Fourier series Definition and 750 Threads

Homework Statement Calculate the following integral: \int_{0}^{2\pi}(\sum_{k=0}^{\infty} \frac{\cos(kx)}{3^k})^2 dx Homework Equations Parseval's identity: \frac{1}{2 \pi} \int_{-\pi}^{\pi} {|f(x)|^2 dx} = \sum_{n=0}^{\infty} {|a_n|^2+|b_n|^2} Where a_n, and b_n are the trigonometric...

Homework Statement f(x) = 1, 0<x<1 Extend f(x) t generate an even function P(x) and find Fourier coefficients Homework Equations an = 2/T ∫ P(x)cos(2nx/T) dx The Attempt at a Solution P(x) = 1, -1<x<1 0, -2<x<-1 , 1<x<2 even function so b0 = 0 Average of P(x)...

Hey guys! if anyone can help me I guess it is you! :) I'm trying to find the Fourier Series demonstration to continuous and periodic functions. I don't understand why people keep using X(jw) and X[e^jw] and even sometimes X(w) and X(f) If anyone can help me I'm really not understanding that...

First, I've had to find the Fourier series of F(t) = |sin(t)|, which I've calculated as f(t) = \frac{2}{\pi} + \sum_{n=1}^{\infty}\frac{4cos(2nt)}{\pi-4\pi n^2} I'm pretty sure that's right, but now I need to evaluate the sum using the above Fourier series...

Homework Statement Find the Fourier coefficients for the function *Should be a piecewise function, not sure how to write one in [itex /itex] tags* f(x) = |x|, |x| < 1, 1, 1≤|x|< 2; f(x+4) = f(x) and Find the Fourier series for f(x) = cos1/2\pi x, -1≤x<1...

I have some rather technical questions about the complex exponential form of the Fourier series: 1) What is the motivation behind the complex exponential form? Why not just use the real form (i.e. with sine and cosines)? 2) Surely the complex exponential form is an orthogonal set, i.e...

Homework Statement I must calculate the Fourier Series of f(x) = 0, when -∏< x < 0 and f(x) = sinx, 0 < x < ∏ Homework Equations The Attempt at a Solution Using the formulae, I calculated a0 = 2/pi, an = [ (-1)^n + 1 ] / [ ∏(1 - n^2) ], and bn = 0, so my Fourier series goes...

When creating a Fourier series for a function f(x), I consider whether the function is odd or even first. Yet, often these functions are in the positive region [0, L] . Since f(x) is only defined in this region, can I change the function to get a desired parity? By example, my concern...

Homework Statement I'm trying to find a Fourier series for the piecewise function where f(x)= 0 \in -\pi \leq x \leq 0 -1 \in 0 \leq x \leq \frac{\pi}{2} 1 \in \frac{\pi}{2} \leq x \leq \pi Homework Equations a_{n} = \frac{1}{\pi} \int_{0}^{2\pi}\cos(nx)y(x)\,dx b_{n} = \frac{1}{\pi}...

Homework Statement Find the Fourier series representation of: f(t)={-t , -∏<t<0 f(t)={0 , 0<t<∏ This is a piecewise function. T=2∏ (the period) Homework Equations a_{0}=\frac{2}{T}*\int_0^T f(t),dt The Attempt at a Solution I need help only with calculating the DC...

The sine-cosine (SC) Fourier series: $$f(x) = \frac{A_0}{2} + \sum_{j=1}^{+\infty} A_j cos(jx) + \sum_{j=1}^{+\infty} B_jsin(jx) $$ This form can also be expanded into a complex exponential (CE) Fourier series of the form: $$ f(x) = \sum_{n=-\infty}^{+\infty} C_n e^{inx} $$ and vice versa...

Homework Statement What is the percentage of power (out of the total power) contained up to the third harmonic (power in DC component, a1 , a-1 , a2 , a-2 , a3 , a-3 ) of the square waveform shown above? (the duty cycle = D = τ/T0= 0.5) Homework EquationsThe Attempt at a Solution Hey all...

Homework Statement Compute the power contained in the periodic signal x(t) = 10.0[cos(160.7*pi*t)]^4 Homework Equations The Attempt at a Solution Hey guys, I have just started Fourier Series and am struggling with this one. Without writing all my calculations, -I start with inverse Euler...

Homework Statement Length of rod = 1 Initial Conditions: u(x,0)=sin(πx) Boundary conditions: u(0,t)=0 and u(1,t)=5. Alright I am supposed to find the temperature at all times, but I am curious about the setup of the problem itself. When x = 1, the boundary condition says...

This is A and B my friend is telling me that Co is actually 0 and I am getting 1/2 and i don't see exactly what I am doing wrong if i indeed am doing something wrong hopefully someone here can check this out and let me know exactly where i went wrong.. Thanks

If f(x)=x+1, expand f(x) in Fourier series and hence show that \sum_{n=0}^\infty \frac{1}{(2n-1)^2}=\frac{\pi^2}{8}This question was set in an exam. I am in a position to try it if there is some interval say [-\pi \quad \pi] or like that. But there is no interval in the question. Please give...

1. http://imgur.com/UoUb27B 2. none? 3. not really sure what this question is asking. I thought that n=1 because its the fundamental frequency and the DC value should just be 120 V. I looked at some other questions and the answers were not found using that method.

[SOLVED] Fourier Series Involving Hyperbolic Functions Hello everyone! Sorry if this isn't the appropriate board, but I couldn't think of which board would be more appropriate. I was running through some problems I have to do as practice for a test and I got stuck on one I'm 99% sure they'll...

Homework Statement Given: https://www.physicsforums.com/attachments/56653, show that this can be written as: https://www.physicsforums.com/attachments/56651. Homework Equations Hint: https://www.physicsforums.com/attachments/56652 The Attempt at a Solution Quite confused by this...

Homework Statement Hi - as part of my revision I have been looking at previous exam papers and came across this question on Fourier Series. I have scanned the question in as well as my attempt at a solution. I am hoping that my solution is correct, but if anyone can find an error in it or...

Homework Statement See the second bullet point on this page: http://facweb.northseattle.edu/rjenne/e240w13flr/hwflr/feb21/e240w13hwfeb21.pdf Homework Equations So I know that fft(x) for a bunch of sample points x={x1, x2, ..., xn} returns the n Fourier coefficients for a function...

Homework Statement The signal g(t) is band limited to B Hz and is sampled by a periodic pulse train ##PT_{s}(t)## made up of a rectangular pulse of width ##1/8B## second (centered at the origin) repeating at the nyquist rate (2B pulses per second). Show that the sampled signal ##\bar{g}(t)##...

$$ -\sum_{n = 0}^{\infty}n^2\omega^2C_ne^{in\omega t} + 2\beta\sum_{n = 0}^{\infty}in\omega C_ne^{in\omega t} + \omega_0^2\sum_{n = 0}^{\infty}C_ne^{in\omega t} = \sum_{n = 0}^{\infty}f_ne^{in\omega t} $$ How can I justify removing the summations and solving for $C_n$? $$...

Hi all, How do I compute the Fourier series coefficients for unit periods for cos(pi)x, the interval is from -1/2 to 1/2. I know the formula but I am getting a wrong answer ?

Homework Statement what values does the Fourier series for f(t) converge to if t = 0 and t = 2? Homework Equations The Attempt at a Solution My answers the red rectangles for the even function t=0 >> 1 and t=2 -->1.5 and odd function t=0 >> 0 and t=2 -->1.5 because at t=0 is continuity...

So I know that sec(x) has period 2Pi, and it's even so I don't need to figure out coefficients for bn. Let's take the limits of the integral to go from -3/2 Pi to 1/2 Pi. How do I integrate sec(x) sin(nx) dx?! Am I on the right path? PS: I know that this doesn't satisfy the Dirichlet...

Homework Statement these two functions will give the same Fourier series? because when I write the graph they look the same? Homework Equations The Attempt at a Solution in the picture thank you

Hey guys. I just started a class on Fourier Analysis and I'm having a difficult time understanding this question. Any help would be much appreciated! Homework Statement Verify that the Fourier Isometry holds on [−π, π] for f(t) = t. To do this, a) calculate the coefficients of the orthogonal...

Somebody posted a question about Fourier series yesterday that got me thinking about an argument I heard some time before. If we have a (complex-valued) analytic function f, then any closed loop in the complex plane will be mapped by f to another closed loop. (If the loop doesn't enclose any...

Hi guys, I was studying the proof below and just can't figure out the the first highlighted step leads to the second and I was wondering if you guys can help me to fill that in. (: Thank you so much for your help in advance guys!

Homework Statement Third question of the day because this assignment is driving me crazy: Suppose that \left\{ f_{k} \right\} ^{k=1}_{\infty} is a sequence of Riemann integrable functions on the interval [0, 1] such that \int ^{0}_{1} |f_{k}(x) - f(x)|dx \rightarrow 0 as k \rightarrow...

Hi, The Fourier series can (among others) expressed in terms of sines and cosines with coefficients a_n and b_n and solely by sines using amplitudes A_n and phase \phi_n. I want to express the latter using a_n and b_n. Using a_n = A_n \sin(\phi_n) \\ b_n = A_n \cos(\phi_n) I...

The Fourier series of a delta train is supposedly (1/T) + (2/T ) Ʃcos(nωt) ... where T is period and ω=2*Pi/T ...but when I plot this, it doesn't give me just a spike towards positive infinity, but towards negative infinity as well (see attached pic), so this does not seem to converge to the...

Is there a way to "crudely" approximate PDEs with Fourier series? By saying crudely, I meant this way: Assuming I want a crude value for a differential equation using Taylor series; y' = x + y, y(0) = 1 i'd take a = 0 (since initially x = 0), y(a) = 1, y'(x) = x + y; y'(a)...

The problem statement: Obtain a Fourier Series Expression Form from the above graph: I can't post the graph, so I will describe it. It's a periodic function with period 1 and magnitude 5. The equation is the following: f(x) = -x, -1/2<x<1/2 I'm really stuck at trying to obtain a series...

Hey, thanks for taking the time to look ay my post (: I have attached a file which shows the question I am stuck on, and my attempt at working it out. My problem is the answer I get, is different to what my Lecturer gets (shown in the attachment). He worked it out a different way to me, he...

semi urgent Fourier series question (small) Homework Statement Hi, I have x(t) = 1/2 + cos(t) + cos(2t) so I can see that a0 = 1/2 and that it is an even function so there is no bn Also that T = 2pi so an = 2/2pi ∫02pi x(t).cos(nω0t) dt but when I integrate this I get an = 0 yet...

Homework Statement Please see picture attached Homework Equations The Attempt at a Solution ck = 1/T ∫ a-a x(t).e-jk2pit/T So x(t) = Ʃk=-∞∞ (sin(k.a.2pi/T).a-a e-jk2pit/T)/k.pi but is is supposed to be: So x(t) = Ʃk=-∞∞ (sin(k.a.2pi/T).a-a e-jk2pit/T)/k.pi.2.a but I...

Hi, I have x(t) = 1/2 + cos(t) + cos(2t) so I can see that a0 = 1/2 and that it is an even function so there is no bn Also that T = 2pi so an = 2/2pi ∫02pi x(t).cos(nω0t) dt but when I integrate this I get an = 0 yet I've been told that the answer is x(t) = 1/2 + Ʃn = 12 cos(nω0t) which...

Why is Fourier sine series of any function satisfying Dirichlet's theorem, not defined on the discontinuous points whereas we define it for Fourier cosine series? ex - sine series of f(x) = cosx, 0<=x<=∏ is defined on 0<x<∏ whereas cosine series of f(x) = sinx, 0<=x<=∏ is defined on 0<=x<=∏

$$ \sum_{n = 1}^{\infty}\sum_{m = 1}^{\infty}A_{nm}\sin\frac{n\pi x}{L}\sin\frac{m\pi y}{H} = -\frac{4}{\pi}\sum_{k = 1}^{\infty}\frac{1}{(2k-1)\sinh\frac{\pi(2k-1)H}{L}}\sin\frac{\pi(2k-1)x}{L}\sinh\frac{\pi(2k-1)y}{L} $$ If I start with x on the left, can I then end up with: $$...

Homework Statement Let f(t) be defined on [-π,π] and by f(t) = { π2 - t2 if t ≠ 1/2n, n \in N t2 if t = 1/2n, n \in N } Find its Fourier series F(t) and comment the result (type of series, type of convergence, F(t) = f(t)?, ...). Homework Equations The Attempt at a...

If you have a function with countable discontinuities on an interval, I know that the Fourier series will converge to that function without those discontinuities. But how could you explain that formally? If the basis of the Fourier series span the space L^2[a,b], that would include functions...

Homework Statement This is a question related to finding the velocity field of an incompressible fluid in a square pipe with sides at y = ±(a/2) and x = ± (a/2). It comes down to solving a homogenous equation which is also Laplace's equation \frac {δ^2 w(x,y)^H}{δ x^2} + \frac {δ^2...

can we simply truncate a Fourier series if it is divergent?? given a Fourier series of the form \sum_{n=0}^{\infty}\frac{cos(nx)}{\sqrt{n}} can i simply truncate this series up to some number finite N so i can get finite results ?? thanks.

Homework Statement Let f(x)=x, 0≤x≤p (a.) Compute the half-range sine series (b.) Use the series to show that 1-(1/3)+(1/5)+...=π/4 Homework Equations bn=(2/L)*int(from 0 to L) f(x)*sin(nπx/L) dx The Attempt at a Solution bn=(2/p)*int(from 0 to p) x*sin(nπx/p) dx Using...

Homework Statement Two similar problems, but once I find out how to do the first one, I can figure out how to do the second. My signals book tells me the answers to the following "Dn"s are: First problem: Dn = (1/∏) ∫ sin(t) * e^(-j2nt) dt = 2/(∏ (1-4n^2) ) if x(t) = rectified sin(t)...

I am SO annoyed with this problem. Ready to jump out a window. Homework Statement Find the first three terms of the Fourier series that approximates f(θ) = tan(θ) from θ = -π/2 to π/2. The Attempt at a Solution So, I know that for an equation on [\frac{-b}{2}, \frac{b}{2}], to...

I am SO annoyed with this problem. Ready to jump out a window. Homework Statement Find the first three terms of the Fourier series that approximates f(θ) = tan(θ) from θ = -π/2 to π/2. The Attempt at a Solution So, I know that for an equation on [\frac{-b}{2}, \frac{b}{2}], to define the...

Homework Statement Given the function 10sin^2(10t) Find the fundamental frequency and period. Find the exponential and trigonometric coefficients of the Fourier Series. Homework Equations The Attempt at a Solution I really have no idea how to start this problem. The sin^2...