Page 296 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Fourier series and Fourier integral 287

we see that equations (B.1) and (B.3b) are identical.

Other variables
The Fourier series (B.1) and (B.3) are expressed in terms of an angle è.However,in
many applications the variable may be a distance x or the time t. If the Fourier series is
to represent a function f (x) of the distance x in a range ÿl < x < l, we make the
substitution
ðx
è ˆ
l
in equation (B.1) to give
1
a 0 X nðx nðx
f (x) ˆ ‡ a n cos ‡ b n sin (B:4)
2 l l
nˆ1
with a n and b n given by
1 l nðx
a n ˆ f (x)cos dx, n ˆ 0, 1, 2, .. . (B:5a)
l ÿl l
l
1 nðx
b n ˆ f (x)sin dx, n ˆ 1, 2, .. . (B:5b)
l ÿl l
If time is the variable, then we may make either of the substitutions
2ðt
è ˆ ˆ ùt
p
where p is the period of the function f (t) and ù is the angular frequency, so that
equation (B.1) becomes

1 1
a 0 X 2ðnt 2ðnt a 0 X
f (t) ˆ ‡ a n cos ‡ b n sin ˆ ‡ (a n cos nùt ‡ b n sin nùt)
2 p p 2
nˆ1 nˆ1
(B:6)
The constants a n and b n in equations (B.2) for the variable t are
p=2 ð=ù
2 2ðnt ù
a n ˆ f (t)cos dt ˆ f (t)cos nùt dt (B:7a)
p ÿ p=2 p ð ÿð=ù
2 p=2 2ðnt ù ð=ù
b n ˆ f (t)sin dt ˆ f (t)sin nùt dt (B:7b)
p ÿ p=2 p ð ÿð=ù

Complex form
The Fourier series (B.1) can also be written in complex form by the substitution of
equations (A.32) and (A.33) for cos nè and sin nè, respectively, to yield
1
X inè
f (è) ˆ c n e (B:8)
nˆÿ1
where