Page 48 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

38 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS

1

-1 1 2 3

FIGURE 3.3: The periodic continuation of the series in Example 3.2

It should be clear from the above examples that in general a Fourier sine/cosine series of
a function f (x) defined on 0 ≤ x ≤ 1 can be written as

∞ ∞
c 0
f (x) = + c n cos(nπx) + b n sin(nπx) (3.38)
2
n=1 n=1
where

1 f (x)cos(nπx)dx
x=0
c n = n = 0, 1, 2, 3,...
2
1 cos (nπx)dx
x=0
1 f (x)sin(nπx)dx
x=0
b n = n = 1, 2, 3,... (3.39)
1 sin (nπx)dx
2
x=0
Problems
1. Show that
π

sin(nx)sin(mx)dx = 0
x=0

when n = m.
2. Find the Fourier sine series for f (x) = 1 − x on the interval (0, 1). Sketch the periodic
continuation. Sum the series for the first five terms and sketch over two periods. Discuss
convergence of the series, paying special attention to convergence at x = 0and x = 1.
3. Find the Fourier cosine series for 1 − x on (0, 1). Sketch the periodic continuation.
Sum the first two terms and sketch. Sum the first five terms and sketch over two periods.
Discuss convergence, paying special attention to convergence at x = 0and x = 1.