1. ## Fourier Series Approximation

How do I find the fourier series approximation to the even extension of g(t), where g(t) is given as shown below using the fact that :

$sin(2pir/3)=(-1)^r+1sin(pir/3)$

2. Hi

I am struggling with this too. I can't make sense of the writing down the odd and even extensions. Have you got that far yet?

3. I was thinking that perhaps since the period is 6 that we need to change the intervals so they are equal to something like 0<t<1/2

Alternatively, it may have something to do with the range -3<t<3

In which case the even extensions will be those that have positive y and the odd will have negative y.

Its really difficult - hope someone on this forum can help out with this one.

4. I think you are right about the range being -3<t<3, as in part a of the question the function f(t)=f(t + 2pi)

2pi was the period of the function, so if you look at the formulas for even and odd extension they use the letter $L$ and $2L$ is defined as the period.

If $L$ is equal to 3 as the fundamental interval is $[-L,L]$ then 2x3 equals 6 giving us the desired period...... i think.

5. That makes sense - though when i substitute this into formula

$F(t)=A_0 + \sum_{n=0}^{\infty}A_rcos(2r\pi/T)$

and then try and work our $A_0 and A_n$

I still can see how we should use

$sin(2r\pi/3)=-1^r+1sin(r\pi/3)$

6. Am I right to assume that the use of sin refers to an odd extension?

Is g(t) an odd function?

I'm so confused!!

7. The sin function is odd because of the shape of the curve (i.e. not symmetrical about y axis). Whereas the cos function is even since if can be reflected in y axis. See bottom of page 9 of unit 21 for explanation

8. have you completed b)i) yet?

9. Nope - still waiting for some help on that. Have you got any ideas

10. Yeah

I think for part i) it is only asking you to write the extensions in function form.

So I have something like

G(odd) (t) =
-1 (-3<= t <-2)
0 (-2<= t < -1)
1 (-1<= t < 0)

then

G(even) (t) =
1 (-3<= t <-2)
0 (-2<= t < -1)
-1 (-1<= t < 0)

Then draw each graph...

Does that make some sense? (sorry i'm still not up to speed on Latex)

11. That looks good too me - thanks for starting me off on part b

12. No worries...

How have you got on for question 2?

13. Just had a thought - wouldnt the even extensions be positive i.e. range from 0<=t<3

14. I don't think so. If you look at the odd and even functions on pg 61 of handbook - it shows the range to be $-L

15. For Q21i have a look at page 27. The answer is virtually the same

For part ii - I think to have non-trivial solutions it needs to be such that u<0. i.e. negative. There is something in the text about this but I havent got it in from of me

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