Fourier series Definition and 706 Threads

Hi folks, I found the Fourier Series for Cosh(x) in the range -1 to +1 and discovered that: f(x)=sinh(1)(1+\sum_{n=1}^\infty\frac{(-1)^n}{1+n^2\pi^2}cos(n\pi x)) Now, I have to integrate the series twice to prove that: \sum_{n=1}^\infty\frac{(-1)^n}{n^2\pi^2(1+n^2\pi^2)}cos(n\pi...

HEllo , Plz help me in solving these problems regarding Fourier Series: 1):Find Fourier Sine Series for f(x)=x^2 -Π(pie)<x<Π(pie) and deduce Π^2/12=1-1/4+1/9-1/16 +------------------- Π indicates PIE 2):With the help of Fourier Sine series and Fourier Cosine series...

can anyone help me solve this question? find the Fourier series for the function f(x)= -2-x for -2<x<0 and f(x)= 2-x for 0<x<2 and f(x+4)=f(x)thank you very much

I have a question about Fourier series that I would like some help with. If there is a function f(t) which does not satisfy all of Dirichlet's conditions then can its Fourier series still represent it? All I've got is that if all of Dirichlet's conditions are satisified by f(t) then the Fourier...

Not sure if this should go into a math section, but I am trying to understand it in order to understand the Hesenberg uncertainy principle. I can't find a simple introduction to Fourier series to answer this question. In my modern physics book, it does a quick introduction to Fourier series...

Anyone have any clues what function f(x) satisfies the following integral (for an integer n > 0): \int^{\pi}_{0} f(x)sin(nx)dx=(-1)^{n+1}

Hi Given the function f(t) = t^2, were t \in ]- \pi, pi[, and is continious find the Fourier series for f(t). L = 2 \pi. Then A_0 = \frac{1}{2 \pi} \int \limit_{-\pi} ^{\pi} t^2 dt = \frac{\pi ^2}{3} A_n = \int \limit_{-\pi} ^{\pi} t^2 \cdot cos(\matrm{n} \pi \mathrm{t}) dt A_n = \int...

Can anyone help me out with this? Find the steady state periodic solution of the following differential equation. x''+10x= F(t), where F(t) is the even function of period 4 such that F(t)=3 if 0<t<1 , F(t)=-3 if 1<t<2. Im basically just having a problem findind the general Fourier...

Can anyone help me out with this? Find the steady state periodic solution of the following differential equation. x''+10x= F(t), where F(t) is the even function of period 4 such that F(t)=3 if 0<t<1 , F(t)=-3 if 1<t<2. Im basically just having a problem findind the general Fourier...

The book for the class that I'm currently taking is "Introduction to Applied Mathematics" by Gilbert Strang. Things have been good with this class until this chapter. If you have used this book, you will understand what I mean when I say it is different. Things have been ok, because I've been...

Ok, I'm going to bed. But I have to ask ANOTHER question about my homework... so I can get up early and work on it. Q: Around the unit circle suppose u is a square wave: u_0 = \left\{\begin{array}{c} +1 \,\,\,\, on\,the\,upper\,semicircle \,\,\,\, 0<\theta < \pi \\ -1 \,\,\,\...

Here is the question: At x= \frac{\pi}{2} the square wave equals 1. From the Fourier series at this point find the alternating sum that equals \pi . \pi = 4(1 - \frac{1}{3}+\frac{1}{5}-\frac{1}{7} + \ldots I do not understand what the question is asking. I'm not knowledgeable enough...

ok, i wasn't sure if i ought put this in math or phys, we're going over it my phys class, but its just math... whatever.. So i had to find the Fourier series representation of x^2 in the intervals (-pi, pi) and (0, 2pi). i haven't even started the (0, 2pi) one, cause i can't get the first...

hey guys, I've got to do some Fourier series work using matlab, but I have no idea what to do. Ive found the coefficients by hand, but now I need to use MATLAB to evaluate and tabulate the first 8 terms. I then have to evaluate the series at 1000 points over a certain range. Does...

So I'm working this HW problem, namely Suppose f is a continuous function on \mathbb{R}, with period 1. Prove that \lim_{N\rightarrow\infty} \frac{1}{N}\sum_{n=1}^{N} f(\alpha n) = \int_{0}^{1} f(t) dt for every real irrational number \alpha. The above is for context. The hint says...

Hi, can someone help me out with the following question? Q. Show that the Fourier series for the function y(x) = |x| in the range -pi <= x < pi is y\left( x \right) = \frac{\pi }{2} - \frac{4}{\pi }\sum\limits_{m = 0}^\infty {\frac{{\cos \left( {2m + 1} \right)x}}{{\left( {2m + 1}...

I have this problem. I would appreciate it if anyone can help me get started. Question: Consider the differential equation: \frac{d^2 y(x)}{dx^2} + y(x) = f(x) \ \ ; \ \ 0 \leq x \leq L \\ The boundard conditions for y(x) are: y(0) = y(L) = 0 \\ Here f(x) is assumed to be a known function...

http://www4.okfoto.co.kr/S_storage4/314500/A05120618494148_t.jpg http://www4.okfoto.co.kr/S_storage4/314500/A05120618494175_t.jpg It's solution is...

Please check my solution and I need help on understanding the second part of the question. Q:Obtain the complex form of the Fourier series of the sawtooth function. f(t) = \frac{2t}{T} \ \ \ 0 < t < 2T\\ So if the period is 2l = 2T then l = T \\ c_n = \frac{1}{2l} \int_{-l}^{l} f(x) e^{in\pi...

I have some problems which says show that (i) \sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90} and (ii) \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} = \frac{\pi^2}{12} And another one which says, show that for 0<x<\pi sin x + \frac{sin 3x}{3} + \frac{sin 5x}{5} + ... = \frac{\pi}{4} The...

Find the Fourier series for f(x)=1 on the interval 0 <=x <= pi in temrs of phi = sin nx. By integrating thi series find a convergent series for hte function g(x) =x oin this interval assuming that the set {sin nx} is complete i can find the Fourier series for f(x) =1. But i would like to know...

Find the FOurier Series in terms of \phi_{n} = \sin(nx) of the step function f(x) = 0 for 0 \leq x \leq \frac{1}{2} \pi [/tex] f(x) =1 for [itex] \frac{1}{2} \pi < x \leq \pi now i have no problem finding the series for each branch. But how would i combine them? for the 0 to 1/2 pi...

Show that \lim_{n \rightarrow \infty} \int_{0}^{\pi} Ln(x) Sin(nx) dx i was told to use this identity given that int f^2 \rho dx is finite then c_{n}^2 \int_{a}^{b} \phi_{n}^2 \rho dx = \frac{(\int_{a}^{b} f \phi_{n} \rho dx)^2}{\int_{a}^{b} \phi_{n}^2 \rho dx} \rightarrow 0 and n...

Find the Fourier Series in terms of \phi_{n} = \sin{nx} of the step function f(x) = 0 for 0 \leq x \leq \frac{1}{2} \pi = 1 for \frac{1}{2} \pi < x \leq \pi Solution Fourier Series is \Sigma c_{n} \phi_{n} and for hte interval for x between 0 and 1/2 pi c_{n} =...

say if you get a result for a0 but 0 for an and bn(using my book's notation where the Fourier series is a0+an*cos(nx)+bn*sin(nx)

I've got parts of this problem but I'm stuck on some of the integration. See attached. Thanks!

GIVE ME A HINT! Fourier series / Kepler's equation By expanding e \sin\psi in a Fourier series in \omega t, show that Kepler's equation has the formal solution \psi = \omega t + \sum_{n=1}^{\infty}{\frac{2}{n}J_{n}(ne)\sin{\omega t}} where J_{n} is the Bessel function of order n. For small...

f(x) = cos(x) x from [-PI, 0] f(x) = -cos(x) x from ] 0, PI] I'm not sure how to deal with getting a Fourier series for this function. (don't bother explaining theory, I know that, just can't apply it in this case) Could anyone help me out?

Hello, My QP homework involves (not is) Fourier expansion. i think I'm done with the physics part and for the answer, i need to expand a function to Fourier series and solve it. So far well, but I couldn't solve that simple function: f(x) = x (in -1,1 interval) I've found various...

I don't solve Fourier series of 1) sinh t :-1<t<1. 2) 1+ltl : -p<t<p anyone please suggest to me. Thank you very much

Hi, i got a task in school, in which I shall find as many application of the Fourier series And Fourier Transforms as possible. Any suggestion? Kindly Paul-Martin

I am not sure I am doing this correctly, so here it is. Problem: Find the Fourier Series f(s)\,=\,\left\{\begin{array}{ccc}x^2&-\pi\,<\,x\,<\,0\\0 &0\,<x\,<\,\pi \\}\end{array}\right Answer(supposedly): a_0\,=\,\frac{\pi^3}{3} a_n\,=\,-\frac{2}{n^2}...

I got this question out of a book, but I can't get the book's answer. Since I can't draw, I'll just describe the graph given. Express q(t) as a Fourier series expansion. The charge q(t) on the plates of a capacitor at time t is shown as a saw-tooth wave with period 2\pi and its peak is at t =...

I've been trying pretty consistently to work out the Fourier series for a number of functions, but continually fail to find the correct series. My book is terrible in that it only has one poorly explained example of how to do a Fourier series. I was wondering if anyone has any links to worked...

Hi, I need help on the following problem on Fourier series: Let phi(x)=1 for 0<x<pi. Expand 1 = \sum\limits_{n = 0}^\infty B_n cos[(n+ \frac{1}{2})x] a) Find B_n. b) Let -2pi < x < 2pi. For which such x does this series converge? For each such x, what is the sum of the series? c) Apply...

Prove only using Fourier Series! By considering the Fourier Series of f(x)=x^2 prove that \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}

Hi all, I've been having little problems getting Fourier series of e^x. I have given f(x) = e^{x}, x \in [-\pi, \pi) Then a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi} e^{x}\ dx = \frac{2\sinh \pi}{\pi} a_{n} = \frac{1}{\pi}\int_{-\pi}^{\pi} e^{x}\cos (nx)\ dx =...

I'm having a hard time grasping exactly what a Fourier series is. I know the book definition and that it represents any periodic function as an infinite series. I also know that the a0 term is the average value of the function over one period. I can calculate the terms and everything but I...

Hey guys i was working on an algorithm for one of my CS classes that included working out the Fourier series for the function f(x) = (sin(x))^2. it's been a few years since I've done anything like this, so I did some googling to refresh my memory of how to determine the Fourier coefficients...

Hello there, Im sure someone on this forum must know how to go about this. It is part of an exam question. Firstly I must draw a sketch of this pulse: v=0 when |t| > a v=V0( 1 + t/a ) when -a < t <= 0 v=V0( 1 - t/a ) when 0 < t < a v represents amplitude, V0 represents peak...

I have tried to find the Fourier series for a function u(x): u(x)=\sin((1+3\cos(t))t) The function is odd, hence the Fourier coefficients a_n equal zero and the b_ns are given as b_n=\frac{4}{T}\int_0^{T/2}u(t)\sin(n\omega t)\,\text{d}t where T=2\pi and \omega=2\pi/T=1...

In quantum mechanics, a free particle in an infinite potential well has the wave function (ie. overlap <x/phi>). Its eigenfunctions take the form: (2/a)^1/2 * sin(n*pi*x/a), n is ofcourse an integer. My question is that do all eigenfunctions form a basis? And if so how can you represent an...

I'm a little confused about the difference between the half range Fourier series and the full range Fourier series. What is the difference between the two in an odd function like f(x)=x and an even function like f(x)=x^2 ? Maybe an example to clear things up. Thank you.

Can someone give me some hints on this problem please? A string (length L) clamped at both ends and initially at rest, the boundary conditions for the wave function y(x,t) are: y(x,0)=y(0,t)=y(L,t)=dy/dt(x,0)=0 A note is obtained by striking the string with a hammer at some point a...

For a given function with a certain finite period, is there only one set of Fourier series coefficients a_n and b_n? The reason I ask is, I was doing a problem where it asked for the coefficients for a certain odd function, and then it asked for the coefficients for that same function shifted...

What is the best and simplest way to solve Fourier series problems?

Hi, could someone give me an explanation of Fourier Series' please or a link that would give someone who has no idea about them a working grasp of what they are.

Mathematically, these are three distinct, although related beasts. Laplace transform (function f(x) defined from 0 to inf) integral of f(x)e-xt, defined for t>=0. Fourier transform (function f(x) defined from -inf to inf) integral of f(x)e-itx defined for all real t. Complex Fourier series...

What is the best way to (quickly) check if a calculated Fourier series is the correct one?

Can anyone help me with this? Show that \sum_{n=1}^{\infty}(-1)^{n+1}\frac{\cos{nx}}{n^2} = \frac{\pi^2-3x^2}{12} \quad , \quad x \in [-\pi,\pi]. I have tried writing the right-side expression as a Fourier series, but it leads nowhere. What should I do?