1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Strike441 [17]
3 years ago
15

Solve using Fourier series.

Mathematics
1 answer:
Olin [163]3 years ago
5 0
With 2L=\pi, the Fourier series expansion of f(x) is

\displaystyle f(x)\sim\frac{a_0}2+\sum_{n\ge1}a_n\cos\dfrac{n\pi x}L+\sum_{n\ge1}b_n\sin\dfrac{n\pi x}L
\displaystyle f(x)\sim\frac{a_0}2+\sum_{n\ge1}a_n\cos2nx+\sum_{n\ge1}b_n\sin2nx

where the coefficients are obtained by computing

\displaystyle a_0=\frac1L\int_0^{2L}f(x)\,\mathrm dx
\displaystyle a_0=\frac2\pi\int_0^\pi f(x)\,\mathrm dx

\displaystyle a_n=\frac1L\int_0^{2L}f(x)\cos\dfrac{n\pi x}L\,\mathrm dx
\displaystyle a_n=\frac2\pi\int_0^\pi f(x)\cos2nx\,\mathrm dx

\displaystyle b_n=\frac1L\int_0^{2L}f(x)\sin\dfrac{n\pi x}L\,\mathrm dx
\displaystyle b_n=\frac2\pi\int_0^\pi f(x)\sin2nx\,\mathrm dx

You should end up with

a_0=0
a_n=0
(both due to the fact that f(x) is odd)
b_n=\dfrac1{3n}\left(2-\cos\dfrac{2n\pi}3-\cos\dfrac{4n\pi}3\right)

Now the problem is that this expansion does not match the given one. As a matter of fact, since f(x) is odd, there is no cosine series. So I'm starting to think this question is missing some initial details.

One possibility is that you're actually supposed to use the even extension of f(x), which is to say we're actually considering the function

\varphi(x)=\begin{cases}\frac\pi3&\text{for }|x|\le\frac\pi3\\0&\text{for }\frac\pi3

and enforcing a period of 2L=2\pi. Now, you should find that

\varphi(x)\sim\dfrac2{\sqrt3}\left(\cos x-\dfrac{\cos5x}5+\dfrac{\cos7x}7-\dfrac{\cos11x}{11}+\cdots\right)

The value of the sum can then be verified by choosing x=0, which gives

\varphi(0)=\dfrac\pi3=\dfrac2{\sqrt3}\left(1-\dfrac15+\dfrac17-\dfrac1{11}+\cdots\right)
\implies\dfrac\pi{2\sqrt3}=1-\dfrac15+\dfrac17-\dfrac1{11}+\cdots

as required.
You might be interested in
Solve 6x(6x) + 11x - 35 = 0
Murljashka [212]
36x^2 + 11x - 35=0\\\\ \frac{-b \pm  \sqrt{ b^2 -4( a)(c)} }{2(a)}\\\\  \frac{-11\pm  \sqrt{11^2 -4(36)(-35)} }{2(36)}\\ \\   \frac{-11\pm  \sqrt{5161 } }{72}\\\\\\ x = 0.8450013919812116\\\\ \text{   } x  = -1.1505569475367672
4 0
3 years ago
Read 2 more answers
Hey You <br> YES YOU CAN ANSWER THIS PLEASE<br> solve for T: 4 j= -9T
OLEGan [10]
-9T means -9 x T
So if T is equivalent to 4,
you should do:
-9 x 4 = -36
J= -36
8 0
3 years ago
34.1 divided by what equals 22
allochka39001 [22]
I hope this helps you

3 0
3 years ago
Read 2 more answers
Jocelyn estimates that a piece of wood measures 5.5 cm. If it actually measures 5.62 cm, what is the percent error of Jocelyn's
morpeh [17]
The percent error is -2.1352% of Jocelyn's estimate. May I have Brainless
8 0
3 years ago
Read 2 more answers
Make C the subject of the formula:<br> F=9/5 C + 32​
Dafna1 [17]
1)-9/5C=-F+32
2)9/5C=F-32
3)9C=5(F-32)
4)C=5/9(F-32)
4 0
3 years ago
Other questions:
  • Calculer les expressions suivantes (après les avoir simplifiés) aidez moi svp urgent ❤️
    12·1 answer
  • Where did the 0.8 come from?
    14·2 answers
  • slader How many partitions of the set {1, 2, 3, . . . , 100} are there such that there are exactly three parts and elements 1, 2
    14·1 answer
  • 2. A rectangular prism has e flat surfaces . 3, A cone h as -flat surface. A cube has 요 flat surfaces.
    5·1 answer
  • forty percent of the first 20 whole numbers are prime. how many of the first 20 whole numbers are prime​
    5·1 answer
  • What is the slope of the line that contains the points -4,2 and 6,3
    13·1 answer
  • 3. Find the rate of change in the represented table.
    8·1 answer
  • I am a bit confused on a derivative is in Calculus. From what I understand, I derivative is the limit of a tangent line on a cur
    8·1 answer
  • Solve plssss skejjddhhshd
    8·1 answer
  • Riley makes a mistake in step 2 while doing her homework. what was her mistake? startfraction x over x squared minus 5 x 6 endfr
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!