Why do Fourier series require specific limits for integration?

[ranju]

Homework Statement

The major problem I am facing while solving for Fourier series is about the limits to be taken while integrating..!
In the general equation of Fourier series the upper & lower limits are t1 & t1+T respectively..while solving for even functions we take t1 =-T/2..! Why is it so..?? does this have something to do with the symmetry across y-axis??
In the 2 attached waveforms , in the first one , limits were like -pi to pi..while in 2nd limits are 0 to T/2.>! I am not getting this..![/B]

Homework Equations

The Attempt at a Solution

The limits are t1 to t1+T whre T is the time period..but I am not getting how to decide value of t1..!
[/B]

 

[DrClaude]

In the general equation of Fourier series the upper & lower limits are t1 & t1+T respectively..while solving for even functions we take t1 =-T/2..! Why is it so..?? does this have something to do with the symmetry across y-axis??

You have a function that is periodic with a period $T$. From the point of view of the theory, it makes no difference what value of $t_1$ you use, it will not affect the result (try it for yourself). Therefore, it is best to take the most convenient limits for the integration. If the function is even, using $t_1 = -T/2$ could save you from the full integral, by taking twice the integral from $t_1$ to $t_1 + T/2$.

In the 2 attached waveforms , in the first one , limits were like -pi to pi..while in 2nd limits are 0 to T/2.>! I am not getting this..!

The second case is somewhat similar to what I mentionned. Since the function is 0 for half the period, starting from 0 allows you to end at $T/2$, since the integral from $T/2$ to $T$ results in 0.

 

[ranju]

Ohkk.. I have got the difference..! thanks for the help...