What is Fourier series: Definition and 750 Discussions
In mathematics, a Fourier series () is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.
Homework Statement
Let f(x)=x on [-\pi,\pi) and peridically extended. Compute the Fourier series and hence show:
\sum_{n \geq 1,nodd} \frac{1}{n^2} = \frac{\pi^2}{8} and \sum_{n \geq 1} \frac{1}{n^2} = \frac{\pi^2}{6} Homework Equations
Parseval's equality
The Attempt at a Solution
I...
Homework Statement
Let f be an integrable, periodic function whose Fourier coefficients satisfy \sum_{-\infty}^{\infty} n^6 |\hat{f}(n)|^2 < \infty . Prove that f is continuous.
Homework Equations
Looking at my notes, the only relevant things i have for this question (i think) are...
Say you have two functions, F(x,y), and G(x,y), and you want to expand them in finite Fourier series. Let their coefficients be designated as F_ij and G_ij. When you multiply the two functions, you get X=FG, and this should also have its own Fourier series, call its components X_mn. What is the...
Homework Statement
What is the fundamental period of f(x) = eax+ibx where a, b are real numbers greater than zero? Find the Fourier series for f(x).
Homework Equations
The Attempt at a Solution
I am able to find the Fourier series for trig based functions but am not sure how to start...
\sum _{n=1}^{\infty } \frac{1}{n^4}
I can do 1/n^2 easily by using x^2 as a function but for this I try
x^4=\frac{1}{2\pi }\int_{-\pi }^{\pi } x^4 \, dx+\sum _{n=1}^{\infty } \left(\frac{2}{\pi }\int _0^{\pi }x^4\text{Cos}[n x]dx\right)\text{Cos}[n x]
I arrive at:
\pi...
Homework Statement
For positive integers m and n, calculate the two integrals:
\frac{1}{L}\int^{L}_{-L}sin(\frac{n \pi x}{L})sin(\frac{m \pi x}{L})dx and \frac{1}{L}\int^{L}_{-L}sin(\frac{n \pi x}{L})cos(\frac{m \pi x}{L})dxHomework Equations
\int u v' dx = [u v] - \int u' v dxThe Attempt at a...
Homework Statement
expand the function in Fourier series and calculate the integral
f(x)= (sinx)^2(cosx)^3, 2\pi is the period
calculate the integral
\int_{0}^{2\pi}f(x)dx
please help...have absolutely no idea how to calculate it...
Homework Equations
The Attempt at a Solution
[b]1. Homework Statement [/b
expand the function in Fourier seriesf(x)= x^2+xcosx
I divided the function in separate part and try to expand it in Fourier series separately
i have started with
x^2
a_{0}= \frac{1}{2\pi}\int_{-\infty}^{\infty}x^2dx=\frac{\pi^2}{3}
a_{n}=...
Homework Statement
Assume that a pressure wave produces a change in pressure at a point in space \Delta P(t) which is proportional to a sawtooth function of frequency f = 1/2 Hz.
(i) If the amplitude of the pressure wave is \Delta P_{0}, write down an expression for \Delta P(t).
(ii) Two...
For finding series expansion solution of problems like
f(x) = h(x) for 0<x<1
f(x) = 0 for 1<x<2
0<x<2
Where the Fourier series expansion only integrate from x=0 to x=1 only and totally ignor the portion of x=1 to x=2.
This is also true for Fourier bessel series expansion...
Homework Statement
If f(x) is a piecewise-continuous function in [-L,L], show that its indefinite integral F(x) = \int_{-L}^x f(s) ds has a full Fourier series that converges pointwise. Homework Equations
Full Fourier series: f(x)=\frac{1}{2}A_0 + \sum_{n=1}^\infty A_n \cos (\frac{n \pi }{L}x)...
Homework Statement
Consider the following equation for the Fourier series:
f(t) = SUM k=1->infinity (Ck * sin(k*pi*t)
What is the meaning of the Ck terms?
What is the importance of the K=1 term?
Homework Equations
The Attempt at a Solution
Ck = Fourier Coefficents. They are...
Hi all. I am currently trying to find the first few terms of a sine expansion of
q(t)= (A+B*cn(t,m))/(C+D*cn(t,m))
where m is the modulus and cn is the jacobic elliptic cnoidal function and A,B,C,D real and C>D implying no poles. I realize that I should start with a simpler problem. Do...
I'm having problem finding the representation for the Fourier series with
function f of period P = 2*pi such that f (x) = cosαx, −pi ≤ x ≤ pi , and α ≠ 0,±1,±2,±3,K is a
constant.
Any help is appreciated...
Homework Statement
I'm trying to find the sum of the infinite series:
1/n^4
using the Fourier series of x^2 on [-pi, pi] which I have as
PI^2/3 + 4*sum(n=1..infinity) (-1)^n/n^2*cos(nx)
Homework Equations
The Attempt at a Solution
So far all my attempts have been focused...
Homework Statement
Determine the Fourier series of f(x) = pi + x
Homework Equations
The Attempt at a Solution
I see you have to calculate the two "series" separately and then add them. I know that the Fourier series of pi is just pi - but i was wondering why (i know that sounds...
Homework Statement
Let f be the 2 periodic function defined on [−pi,pi ) by
f(x) = sin(x/3)
Find the Fourier coefficients of f.
Homework Equations
.5[cos(A-B)-cos(A+B)]=sinAsinB
The Attempt at a Solution
After much work using trigonometry and integration by parts I have...
Homework Statement
Hi
First of all this a textbook question from Stroud Advanced Engineering Mathematics, solution is given, but no steps are shown.
A question I just can't seem to solve at the moment, as below.
A function f(x) is defined by f(x) = \pi - x:0 < x < \pi
f(x + 2\pi) = f(x)...
Homework Statement
Show that the Fourier series f(x) = \sumansin(nx) + bncos(nx) can be written as \sumkn(cos(nx+\vartheta)) and define kn and \vartheta
where the summation is from 0 to \infty
Homework Equations
sin \vartheta = cos (90 - \vartheta) ??
The Attempt at a Solution
Well what I...
Homework Statement
Find the complex Fourier series for:
f(t)=t(1-t), 0<t<1
Homework Equations
f(t)=\sum_{n=-\infty}^{\infty}c_n{e^{i\omega_n{t}}}
c_n=\frac{1}{\tau}\int_{t_0}^{t_0+\tau}e^{-i\omega_n{t}}f(t)dt
\omega_n=2\pi{n}\quad\tau=1
The Attempt at a Solution
I solved...
I know that if
\sum_{n=-\infty}^{\infty} |n| |\hat{f}(n)| < \infty
then
\sum_{n=-\infty}^{\infty} \hat{f}(n) e^{2\pi i nx}
is continuously differentiable as a function of x.
Now I'm interested to know what kind of conditions exist for Fourier coefficients such that they...
Homework Statement
Determine a general Fourier series representation for f(x) = x^3 -1<x<1Homework Equations
The Attempt at a Solution
May seem like a stupid Q, but would i have to calculate a0, an, bn or since i know that x^3 is an odd function, could jump straight into calculating the...
Homework Statement
Suppose, in turn, that the periodic function is symmetric or antisymmetric about the point x=a. Show that the Fourier series contains, respectively, only cos(k_{n}(x-a)) (including the a_0) or sin(k_{n}(x-a)) terms.
Homework Equations
The Fourier expansion for the...
Homework Statement
Find the Fourier series for ex for x in (-1,1).
Find the Parseval identity.
Homework Equations
The Attempt at a Solution
c_{n}=\frac{1}{2}\int_{-1}^{1}e^{x}e^{-i n x}dx
Where cn are the coefficients of the Fourier series.
I tried plotting
\sum_{k=-\infty}^{\infty}c_{n}e^{i...
Homework Statement
Find the Fourier series for y(x)=\begin{cases}
A\sin(\frac{2\pi x}{L}) & 0\leq x\leq\frac{L}{2}\\
0 & \frac{L}{2}\leq x\leq L\end{cases}Homework Equations
B_{n}=\frac{2}{L}\int_{0}^{L}y(x)\sin(\frac{n\pi x}{L})dxThe Attempt at a Solution...
Homework Statement
Since I don't know how to insert equations into a message here, I've scanned both the problem and my attempt at a solution. Where I run into problems is how to find an. I'm not completely sure how to treat that integral and was hoping somebody could nudge me in the...
Homework Statement
Find a sine Fourier series for the function f(x)=1 define on 0<x<1. use this series to show that \Sigma\stackrel{(-1)^k}{2k+1} =\stackrel{\pi}{4} betwen k=0 and infinity
Homework Equations
The Attempt at a Solution
i found the Fourier series to be\Sigma...
Hi all!
I wanted to have a little clarification about this line in relation to Fourier Series: (It's about a periodic and symmetrical signal)
"x(t) is a periodic signal. As cos nwt is an even function and sin nwt is an odd function. So, if x(t) is an even function of t, then x(t) cos nwt...
Homework Statement
f(x) = sin(x) for 0\leqx<\pi. Extend f(x) as an even function . Obtain a cosine Fourier series for f.
Homework Equations
a_{0}/2 + \sum a_{n}cos(nx)
The Attempt at a Solution
So as far as I know, to extend sin(x) as an even function you have to make f(x)=-sin(x)...
Homework Statement
This isn't really a homework question, but something I've been wanting to know out of curiosity in David Griffiths' Introduction to Electrodynamics.
On pages 131 and 132, there is a Fourier series,
V(x,y) = \frac{4V_0}{\pi}\sum_{n=1,3,5...}\frac{1}{n}e^{\frac{-n \pi...
Homework Statement
An oscillator is driven by a triangular periodic force (if that makes sense), which has period \tau = 2.
(a) Find the long-term motion x(t), assuming the following parameters: natural period \tau[naught] = 2 (that is, \omega[naught] = π), damping parameter ß = 0.1, and...
Homework Statement
Find the Fourier series for: sin(a*pi*t). Consider what happens when a -> 1/L
Homework Equations
The Attempt at a Solution
I keep getting zeros for a_o, a_n, and b_n.
I though that atleast b_n should give me something, can anyone also confirm this?
Homework Statement
i can find the Fourier coefficent of the signal but i could't understand how to find the frequencies which are present in the signal.can anyone help me?
Complex Fourier Series Coeffcients; what are they? what do they represent?
Homework Statement
I'm not sure if this is the right place for this but it seems appropriate. I just started an intro signals and systems course at my university at the beginning of this semester. We started Complex...
Homework Statement
Homework Equations
The Attempt at a Solution
Ive numbered the solution steps, the ones that are giving me trouble are from 1 to 2 and from 3 to 4
From 1 to 2 i don't understand how there can be an exp(x) term taken out of the bracket and still be in the...
Homework Statement
Express the function plotted in the figure below as a Fourier series.
Homework Equations
The Attempt at a Solution
I have the fully worked out solution infront of me and I am ok with working out the a0, an and bn parts but what i want to know is why is the function...
P_r is defined as:
P_r(x)=\frac{1-r^2}{1-2r\cos x +r^2}
and
P_r(x)=\frac{1-r^2}{1-2r\cos x +r^2}=\sum_{n=-\infty}^{\infty}r{|n|}e^{inx}
and
f(x)=\sum_{-\infty}^{\infty}c_ne^{inx}
which is continues
i need to prove that...
Homework Statement
Claim: If f(x) is a REAL-valued function on x E [-L,L], then the full Fourier series is exactly equivalent to the complex Fourier series.
This is a claim stated in my textbook, but without any proof. I also searched some other textbooks, but still I have no luck of...
I was finally able to figure out how to find the sine series for cos(x), but only for [0,2pi]. A question i have though is what is the interval of validity? is it only [0,pi]?
Ie if I actually had to sketch the graph of the sum of the series, on all of R, would I have cosine or just a periodic...
Homework Statement
f(x) = 1 0<t<1
= -1 1<t<2
How can I simplify this given that function(on the attachment).
What I mean is that how can I write the function in any other way?
In addition, How can I know if the function can be written in other form?
How can I...
I need to find the Fourier series for the function f;
0 if -\pi \prec x \leq -\frac{\pi}{2}
1+x if -\frac{\pi}{2} \prec x \prec \frac{\pi}{2}
0 if \frac{\pi}{2} \leq x \leq \pi
I've never done a Fourier series computation before so I don't really know if any of what I'm...
Homework Statement
The function g(x) is defined as follows:
g(x) = \left\{ \begin{array}{rcl}
{-\pi e^x} & \mbox{for}
& -\pi < x < 0 \\
{\pi e^{ -x}} & \mbox{for} & 0 < x < \pi
\end{array}\right.
And the Fourier series for g(x) is as follows:
\sum_{n=0}^\infty...
Homework Statement
Using the Fourier trigonometrical series for f(x) = {x^2},{\rm{ }}0 \le x < 2\pi , prove that \sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}}}} = \frac{{{\pi ^2}}}{6}
The Attempt at a Solution
This is more of a "what am I doing wrong question". First, because I'm not...
Homework Statement
I am given this function
f(x) = \left\{\begin{array}{cccc} x^2 \ \mathrm{where} \ \frac{-\pi}{2} < x < \frac{\pi}{2} \\ \ \frac{1}{4}\pi^2 \ \mathrm{where} \ \frac{\pi}{2} < x < \frac{3\pi}{2} \end{array}
Doesn't this mean that the function is periodic Fourier...
Homework Statement
Suppose that f is an integrable function (and suppose it's real valued) on the circle with c_n=0 for all n, where c_n stands for the coefficient of Fourier series. Then f(p)=0 whenever f is continuous at the point p.
Homework Equations
The Attempt at a Solution...
Is a Fourier series essentially the analogue to a Taylor series except expressing a function as trigs functions rather than as polynomials? Like the Taylor series, is it ok only for analytic functions, i.e. the remainder term goes to zero as n->infinity?