1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

1.2 Arbitrary Period and Half-Range Expansions 71

Figure 5 Odd periodic extension (period 2) off(x)=x,0<x<1.

Figure 6 Even periodic extension (period 2) off(x)=x,0<x<1.

The following six correspondences (we will later show them to be equalities)
follow from the ideas of this section. Note that the inequalities showing the
applicable range ofxare crucial.

∑∞

n= 1

−2cos(nπ)

nπ

sin(nπx)∼







f(x)=x, 0 <x<1,

fo(x)=x, − 1 <x<1,

f ̄o(x), −∞<x<∞,

1

2

−

∑∞

n= 1

2 ( 1 −cos(nπ))

n^2 π^2

cos(nπx)∼







f(x)=x, 0 <x<1,

fe(x)=|x|, − 1 <x<1,

f ̄e(x), −∞<x<∞.

EXERCISES

1.Find the Fourier series of each of the following functions. Sketch the graph

of the periodic extension offfor at least two periods.

a. f(x)=|x|, − 1 <x<1;

b. f(x)=

{− 1 , − 2 <x<0,

1 , 0 <x<2;

c. f(x)=x^2 , −^12 <x<^12.