1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

342 Chapter 5 Higher Dimensions and Other Coordinates

Example.
Let

f(x)=

{− 1 , − 1 <x<0,

1 , 0 <x<1.

The Legendre series will contain only odd-indexed polynomials, and their co-
efficients are

bn=( 2 n+ 1 )

∫ 1

0

Pn(x)dx (nodd)

=−n^2 (nn++^11 )

[(

1 −x^2

)

Pn′(x)

] 1

0

=n^2 (nn++ 11 )Pn′( 0 )=^2 nn++ 11 Pn− 1 ( 0 )

=(− 1 )(n−^1 )/^2

1 · 3 · 5 ···(n− 2 )

2 · 4 · 6 ···(n− 1 )·

2 n+ 1

n+ 1 (n=^3 ,^5 ,^7 ,...).

Specifically we findb 1 = 3 /2 (by a separate calculation),b 3 =− 7 /8,b 5 =
11 / 16 ,....Becausef(x)is indeed sectionally smooth,

f(x)=

3

2 P^1 (x)−

7

8 P^3 (x)+

11

16 P^5 (x)−···.

See Fig. 13 for graphs of the partial sums of this series.

(a) (b)

Figure 13 Graphs of a function and a partial sum of its Legendre series:
(a) throughP 9 (x)for the functionf(x)in the example; (b) throughP 6 (x)for
f(x)=|x|,− 1 <x<1. Compare with the partial sums of the Fourier series, Figs. 9
and10ofChapter1.