Mathematical Physics Vol 1

6.1 Periodic functions

309

6.1.2 Extension of non-periodic functions

Observe an arbitrary non-periodic function f ( x ) , defined on the interval a ≤ x ≤ a + T . Let us now construct a periodic function F ( x ) , with period T , which coincides with the function f ( x ) on the interval a ≤ x ≤ a + T . The graph of the new function is obtained by moving the function f ( x ) translationally along the x - axis, to the left or to the right for ± T , ± 2 T ,..., ± nT ,... (see Fig. 6.1).

Figure 6.1: Extension of a non-periodic function.

6.1.3 Sum (superposition) of harmonics

Observe the array of harmonics A k sin 2 π k T The period of the k -th harmonic is 2

x + ϕ k , k = 1 , 2 ,..., − ∞ < x < ∞ , T > 0 .

(6.5)

T k

T k =

(6.6)

.

The sum (superposition) of a finite number of harmonics is a function of the form

A k sin

x + ϕ k .

N ∑ k = 1

2 π k T

f N ( x )= A 0 +

(6.7)

This function is periodic, with period T (see Note 2). When N → ∞ we obtain an infinite series, namely the function f ( x )= A 0 + ∞ ∑ k = 1 A k sin 2 π k T x + ϕ k ,

(6.8)

which is also periodic, with period T . It is well known from trigonometry, that sin ( α + β )= sin α cos β + sin β cos α . Using this relation on the previous series, we obtain

N ∑ k = 1

ℓ

k π x ℓ

k π x

a 0 2

f N ( x )=

a k cos

+ b k sin

(6.9)

+

2

sin

T

x + + ϕ k = sin ( x + T )+ ϕ k = sin T k

x + ϕ k + 2 π = sin

x + ϕ k ,

2 π k

2 π k T 2 π k T

2 π k T

sin

T = sin

x + ϕ k + 2 π k = sin

x + ϕ k .

2 π k T

2 π k T

2 π k T

2 π k T

x + ϕ k +