Fourier Series of Periodic Function

[he1senberg]

Homework Statement

http://imageshack.us/photo/my-images/824/50177563.png/
I need to find the Fourier coefficients and estimate the series for certain values of n. (4, 20 and 100)

Homework Equations

http://imageshack.us/photo/my-images/839/32591148.png/

The Attempt at a Solution

I was unsure about what to do and found the equation above. So I used it and b coefficient was 0. I set the period 2L as 3, and set -L -3/2 and L as 3/2. So the total value I found was
a0 + Σ(3/(n*pi) - 27/(2*(n*n*n)*(pi*pi*pi)))*sin(2*n*pi/3)+ 9*cos(2*n*pi/3)/(2*(n*n)*(pi*pi))*(cos(2n*pi*x/3)

I managed to get some result but I am not sure if my results are correct, as the graph of the function is discontinuous at the ends of parabola. So, do you think my answer is correct? If not, how can I fix it?

 

[vela]

I have Mathematica crank out the integral, and it produced almost the same result. The only difference is that there shouldn't be 2 in the denominator of the last term.

What did you get for a₀?

 

[he1senberg]

Thanks for reply.

I think a0 was 2/9 or 4/9, I don't have the paper with me right now. I think that last 2 should be 4. Other than that, do you think my procedure to deal with problem is correct? Because I couldn't really make sure that I solved the problem correctly.

 

[vela]

Yes, your method sounds fine. It's just getting the math right now. a₀ is the average value of the function over one period, so it should be 2/9.

 

[paranormal]

As a scientist, it is important to ensure the accuracy and validity of our results. In this case, it is important to double check your calculations and make sure that you have correctly applied the Fourier series equations. It is also important to consider the properties and behavior of the function you are analyzing, such as continuity and periodicity. In this case, the discontinuity at the ends of the parabola may affect the accuracy of your results.

To ensure the accuracy of your results, you can try using different values of n and see if the results converge to a similar pattern. You can also compare your results with known solutions or use software or online tools to verify your calculations.

Additionally, it is important to carefully interpret and analyze the results. Are the coefficients and series you obtained physically meaningful in the context of the problem? Are there any limitations or assumptions in the Fourier series that may affect the accuracy of your results?

Overall, it is important to approach scientific problems with a critical and analytical mindset, and to seek help or clarification if needed.