e^{i\cdot \frac{2\pi n}{L}x} where L is the period of the...

Hi. My problem is that even though I can find the Fourier series, its coefficients etc. I have trouble determining the period of equation. For example let's say we have f(t) = t, t \in [-pi,pi]. I thought that the period was 2pi but in the solution it says that the period is pi. This isn't the...

Homework Statement The function f has period 4 and is such that f(x)=2+x, -2<x\leq0 f(x)=2, 0<x<2 Sketch the graph of f for x∈[−4, 4] and obtain its Fourier series. Homework Equations The Attempt at a Solution Okay so I've pretty much sketched the graph, but I've been...