Confirm Fourier co-efficient is zero despite not being odd/even function

Hi, my function is defined as

f = 1, -1<t<0
f = cos(pi*t),0<t<1
f(t+2) = f(t)

To what value does this series converge when t=1

So I need to find the Fourier series expansion and set it equal to one. Sooooo

I found the DC Value to be 1/2. Then I found a_{n} = 0 as well. Is this correct. Can a = 0 despite the function not being odd?

Thanks
Thomas

LCKurtz
Homework Helper
Gold Member
Hi, my function is defined as

f = 1, -1<t<0
f = cos(pi*t),0<t<1
f(t+2) = f(t)

To what value does this series converge when t=1

So I need to find the Fourier series expansion and set it equal to one. Sooooo

You should be able to answer that question without calculating the FS at all.

I found the DC Value to be 1/2. Then I found a_{n} = 0 as well. Is this correct. Can a = 0 despite the function not being odd?

Thanks
Thomas

Yes to both. 1 + a sum of sine terms is not an odd function and neither is your original function.

Okay at t=1, the series is attempting to converge to -1 as cos(pi*1) = -1

Are you saying that DC value of a 1/2 is correct and does infact a_{n} = 0?

Is that it? Thanks

LCKurtz
Homework Helper
Gold Member
Okay at t=1, the series is attempting to converge to -1 as cos(pi*1) = -1

No. The value of cos(pi*t) at t = -1 has nothing to do with this problem.

Are you saying that DC value of a 1/2 is correct and does infact a_{n} = 0?

Is that it? Thanks

Yes, a0 = 1/2 and the other an = 0 as you have calculated. But you are missing the point about what the series converges to when t = 1. What you need to do is draw a couple periods of your given function. Graph it from t = -1 to t = 3. Look at that graph at t = 1 and think about your theorem about convergence of FS. See if you can figure out whether a0 = 1/2 makes any sense.

uart
Okay at t=1, the series is attempting to converge to -1 as cos(pi*1) = -1

Are you saying that DC value of a 1/2 is correct and does infact a_{n} = 0?

Is that it? Thanks

1. The DC value of $a_0=1/2$ is correct.

2. The assertion that all other $a_n=0$ is NOT correct.

LCKurtz
Homework Helper
Gold Member
1. The DC value of $a_0=1/2$ is correct.

2. The assertion that all other $a_n=0$ is NOT correct.

You are correct.

Last edited:
vela
Staff Emeritus
Homework Helper
$$f(t) - a_0 = \sum_{n=1}^\infty b_n \sin(n\pi t)$$