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where the last equality is a consequence of the property (8.72) of the sine function.
Thus we have derived an amazingly simple formula for the coefficients in (8.70),
D
2
Z
1
0
c
k
f.x/sin.kx/
dx
for k
D
1;2;::::
(8.73)
It is important that you get this right. Our argument shows that if the function f
can be written in the form (8.70) then the coefficients must satisfy (8.73). It does
not
provide any information about which functions can be expressed as a series of sine
functions. This topic is treated in more advanced texts on Fourier analysis, see, e.g.,
Weinberger [29].
Roughly speaking, every “well-behaved”
24
function can be written in the form
(8.70), with Fourier coefficients given by (8.73). We will not pursue this question
any further. Throughout this text every initial condition f will be such that the
Fourier sine series converges and that (8.70) holds.
8.2.7
Summary
Let us summarize our findings so far.
The Fourier coefficients for a well-behaved function f.x/, x
2
.0; 1/,isdefinedby
D
2
Z
1
0
c
k
f.x/sin.kx/
dx
for k
D
1;2;:::;
(8.74)
and the associated Fourier sine series by
X
f.x/
D
for x
2
.0; 1/:
c
k
sin.kx/
(8.75)
k
D
1
Furthermore, if f satisfies (8.75), then
X
c
k
e
k
2
2
t
sin.kx/
u
.x; t /
D
(8.76)
k
D
1
24
Typically, continuous functions and functions with a finite number of jump discontinuities.