An introduction to mathematics, by A. N. Whitehead.

TRIGONOMETRY 191
we mean by a periodic function. The function f(x) is periodic, with the period a, if (i)
for any value of x we have f(x) =f(x +a), and
(ii) there is no number b smaller than a such
that for any value of x, f(x) =f(x +b).
The second clause is put into the definition
because when we have sin,r it is not only
a
periodic in the period a, but also in the periods
2a and 3a, and so on; this arises since. 27r(x+3a). /27r \. in 27rX
sin — (x =sin (x+67r sm
a a a
So it is the smallest period which we want to
get hold of and call the period of the function.
The greater part of the abstract theory of
periodic functions and the whole of the applications of the theory to Physical Science are
dominated by an important theorem called
Fourier's Theorem; namely that, iff(x) be a
periodic function with the period a and if f(x)
also satisfies certain conditions, which practically are always presupposed in functions
suggested by natural phenomena, then f(x)
can be written as the sum of a set of terms
in the form
co +c sin ( +e) +2 sin ( +e2)
+C3 sin (6 +e3 ) + etc.
a