Finding the Sum of Sin2(na)/n2 Using Fourier Series for f(x)

In summary, the student is trying to find the sum of sin(na)/n2 for various values of na, but is having trouble with the formulas.
  • #1
Ratpigeon
56
0

Homework Statement


Use the Fourier series of
f(x) = { 1 |x|<a
{ 0 a<|x|<[itex]\pi[/itex]
for 0<a<[itex]\pi[/itex]
extended as a 2-Pi periodic function for x [itex]\in[/itex]R
to find
[itex]\sum[/itex] Sin2(na)/n2
2. Homework Equations [/b

I got that the Fourier series of f(x) was
a/[itex]\pi[/itex]+[itex]\sum[/itex] (2/(m[itex]\pi[/itex]) sin(ma) sin(mx)


The Attempt at a Solution


I'm not sure what to do. I'm guessing that since its meant to use the Fourier series of f; it should be related to f(a), but f(a) is undefined, and the denominator is wrong...
Could anyone please point me in the right direction?
 
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  • #2
Your f(x) is an even function, so you should have cosines instead of sines:
$$f(x) = \frac{a}{\pi} + \sum_{m=1}^\infty \frac{2}{m\pi} \sin ma \cos mx.$$ Now if you look at the sum you're trying to evaluate, you need another power of m on the bottom and you want to turn the cosine into a sine. Any idea of how to get that?
 
  • #3
Ratpigeon said:
it should be related to f(a), but f(a) is undefined

The formulas you are likely using for the Fourier Series of a function [itex]f(x)[/itex] require the function to be piecewise smooth and periodic over the Reals, as well as being defined at each discontinuity as equal to the mean value of the one-sided limits, so you require

[tex]f(x=\pm a) = \frac{1}{2} \left( \lim_{x \to \pm a^+}f(x) + \lim_{x \to \pm a^-}f(x) \right) = \frac{1}{2}[/tex]
 
  • #4
I integrate it from zero to a to get the extra sine term?
 
  • #5
Sounds good. What do you get?
 
  • #6
a Pi/2. Thanks for the help :)
 
  • #7
Ratpigeon said:
a Pi/2. Thanks for the help :)
Ok, you're on the right track but not quite there yet. I think there's another term.
 
  • #8
I don't think there's another term - the 1/2 A0 term goes to zero when you integrate it from
-a to a.
 
  • #9
Ratpigeon said:
I don't think there's another term - the 1/2 A0 term goes to zero when you integrate it from
-a to a.
No actually it doesn't.

BTW. Previously you said you would integrate from zero to pi? Either way will work, but zero to pi is easier.
 
  • #10
Integrating from zero to a got that it was Pi/2 (a+a^2/Pi); is that right?
Thanks for pointing it out
 
  • #11
Ratpigeon said:
Pi/2 (a+a^2/Pi); is that right?
Almost. Check your "signs".
 
  • #12
Minus, sorry, thanks - I need sleep...
 
  • #13
Yep, minus.

And as a nice little "sanity" check, notice that the sum is now zero when [itex]a = \pi[/itex]. :smile:
 
  • #14
Thank you :)
 

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions with different frequencies and amplitudes.

2. How is a Fourier series used?

Fourier series are used in various fields such as physics, engineering, and signal processing to analyze and represent periodic functions and signals. They are also used in solving differential equations and data compression.

3. What are the main components of a Fourier series?

The main components of a Fourier series are the coefficients, which represent the amplitudes of the sinusoidal functions, and the frequencies, which represent the rate of oscillation of the functions.

4. What is the difference between a Fourier series and a Fourier transform?

A Fourier series is used to represent a periodic function while a Fourier transform is used to represent a non-periodic function. Fourier transforms also use complex numbers while Fourier series use only real numbers.

5. What are some real-world applications of Fourier series?

Some real-world applications of Fourier series include sound and image processing, heat transfer analysis, and vibration analysis in mechanical systems. They are also used in music and signal analysis, as well as in solving partial differential equations in physics and engineering.

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