6.5 Reduction formulas 177
More important still, the recurrence relation is ideally suited for the computation of
one or several members of a family of integrals.
Reduction formulas are particularly simple for some important definite integrals in
the physical sciences. For example, the integral
(a 1 > 1 0) (6.16)
where nis a positive integer or zero, occurs in the quantum-mechanical description of
the properties of the hydrogen atom. Integration by parts gives
Whenn 1 ≠ 10 , the quantitye
−ar1 r
nis zero at both integration limits,r 1 = 10 andr 1 → 1 ∞.
Therefore
n 1 ≥ 11
and it follows that
(6.17)
wheren! 1 = 1 n(n 1 − 1 1)(n 1 − 1 2)1-1 1 is the factorial of n.
EXAMPLE 6.14Determine a reduction formula for where nis a
positive integer.
Write the integrand as cos
n− 11 x 1 cos 1 x, and let u 1 = 1 cos
n− 11 x and Then
v 1 = 1 sin 1 x, and
and, becausesin
21 x 1 = 111 − 1 cos
21 x,
= 1 cos
n− 11 x 1 sin 1 x 1 + 1 (n 1 − 1 1)I
n− 21 − 1 (n 1 − 1 1)I
nI x xn xdxn xd
nnnn=+− −−
−−cos sin ( ) cos ( ) cos
1211 ZZxx
Ixxn xxdx
nnn=+−
−−cos sin ( ) cos sin
1221 Z
d
dx
x
v
=cos.
Ixdx
nn=Zcos
Ierdr
n
a
nar nn==
!
−+Z
01∞I
n
a
I
nn=,
− 1I
a
er
n
a
I
nar nn=− +
−−011
∞Ierdr
nar n=
−Z
0∞