Finding Fourier coefficients and Fourier Series

[Smazmbazm]

Homework Statement

Find the Fourier coefficients for the function

*Should be a piecewise function, not sure how to write one in [itex /itex] tags*

f(x) =
|x|, |x| < 1,
1, 1≤|x|< 2;

f(x+4) = f(x)

and

Find the Fourier series for

f(x) = cos1/2\pi x, -1≤x<1; f(x+2)=f(x)

Would be great if someone could explain how to solve these. It feels like our lecturer rushed over Fourier series / coefficients, without giving any similar examples. Thanks for any help

 

[HallsofIvy]

The Fourier coefficients for a function, f, defined on interval [a, b]. are the coefficients A_i and B_i[/tex] such that <br /> f(x)= A_0+ A_1 cos((2\pi/(b-a))x)+ B_1 sin((2\pi/(b-a))x)+ A_2 cos((4\pi/(b-a))x)+ B_2sin((4\pi/(b-a))x)+ \cdot\cdot\cdot= \sum_{i= 0}^\inft A_i cos((2i\pi/(b-a))x)+ B_i sin((2i\pi/(b-a))x)<br /> <br /> The &quot;theoretical&quot; point is that the sines and cosines form an &quot;orthonormal basis&quot; for such functions and so we can find the coefficients by taking the &quot;inner product&quot; of f with each of those functions:<br /> <br /> A_i= \frac{2}{b- a}\int_a^b f(x)cos((2\pi/(b-a))x) dx<br /> B_i= \frac{2}{b- a}\int_a^b f(x)sin((2\pi/(b-a))x) dx<br /> <br /> Surely those formulas are in your textbook?

 

[Smazmbazm]

These are the equations we are given in our notes

f(x) = \frac{a_{0}}{2} + \sum^{∞}_{m=1} [a_{m}cos\frac{mx\pi}{L} + b_{m}sin\frac{mx\pi}{L}]

where a_{m} and b_{m} are the Fourier constants.

To determine the Fourier coefficients, we are given

a_{m} = &lt;cos\frac{mx\pi}{L},f(x)&gt;_{L} = \frac{1}{L} ∫^{L}_{-L}cos\frac{mx\pi}{L}f(x)dx
b_{m} = &lt;sin\frac{mx\pi}{L},f(x)&gt;_{L} = \frac{1}{L} ∫^{L}_{-L}sin\frac{mx\pi}{L}f(x)dx

L is calculated from T = 2L where T is the period which is found from f(x+T) = f(x) So L must be 2.

Since the function involves a piecewise function, am I supposed to take the integral of the function over the intervals [0,1] and [1,2]?

 

[micromass]

Since the function involves a piecewise function, am I supposed to take the integral of the function over the intervals [0,1] and [1,2]?

Yes. Except that your domain also has negative numbers. So you should also include [-1,0] and [-2,-1] somewhere.

 

[Smazmbazm]

Ok so here is my attempt for a_{0}

a_{0} = \frac{1}{2}∫^{-1}_{-2}1dx + \frac{1}{2}∫^{0}_{-1}xdx + \frac{1}{2}∫^{1}_{0}xdx + \frac{1}{2}∫^{2}_{1}1dx

Evaluated, I get

a_{0} = 1

Correct? Should be able to do the rest fine now if that's correct

 

[Smazmbazm]

Can someone please confirm that this is incorrect / correct so that I know that I'm on the right track or not. Thanks