# Period of function

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• Dec 16th 2010, 04:54 PM
horan6
Period of function
Considering the function $\displaystyle u(t,x)=\sum_{h=1}^{\infty}\left(a_{n}\cos\frac{n\p i}{L}ct+b_{n}\sin\frac{n\pi}{L}ct\right)\sin\frac{ n\pi}{L}x$

I know that the period of this function is $\displaystyle 2\pi/L$, but I am not sure how to prove it. I tried plugging in the period for t and reducing, but didn't get very far. Is this the right approach? Can anyone help with this?

Thanks in advance!
• Dec 16th 2010, 05:51 PM
snowtea
If your period is $\displaystyle T$, then $\displaystyle u(t+T,x) = u(t,x)$ for all $\displaystyle t$ and $\displaystyle x$.
• Dec 18th 2010, 08:43 AM
SammyS
Quote:

Originally Posted by snowtea
If your period is $\displaystyle T$, then $\displaystyle u(t+T,x) = u(t,x)$ for all $\displaystyle t$ and $\displaystyle x$.

Actually, if $\displaystyle \displaystyle T$ is the period, then $\displaystyle \displaystyle u(t+kT,x) = u(t,x),$ for any integer, $\displaystyle \displaystyle k$.

So, you need to show that $\displaystyle \displaystyle T$ is the smallest value for which $\displaystyle \displaystyle u(t+T,x) = u(t,x)$ is true.