1. Periodic Functions

2. Originally Posted by chrozer
I think that both (c) is equal, but I do not know why. And I'm not to sure if (a) or (b) is equal or not? Any ideas or explanations?
(a) $f(t-2c) = f(t)$
Let $t'=t-2c$
so $f(t)=f(t'+2c)=f((t'+c)+c)=f(t'+c)=f(t')=f(t-2c)$

(b) $f(t+ \frac {1}{2}c) = f(\frac {1}{2}t)$

(c) $f(\frac {1}{2}(t+c)) = f(\frac {1}{2}t)$
b) is false, it makes nonsense...

c) is false, because it would mean it has a period $\frac c2$. But c is the smallest number such that $f(t+c)=f(t)$

3. Let $t'=t-2c$
so $f(t)=f(t'+2c)=f((t'+c)+c)=f(t'+c)=f(t')=f(t-2c)$
Ok...part (a)...I just don't seek how you got that. Where did you get $f(t'+2c)$ and how does $f((t'+c)+c)=f(t'+c)$?

(b) is false, it makes nonsense...
Why is it nonsense? What makes it false?

4. Originally Posted by chrozer
Ok...part (a)...I just don't seek how you got that. Where did you get $f(t'+2c)$
Because I let $t=t'+2c$
It's easier if you make a substitution, as you'll report to the formula you know
and how does $f((t'+c)+c)=f(t'+c)$?
Because c is the period !
Again, if you can't see it, let T=t'+c, you'd have f(T+c)=f(T), which is f((t'+c)+c)=f(t'+c)
Whatever you have, if you add c, it will give the same !

Why is it nonsense? What makes it false?
Hmmm I've been looking for an explanation... I think I found it

$f\left(\tfrac t2\right)=f \left(\tfrac t2+c\right)$
$=f \left(\tfrac 12 (t+2c)\right)=f \left(\tfrac 12 (t+c)+\tfrac c2\right)$
Can it possibly equal $f \left(\tfrac 12(t+c)\right)$ ?
Only if :
- c=0 (which is not possible otherwise it would not make sense for the periodicity)
- f is c/2 periodic. But as I explained for c), it's not possible.

Thus b) is false

5. Originally Posted by Moo
Because I let $t=t'+2c$
It's easier if you make a substitution, as you'll report to the formula you know

Because c is the period !
Again, if you can't see it, let T=t'+c, you'd have f(T+c)=f(T), which is f((t'+c)+c)=f(t'+c)
Whatever you have, if you add c, it will give the same !

Hmmm I've been looking for an explanation... I think I found it

$f\left(\tfrac t2\right)=f \left(\tfrac t2+c\right)$
$=f \left(\tfrac 12 (t+2c)\right)=f \left(\tfrac 12 (t+c)+\tfrac c2\right)$
Can it possibly equal $f \left(\tfrac 12(t+c)\right)$ ?
Only if :
- c=0 (which is not possible otherwise it would not make sense for the periodicity)
- f is c/2 periodic. But as I explained for c), it's not possible.

Thus b) is false
Ok I see now...thanx for the explanation.