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428 CHAPTER 13 Fourier Series
Fourier found that the functions
2
u n (x,t) = b n sin(nx)e −kn t
satisfy the heat equation and the boundary conditions, for every positive integer n and any number
b n . However, there is no choice of n and b n for which this function satisfies the initial condition,
which would require that
u n (x,0) = b n sin(nx) = x(π − x).
for 0 ≤ x ≤ π.
We could try a finite sum of these functions, attempting a solution
N
2
−kn t
u(x,t) = b n sin(nx)e .
n=1
But this would require that N and numbers b 1 ,··· ,b N be found so that
N
u(x,0) = x(π − x) = b n sin(nx)
n=1
for 0 ≤ x ≤ π. Again, this is impossible. A finite sum of multiples of sine functions is not a
polynomial.
Fourier’s brilliant insight was to attempt an infinite superposition,
∞
2
u(x,t) = b n sin(nx)e −kn t .
n=1
This function will still satisfy the heat equation and the boundary conditions u(x,0) = u(π,0) =
0. To satisfy the initial condition, the problem is to choose the numbers b n so that
∞
u(x,0) = x(π − x) = b n sin(nx)
n=1
for 0 ≤ x ≤ π. Fourier claimed not only that this could this be done, but that the right choice is
1 π 4 1 − (−1) n
b n = x(π − x)sin(nx)dx = .
π 0 π n 3
With these coefficients, Fourier wrote the solution for the temperature function:
∞
4 1 − (−1) n 2
u(x,t) = sin(nx)e −kn t .
π n 3
n=1
The astonishing claim that
∞
4 1 − (−1) n
x(π − x) = sin(nx)
π n 3
n=1
for 0 ≤ x ≤ π was too much for Fourier’s contemporaries to accept, and the absence of rigorous
proofs in his paper led the Academy to reject its publication (although they awarded him the
prize). However, the implications of Fourier’s work were not lost on natural philosophers of
his time. If Fourier was right, then many functions would have expansions as infinite series of
trigonometric functions.
Although Fourier did not have the means to supply the rigor his colleagues demanded, this
was provided throughout the ensuing century and Fourier’s ideas are now seen in many important
applications. We will use them to solve partial differential equations, beginning in Chapter 16.
This and the next two chapters develop the requisite ideas from Fourier analysis.
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October 14, 2010 14:57 THM/NEIL Page-428 27410_13_ch13_p425-464