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I dont understand the meaning of this line in my book -

" $\sin^2x + \cos^2x$ is periodic but the fundamental period is not defined. "

Why is the period not defined? $F(x)$ is $1$ here so it is a constant function which should be periodic ?

Does this mean all constant functions are not periodic?

Please explain

Mikasa
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geek101
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    It is periodic, but every positive real is a period. – André Nicolas Mar 19 '16 at 05:56
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    The fundamental period of a periodic function $f(x)$ is the smallest $T > 0$ (if one exists) such that $f(x +T) = f(x)$ for all $x$. For example, $\cos(x)$ the fundamental period is $T = 2\pi$. For a constant function, this hold for any $T > 0$ so there is no smallest positive $T$ and so there is no fundamental period. – User8128 Mar 19 '16 at 05:56
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    There are other periodic functions with no fundamental period. For example, let $f(t)=0$ if $t$ is rational, and $1$ if $t$ is irrational. Then every positive rational is a period. – André Nicolas Mar 19 '16 at 06:09

3 Answers3

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A function $f:\mathbb{R} \to \mathbb{R}$ is periodic if there exists some number $t > 0$ such that $$ f(x) = f(x + t) $$ A constant function is periodic since you can take $t = 1, t = 2$, etc. (Hint: Hover over the tag "periodic-functions". What do you see?)

The fundamental period of $f$ is the smallest of such $t$'s. Since $t$ cannot be $0$, you are looking for the minimum of $(0,\infty)$, which does not exist.

hmakholm left over Monica
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Henricus V.
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The fundamental period of $f$ is the smallest positive number $T$ so that $f(t\pm T)=f(t)$ for all $t$. Since no smallest number exists for $f(t)=\sin^2(t)+\cos^2(t)=1$, there is no fundamental period.

There are different definitions of periodic (some require a fundamental period, others do not), but I prefer to exclude constant functions from being periodic.

Plutoro
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When i googled the definition of periodic function i got-"A function f(x) is periodic if f(x) = f(x + p) for all x and some fixed p. p is called a period of f. The smallest positive p that works is called the period of f. (A constant function is periodic with any value of p as a period, so there's no such thing as "the period" of such a function." Which explains everything.

geek101
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