Changing periods. Very confusing

[samh]

This is driving me nuts:

Suppose f(x) has period 2L, that is, f(x+2L)=f(x) for all x. If we let t=pi*x/L, and
g(t) = f(x) = f(L*t/pi)
then, as you can verify, g has period 2pi.

How do you show that that's true?!? How do you prove it? For the life of me I can't see how this holds despite the fact that I've wasted the past two hours working at it. I can't think of a technical explanation for it (a proof) OR an intuitive one... Please help.

 

[dextercioby]

It's simple

$$f\left(\frac{Lt}{\pi}+2L\right)=f\left(\frac{L}{\pi}(t+2\pi)\right)=f\left(\frac{L}{\pi}t\right)$$

So you can see very clearly that if you denote by

$$g(t)=f\left(\frac{L}{\pi}t\right)$$

, then g has a period of 2\pi.

Daniel.

 

[d_leet]

This is driving me nuts:



How do you show that that's true?!? How do you prove it? For the life of me I can't see how this holds despite the fact that I've wasted the past two hours working at it. I can't think of a technical explanation for it (a proof) OR an intuitive one... Please help.

To prove it just show that g(t) = g(t+2pi) using the definition of g and the properties of f.