Page 345 - Schaum's Outline of Theory and Problems of Advanced Calculus

Fourier Series

Mathematicians of the eighteenth century, including Daniel Bernoulli and Leonard Euler, expressed
the problem of the vibratory motion of a stretched string through partial differential equations that had
no solutions in terms of ‘‘elementary functions.’’ Their resolution of this difficulty was to introduce
infinite series of sine and cosine functions that satisfied the equations. In the early nineteenth century,
Joseph Fourier, while studying the problem of heat flow, developed a cohesive theory of such series.
Consequently, they were named after him. Fourier series and Fourier integrals are investigated in this
and the next chapter. As you explore the ideas, notice the similarities and differences with the chapters
on infinite series and improper integrals.

PERIODIC FUNCTIONS
A function f ðxÞ is said to have a period T or to be periodic with period T if for all x, f ðx þ TÞ¼ f ðxÞ,
where T is a positive constant. The least value of T > 0iscalled the least period or simply the period of
f ðxÞ.

EXAMPLE 1. The function sin x has periods 2 ; 4 ; 6 ; ... ; since sin ðx þ 2 Þ; sin ðx þ 4 Þ; sin ðx þ 6 Þ; ... all
equal sin x. However, 2 is the least period or the period of sin x.

EXAMPLE 2. The period of sin nx or cos nx, where n is a positive integer, is 2 =n.

EXAMPLE 3. The period of tan x is .

EXAMPLE 4. Aconstant has any positive number as period.
Other examples of periodic functions are shown in the graphs of Figures 13-1(a), (b), and (c) below.

Period Period
f (x) f (x) f (x) Period

x x x

(a) (b) (c)
Fig. 13-1
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