Fourier series Definition and 706 Threads

What is the complex Fourier series of f(x)=x from -pi to pi? I'm in a complex variables class and we have an extra credit assignment to figure out the complex Fourier series of f(x)=x from -pi to pi. We only vaguely covered the topic in class and our book is not very good so I'm not...

Homework Statement f(x) = e^x for -\pi/2 \leq x \leq \pi/2 Homework Equations The Attempt at a Solution Having a bit of trouble getting this out. This is a odd function right so a_n = 0 so i only have to find b_n , right?

Homework Statement Suppose we are given the following information about a continuous-time periodic signal with period 3 and Fourier coefficients a_{k} 1. a_{k} = a_{k+2} 2. a_{k} = a_{-k} 3. \int_{-0.5}^{0.5}x(t)dt = 1 4. \int_{0.5}^{1.5}x(t)dt = 2 Determine x(t) Homework Equations if x(t)...

Homework Statement let f(x)={0;-2\leqx\leq0. x;0\leqx\leq2 find a0 an bn given the period is 4 Homework Equations a0=1/L\intf(x)dx an=1/L\intf(x)cos(n\pix/L) bn=1/L\intf(x)sin(n\pix/L) The Attempt at a Solution so I can get a0 = 1 but I run into trouble with an. so I plug...

i'm going through this question. firstly how would the graph look like? at first i thought it would be http://i.imgur.com/XfVCK.png the top graph drawn, but then i thought maybe it's the bottom one. if it was the bottom one then it'd be easier for me to do the question, since it'd be an odd...

Homework Statement Homework Equations The Attempt at a Solution i'm pretty much learning Fourier series from scratch today after only looking at it in lectures (and my exam is tomorrow lol @ me) and I'm sort of stuck. The solutions have something different to what i have, i think it might be an...

x(t) is periodic with T=3. X(k) is FS coeff. X(k)=X(-k) and X(k)=X(k+2) also integral from t=-0.5 to 0.5 of x(t)dt is 1 integral from 0.5 to 1.5 of x(t) dt is 2. Find x(t) ======== i found that the signal x(t) is even and X(0) is 5/3 (if I'm correct) and integral from t=-1.5 to 1.5 of...

Homework Statement We're given this 'interactive page' that gives us the values, so T=2.8066, W=0.9542 and A=8.5988 and then told to find a0, b0, a1, b1, a3, b3, Total Power and 3rd harmonic power. Homework Equations Cn given above and: a0=1/T \int s(t) dt integrating from 0 to T. Also...

Find the Fourier Series for f(x)=x^2 evaluate f(0) and show that the summation \sum^{\infty}_{n=1}\frac{1}{(2n-1)^2}=\frac{\pi^2}{8} The first part of this problem asked that I find a_{n} and b_{n} Since x^2 is an even function b_n=0 and for a_n I got \frac{4(-1)^2}{n^2} for a_0 I got...

Homework Statement For the function f(t)={t+2, -2<t<0 {2, 0<t<2 {f(t+4) all t Homework Equations a(n)=2/T*Integration[f(t)cos(n*w*t)dt] b(n)=2/T*Integration[f(t)sin(n*w*t)dt] xcos(ax)=...

Homework Statement Hey, This is a question from a sample exam we are given for our engineering maths exam. Firstly given that the Fourier series contains only "sin(x)" doesn't this mean that it is an "odd" function? Can the period be calculated from the given function easily? Secondly is...

Hi. Well, I have to demonstrate that \sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}=\frac{\pi^2}{8} Using a previous result of a Fourier series for f(t)=1 if 0<t<1, f(t)=0 if -1<t<0. I've found that: f(t)\sim{}1/2+\sum_{n=1}^{\infty}\frac{2}{(2n-1)\pi}\sin([2k-1]\pi t) I think this result, the series...

Hi there. I have some trouble with this problem, it asks me to find the Fourier expansion series for the function f(t)=0 if -pi<t<0, f(t)=t^2 if 0<t<pi So I've found the coefficients a_0=\displaystyle\frac{1}{\pi}\displaystyle\int_{0}^{\pi}t^2dt=\displaystyle\frac{\pi^2}{3}...

Hi there. I have this interesting problem which I don't know how to solve. I'll post it here because I think more people will se it, but I'm not sure if this is the proper subforum. The problem says: How can be sure that \sum_{n = 1}^\infty \frac{1}{n}\sin (nx) isn't the Fourier series of...

How can I transform a complex Fourier series into a real one in general? If for example I have the complex Fourier series for K=2m+1 (\frac{1}{2}+\frac{2}{i\pi})\sum\frac{1}{2m+1}e^{i(2m+1)t} what shoudl I do to transform it into a series with real coeff?

I am so stuck on my revision and i really need someones help! I am using the definition of Fourier series as My lecturer has told us that if f is odd. Could someone please tell me how he has derived this because i can't understand how he's got to it, iv tried using trig identities and...

let be the Fourier expansion of the function f(x) = \sum_{m=-\infty}^{m=\infty}c_{m} exp(imx) valid on the interval (-1,1) , from this can we obtain the inverse function f^{-1} (x) by reflection of the Fourier series through the line y=x ??

Hi, All: Given a normed vector space (X,||.||), and an inconsistent system Ax=b, the generalized least squares solution x^ to Ax=b is the point in the span of Ax that is closest to b, i.e., given a fixed matrix A, we define AX={Ax: x in X}, and then: x^:={ x in AX...

Homework Statement how to expand f(x)=x^2, -pi<x<pi as Fourier series answer at back of the book says: f(x)=((pi^2)/3)+4sigma(from 1 to infinity) [((-1)^n)/(n^2)]cos nx i tried the a(n) b(n) stuff but i don't see where they get the n^2 in the denominator from when i integrated to try to...

Hi, I'm in a signal processing class, and I'm having some trouble with complex numbers. As an example, I've attached a pretty simple question about an exponential Fourier series. I don't find these questions particularly hard, it just takes me ridiculously long to do them. I mean, after I...

Hello I'am a little confused. In my textbook it is written that all odd function can be described by a sine series. I have this following equation from an exercise: A_{0}+\sum\limits_{n=1}^\infty (A_{n} cos(n \phi) + B_{n} sin(n \phi))c^{n} = sin(\dfrac{\phi}{2}) It's a standard...

Homework Statement See figure attached. Homework Equations The Attempt at a Solution See figure attached for my attempt. Is this the correct way to sketch the full Fourier series of the given function? If not, what am I doing wrong/misunderstanding? Thanks again!

Homework Statement one period of a function f(t) is given by the piecewise function f(t)= 2t if -pi/2 <t <0 and -1 if 0< t < pi/2 I need to find a sub 0, a sub n, a sub 1,2,3,& 4, and b sub 1,2,3 &4 If I could just get a sub 0 the rest are easy, however, my answer does not look correct...

D_n=\frac{1}{\pi}\int_0^\pi\sin{(t)}\cdot e^{-i2nt}dt=\frac{2}{\pi(1-4n^2)} I have no idea on how they get from one side of the equation symbol to the other, can i get some tips and tricks ? I have try ed writing sint as an exp function, but i don't feel it gets me anywhere close.

Hello. I have to find the Fourier series for f(x) = 1 + cos(pi x / L). My question is about f(x) Is this function even? I plotted it out and it looks even. The question I am completing starts off by saying: assume that any function f for which f and its derivative are piecewise continuous...

Homework Statement I just have a general question about a set of questions given in my textbook. Homework Equations The Attempt at a Solution The questions are given in sequential order, and they both ask the same thing, Find the Fourier series of the function f(x). One...

Homework Statement determine the Fourier Series for f(x)=cos3x Homework Equations f(x)=ao/2+(sum) an cos(nx)+ (sum) bncos(nx) ao= (integral) f(x)dx (from -\pi to\pi) an= (integral) f(x)cos(nx)dx (from -\pi to\pi) bn= (integral) f(x)sin(nx)dx (from -\pi to\pi) The Attempt at a Solution i...

Homework Statement Q1) (dy/dx)= 2x(y2+9); y(0)=0 Q2) (x4+y2) dx - xy dy =0; y(2)=1 Q3) (dy/dt)= 4y+t Q4) y"+2y'+y=0; y(0)=4 and y'(0)=-6 Q5) y"+3y'+2y= 30e2t 3. Solutions found A1) y= tan (x2/3) A2) this is not an exact differential and so cannot be solved A3) couldn't solve this...

Homework Statement http://img684.imageshack.us/img684/4496/fourier.png Homework Equations http://en.wikipedia.org/wiki/Small-angle_approximation The Attempt at a Solution This is part of a larger question, I have underlined in Red the areas I am struggling with and have cut out...

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...

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...

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)=a[SIZE="2"]0/2+∑(from n=1 to∞) (a[SIZE="2"]ncos(n pi x/L)+b[SIZE="2"]nsin(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...