Changing periods. Very confusing

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In summary: This will demonstrate that g has a period of 2pi and therefore, f has a period of 2L. In summary, the given function f(x) with period 2L can be transformed into g(t) with period 2pi by using the substitution t=pi*x/L. This can be verified by showing that g(t) = g(t+2pi), proving that f has a period of 2L.
  • #1
samh
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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.
 
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  • #2
It's simple

[tex] f\left(\frac{Lt}{\pi}+2L\right)=f\left(\frac{L}{\pi}(t+2\pi)\right)=f\left(\frac{L}{\pi}t\right) [/tex]

So you can see very clearly that if you denote by

[tex] g(t)=f\left(\frac{L}{\pi}t\right) [/tex]

, then g has a period of 2\pi.

Daniel.
 
  • #3
samh said:
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.
 

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