Fourier Sine Series of f(x)=x: Solution & Explanation

[DreamSanchez]

Homework Statement

Find the F.S.S. of f(x)=x
0≤x≤1
of period 2

Homework Equations

(uploading photo with equations shown)

The Attempt at a Solution

(uploaded photo)

I have completely worked this problem out in the provided photo but I am unsure as to whether or not I went about it in the correct manner. If anyone could take a look at it and let me know how I did, I would truly appreciate it!

 

[Ray Vickson]

Homework Statement

Find the F.S.S. of f(x)=x
0≤x≤1
of period 2

Homework Equations

(uploading photo with equations shown)

The Attempt at a Solution

(uploaded photo)

I have completely worked this problem out in the provided photo but I am unsure as to whether or not I went about it in the correct manner. If anyone could take a look at it and let me know how I did, I would truly appreciate it!

There is something wrong with your problem statement. If f(x) really does have period 2, you either have to specify it also on the interval (-1,0) or else on the interval (1,2). If all you have is f(x) on (0,1), its Fourier series will have period 1, not 2; and in that case it will not be an odd function, and so will not be expandable only in sines---you will also need cosines.

 

[DreamSanchez]

The way I wrote the problem is how it was given to me in class. The professor is known for making mistakes though when writing the homework on the board. Perhaps neglect the "period of 2" portion. Would my work make sense in that case?

 

[Ray Vickson]

The way I wrote the problem is how it was given to me in class. The professor is known for making mistakes though when writing the homework on the board. Perhaps neglect the "period of 2" portion. Would my work make sense in that case?

Yes, but it is giving the series for the function g(x) = x, -1 < x < 1, with period 2. (To convince yourself of this, try plotting sums of, say, 20 or 30 terms on the interval -1 < x < 1.)