Fourier Series (Half-range expansion)

[galaxy_twirl]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't really understand why my solution is wrong as I think I have substituted everything in correctly..

Is it okay if anyone can help me take a look at my solution? Thank you. :)

My solution: (Only b_n)

My teacher's solution (not the whole solution):

The thing I don't understand is why is my teacher's sin portion "sin nx" and not "sin 2nx" like mine?

Thanks. :)

 

[Orodruin]

Your L should be pi. As it is now you are missing half of the interval and not using all of the basis functions.

 

[galaxy_twirl]

Your L should be pi. As it is now you are missing half of the interval and not using all of the basis functions.

I see. Is this due to the half-range expansion, or is it because of the range from (0, π)?

Thank you. :)

 

[Orodruin]

Well, both. The half-range is 0 to pi so the length of the full range must be 2pi. In the end, you could just as well have considered it to be just an expansion on 0 to pi without making an extension, the basis functions are the same.

 

[galaxy_twirl]

Your L should be pi. As it is now you are missing half of the interval and not using all of the basis functions.

I see. Just wondering, if this is just a normal Fourier series expansion of cos(x) from (0,π), will the L still be the same?