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Finding the Period of a Function
- Thread starter shen07
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chisigma
Well-known member
- Feb 13, 2012
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Setting $\displaystyle 2\ x + 3 = \xi$ You have that...
$\displaystyle \phi(\xi) = 10 - \phi (\xi + 4)\ (1)$
... but is also...
$\displaystyle \phi(\xi + 4) = 10 - \phi(\xi + 8)\ (2)$
Comparing (1) and (2) You arrive to write...
$\displaystyle \phi(\xi) = \phi(\xi + 8)\ (3)$
... so that the period of $\phi(*)$ is 8...
Kind regards
$\chi$ $\sigma$
Hi,
Certainly the previous response, \(\displaystyle \phi(x+8)=\phi(x)\) for all x, is true. However, you asked for the period of the function. Usually when we talk of the period of a function f, we mean the least positive p with f(x + p) = f(x) for all x (sometimes this is called the principal period). For example, the period of sin(x) is \(\displaystyle 2\pi\). Aside: if f is a non-constant continuous function and there is a positive p with f(x + p) = f(x) for all x, then there is a smallest such p; i.e. the period of f exists.
For your specific question, if \(\displaystyle \phi\) is a non-constant continuous function, the period can be an arbitrarily small positive value!
Example:
Let n be a positive integer, \(\displaystyle a={(2n+1)\pi\over4}\) and \(\displaystyle \phi(x)=\text{sin}(ax)+5\). Easily then \(\displaystyle \phi(x)+\phi(x+4)=10\) for all x, and so this function satisfies your equation. The period of this function is \(\displaystyle {2\over 2n+1}\), which can be made as small as desired by taking n sufficiently large.
Certainly the previous response, \(\displaystyle \phi(x+8)=\phi(x)\) for all x, is true. However, you asked for the period of the function. Usually when we talk of the period of a function f, we mean the least positive p with f(x + p) = f(x) for all x (sometimes this is called the principal period). For example, the period of sin(x) is \(\displaystyle 2\pi\). Aside: if f is a non-constant continuous function and there is a positive p with f(x + p) = f(x) for all x, then there is a smallest such p; i.e. the period of f exists.
For your specific question, if \(\displaystyle \phi\) is a non-constant continuous function, the period can be an arbitrarily small positive value!
Example:
Let n be a positive integer, \(\displaystyle a={(2n+1)\pi\over4}\) and \(\displaystyle \phi(x)=\text{sin}(ax)+5\). Easily then \(\displaystyle \phi(x)+\phi(x+4)=10\) for all x, and so this function satisfies your equation. The period of this function is \(\displaystyle {2\over 2n+1}\), which can be made as small as desired by taking n sufficiently large.