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Definition 10.3. Periodic Functions: A function f is said to be periodic if there is a real
number c so that f (t + c) = f (t) for all real numbers t in the domain of f . The smallest positive
number p for which f (t + p) = f (t) for all real numbers t in the domain of f , if it exists, is called
the period of f .
We have already seen a family of periodic functions in Section 2.1: the constant functions. However,
we leave it to the reader as an exercise to show that, despite being periodic, constant functions
have no period. Returning to the circular functions, we see that by Deﬁnition 10.3, f (t) = cos(t)
is periodic, since cos(t + 2π · k ) = cos(t) for any integer k . To determine the period of f , we need
to ﬁnd the smallest real number p so that f (t + p) = f (t) for all real numbers t or, said diﬀerently,
the smallest positive real number p such that cos(t + p) = cos(t) for all real numbers t. We know
that cos(t + 2π ) = cos(t) for all real numbers t but the question remains if any smaller real number
will do the trick. Suppose p > 0 and cos(t + p) = cos(t) for...
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