Understanding Odd, Periodic Functions: Integrals and Periodic Shifts

[Niles]

Homework Statement

Hi all.

Can you confirm these statements:

1) If I integrate an odd, periodic function of period 2L over one period, then the integral equals zero.

2) If I have a function f(x) with period 2L, then f(x+alfa), where alfa is an arbitrary number, will not change it's period.

Thanks in advance.


Niles.

 

[tiny-tim]

Can you confirm these statements:

1) If I integrate an odd, periodic function of period 2L over one period, then the integral equals zero.

2) If I have a function f(x) with period 2L, then f(x+alfa), where alfa is an arbitrary number, will not change it's period.

Hi Niles!

(have an alpha: α )

Hint: prove 2) first (use the obvious substitution) … then use that result to prove 1).

 

[Niles]

Thanks for the α.

Hmm, well, does #2 really need any proof? I mean, isn't it kinda obvious? Adding a constant α will just translate the function on x-axis (either to the left or right), so the period will remain unchanged.

And for #1: I can't see the link between this and the previous question.Niles.

 

[tiny-tim]

oops!

Hi Niles!

Hmm, well, does #2 really need any proof? I mean, isn't it kinda obvious? Adding a constant α will just translate the function on x-axis (either to the left or right), so the period will remain unchanged.

And for #1: I can't see the link between this and the previous question.

oops!

I misread 2) as ending "will not change its integral"

Prove that, and then prove 1).

Sorry!

 

[Niles]

I have proven that the definite integral of a 2L-periodic function is the same over any interval of length 2L.

But I still can't see what the link is between this proof/theorem and my question #1.

 

[Niles]

Ok, I got it now.. you were right.

Thanks!