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Need help solving what seems like a strange integral

  • Thread starter saybrook1
  • Start date
  • #1
101
4

Homework Statement


I need to find the Fourier series for [itex]f(x)=|x|^3[/itex].

Homework Equations


[itex]f(x)=|x|^3[/itex]
[itex]A_n=\frac{2}{L}\int_{0}^{L}|x|^3cos{\frac{n\pi x}{L}}dx[/itex]
[itex]\int_{0}^{L}x|x|sin{\frac{n\pi x}{L}}dx[/itex]

The Attempt at a Solution


Since it is an even function I know that it will be a cosine series and so I set out to find the A coefficient like so:
[itex]A_n=\frac{2}{L}\int_{0}^{L}|x|^3cos{\frac{n\pi x}{L}}dx[/itex]

Through integration by parts I end up with a sine term that goes to zero and some coefficients out in front of an integral that looks like:
[itex]\int_{0}^{L}x|x|sin{\frac{n\pi x}{L}}dx[/itex]

I would really appreciate any help with this particular integral or if someone could point me in the direction of a solution where a Fourier series is calculated for a function with an absolute value to the power>2. Thanks in advance.
 

Answers and Replies

  • #2
92
8
When dealing with |x| it often helps to use the piece wise expansion of it:
=x, x>0
= -x, x<0.
If you were on the interval from [-L,L] you would have 2 integrals one from [-L,0] and one from [0,L]. However, since you are only interested in the function from [0,L] then $|x|^3$= $x^3$. To solve the integral you have integration by parts like normal.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
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Did you mean [itex]\int_0^L |x|^3 cos(\frac{n\pi x)}{L}dx[/itex] of [itex]\int_{L/2}^{Ll/2} |x|^3 cos(\frac{n\pi x)}{L}dx[/itex]? In the first, which is what you wrote, x is always non-negative so |x|= x. For the second, do the integrals from -L/2 to 0 and from 0 to L/2 separately.
 
  • #4
101
4
Did you mean [itex]\int_0^L |x|^3 cos(\frac{n\pi x)}{L}dx[/itex] of [itex]\int_{L/2}^{Ll/2} |x|^3 cos(\frac{n\pi x)}{L}dx[/itex]? In the first, which is what you wrote, x is always non-negative so |x|= x. For the second, do the integrals from -L/2 to 0 and from 0 to L/2 separately.
Yeah, that was my bad; the question wants me to find the fourier series for f(x)=|x|^3 on -L<x<L. So I think I can choose [itex]A_{n}=\frac{2}{L}\int_{0}^{L}x^{3}cos{\frac{n\pi x}{L}}dx[/itex] and then [itex]A_{0}=\frac{1}{L}\int_{0}^{L}x^{3}dx[/itex]?
 
  • #5
101
4
When dealing with |x| it often helps to use the piece wise expansion of it:
=x, x>0
= -x, x<0.
If you were on the interval from [-L,L] you would have 2 integrals one from [-L,0] and one from [0,L]. However, since you are only interested in the function from [0,L] then $|x|^3$= $x^3$. To solve the integral you have integration by parts like normal.
Thanks for your reply. I was unclear in the question statement; It asks me to find the fourier series for the function f(x)=|x|^3 on -L<x<L. I guess I can choose [itex]A_{n}=\frac{2}{L}\int_{0}^{L}x^{3}cos{\frac{n\pi x}{L}}dx[/itex] and then [itex]A_{0}=\frac{1}{L}\int_{0}^{L}x^{3}dx[/itex]?
 

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