Page 45 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

ORTHOGONAL SETS OF FUNCTIONS 35
Noting that

b b
m m m
2 2 2 2 2 2
K (x)dx = a n a j n (x) j (x)dx = a = a + a + a + ··· + a
m n 1 2 3 m
x=a n=1 j=1 x=a n=1
(3.24)
and

b b
∞ m

f (x)K(x)dx = c n a j n (x) j (x)dx
x=a n=1 j=1 x=a
m

= c n a n = c 1 a 1 + c 2 a 2 + ··· + c m a m (3.25)
n=1
b

2 2 2 2
E = f (x)dx + a + ··· + a − 2a 1 c 1 − ··· − 2a m c m (3.26)
1
m
x=a
2
2
2
Now add and subtract c , c ,..., c . Thus Eq. (3.26) becomes
2
m
1
b

2 2 2 2 2 2 2 2
E = f (x)dx − c − c − ··· − c + (a 1 − c 1 ) + (a 2 − c 2 ) + ··· + (a m − c m )
m
2
1
x=a
(3.27)
which is clearly minimized when a n = c n .
3.2.3 Convergence of Fourier Series
We briefly consider the question of whether the Fourier series actually converges to the function
f (x) for all values, say, on the interval a ≤ x ≤ b. The series will converge to the function if
the value of E defined in (3.19) approaches zero as m approaches infinity. Suffice to say that
this is true for functions that are continuous with piecewise continuous first derivatives, that
is, most physically realistic temperature distributions, displacements of vibrating strings and
bars. In each particular situation, however, one should use the various convergence theorems
that are presented in most elementary calculus books. Uniform convergence of Fourier series
is discussed extensively in the book Fourier Series and Boundary Value Problems by James Ward
Brown and R. V. Churchill. In this chapter we give only a few physically clear examples.