Suppose the functions f(t) and g(t) are periodic with periods P and Q, respectively. If the ratio P/Q of their periods is a rational number, show that the sum f(t)+g(t) is a period function.
How to prove this?
Not really a homework question; I typed this sum into Wolframalpha and it gave a nice, compact expression, but I couldn't figure out where to begin finding it. Is there a way to find it using just calc II-level knowledge of infinite sums...
Hello,
Attached are two problems I can not solve, thanks for the help.
The Attempt at a Solution
For the first question, I understand that I need insert A1coswt+A2sinwt into the homogenous equation , but don't know what's then ..
But I'm pretty much lost on both of em :(
Homework Statement
Find the Fourier series of f on the given interval:
EDIT: For the result of an, it should be multiplied by \frac{2}{1+n^{2}}. That has been corrected.
Hello Folks,
I have a problem understanding a step of the complex Fourier series; it’s a step which involves simple addition and subtraction of exponentials (regrettably not simple for me).
I have attached a picture of the step I am having a problem with would really appreciate if someone...
A square wave has amplitude 3 and period 5. calculate its power?
Using Fourier series for this square wave and Parseval’s
theorem, calculate the power in a signal obtained by cutting out frequencies
above 1 Hz in the square wave?
i am able to obtain the Fourier series for the square wave...
Hello,
I know how to get the full Fourier series with complex coefficients and with real coefficients, and I know the relationship between An, Bn and Cn. However, I don't know why the relationship between them is what is it. Can someone either explain to me where the relationship comes from...
i have an exam in these kind of questions in a few days so i was pracitsing a a few problems but I can't do them!
Any help would be appreciated.
Calculate the Fourier series for f(x) when f(x) = 0, on -pi <= x <= 0, and f(x) = coshx, on 0 <= x <= pi.
and show that SUM (from n=1 to...
Homework Statement
f(t) is given as:
from 0 to 0.2s, f(t) = 5
from 0.2s to 0.6s, f(t) = 0
from 0.6s to 0.8s, f(t) = 5,
etcHomework Equations
for an odd function
a0 = 2/p * integral(from -p/2 to p/2) of f(t) dt
bn = 4/p * integral(from 0 to p/2) of f(t)*sin(2*pi*n*t/p) dtThe Attempt at a...
Homework Statement
Trying to find the Fourier series for the function
f(x) = 0 for -pi<x<0 and f(x) = sinx for 0<x<pi
The Attempt at a Solution
im having a little trouble working it out..
are any of the sets of coefficients = 0?
Im getting two non-zero integrals for the...
OK, so I was trying to solve the Heat Equation with Inhomogeneous boundary conditions for a rod through Fourier Series when I got stuck at the solution for the coefficient c_n, the part where I'm stuck is highlighted in red.
The following is just a step-by-step solution of how I got to c_n...
Homework Statement
Show that the orthogonality relation for the "cosine basis functions" used in the Fourier series is
1/L\intcos[(n*pi*x)/L)]cos[(m*pi*x)/L)]dx = {Sin([n-m]*pi)}/[(n-m)*pi] + {Sin([n+m]*pi)}/[(n+m)*pi]
By considering the different integer n and m, show that the right...
Say you have the coefficients a_k of a Fourier series representation of some function x(t). You can easily then give x(t) as
$$x(t) = \sum_{k = -\infty}^{\infty} a_k e^{i k \omega_0 t}$$
But this doesn't do much good in telling you what the actual function looks like. For example, if we have...
Homework Statement
Show that the Fourier series formula F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nwt)+b_{n}sin(nwt)) can be expressed as F(t)=\frac{1}{2}a_{0}+\sum^{\infty}_{n=1}c_{n}cos(nwt-\phi_{n}). Relate the coefficients c_{n} to a_{n} and b_{n}.
Homework Equations
We...
Homework Statement
Obtain the Fourier series representing the function F(t)=0 if -2\pi/w<t<0 or F(t)=sin(wt) if 0<t<2\pi/w.
Homework Equations
We have, of course, the standard equations for the coefficients of a Fourier expansion...
http://img69.imageshack.us/img69/6758/123123123nx.jpg
http://img819.imageshack.us/img819/5390/fsdfsdfsdf.jpg
To calculate the Fourier series, I used the formulae above, and I got:
[PLAIN][PLAIN]http://img831.imageshack.us/img831/2008/xcvxcvxcv.jpg
and i substituted the...
Homework Statement
Find the Fourier series coefficients X_k of the periodic signal:
x(t) = 5cos(6w_0t+pi/2)
(digital or discrete spectrum)
Homework Equations
The Attempt at a Solution
I am really confused with all of this and don't...
Homework Statement
f(t)=sin(|6t|), −pi<t<pi
with f(t) = f(t+2pi)
Homework Equations
Show that the Fourier series for f(t) can be written as (24/pi) time the sum, from n=0 to infinity, of ( 1/( 36 - (2k+1)^2 ) )cos(2k+1)t.
The Attempt at a Solution
I have an answer of a0 being 0...
Homework Statement
f(x)=sin^2(x)
Homework Equations
The Attempt at a Solution
solving for a(0)= i did (1/2Pi)*int(sin^2(x),x,-Pi..Pi)=1/2
b(n)=0 because sin^2(x) is an even function...
Homework Statement
Find the complex Fourier series of the periodic function
f(t)=2sin(πt) 0 < t < 1 and f(t+1) = f(t) for all t. (π is pi)Homework Equations
http://upload.wikimedia.org/math/9/d/7/9d7f73fbcba87cbff485e66646aa541d.png...
Homework Statement
Show that
\sum_{r=0}^\infty\frac{1}{(2r+1)^2}=\frac{\pi^2}{8}Homework Equations
The equation of the function is
F(t)&=&\dfrac{\pi}{4}-\dfrac{2}{\pi}\left(\cos t+\dfrac{\cos3t}{3^{2}}+\dfrac{\cos5t}{5^{2}}+\cdots\right)-\left(\sin...
Hi,
When I solve the diffusion equation for a spherically symmetric geometry in spherical coordinates I obtain the following general solution (after application of the boundary conditions).
T(r,t) = \sum_{n=1}^{\infty}\, \frac{A_n}{r}\sin(\lambda_nr)\exp(-\alpha\lambda_n^2t)
So to...
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...
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...