90 Chapter 1 Fourier Series and Integrals
- Use the series that follows, together with integration or differentiation, to
find a Fourier series for the functionp(x)=x(π−x),0<x<π.
x= 2∑∞
n= 1(− 1 )n+^1
n sin(nx),^0 <x<π.- Letf(x)be an odd, periodic, sectionally smooth function with Fourier
sine coefficientsb 1 ,b 2 ,.... Show that the function defined by
u(x,t)=∑∞
n= 1bne−n^2 tsin(nx), t≥ 0 ,has the following properties:a.∂^2 u
∂x^2 =∑∞
n= 1−n^2 bne−n^2 tsin(nx), t>0;b.u( 0 ,t)= 0 , u(π,t)= 0 , t>0;c. u(x, 0 )=^12(
f(x+)+f(x−))
.
10.Letfbe as in Exercise 9, but defineu(x,y)byu(x,y)=∑∞
n= 1bne−nysin(nx), y> 0.Show thatu(x,y)has these properties:a.∂^2 u
∂x^2 =∑∞
n= 1−n^2 bne−nysin(nx), y>0;b.u( 0 ,y)= 0 , u(π,y)= 0 , y>0;c. u(x, 0 )=1
2
(
f(x+)+f(x−))
1.6 Mean Error and Convergence in Mean
While we can study the behavior of infinite series, we must almost always use
finite series in practice. Fortunately, Fourier series have some properties that
make them very useful in this setting. Before going on to these properties, we
shall develop a useful formula.
Supposefis a function defined in the interval−a<x<a, for which
∫a
−a(
f(x)) 2
dx