Section 6.5
52.Determine a reduction formula for where nis a positive integer.
53.Show that, for integersm 1 ≥ 10 andn 1 ≥ 11 ,
54.Use the results of Exercises 52 and 53 to evaluate
55.Show that, for integersm 1 ≥ 10 andn 1 > 11 ,
Evaluate
60.The probability that a molecule of mass min a gas at temperature Thas speed vis given
by the Maxwell–Boltzmann distribution
where kis Boltzmann’s constant. Find the average speed.
61.For the Maxwell–Boltzmann distribution in Exercise 60, find the root mean square speed
, where.
62.Line shapes in spectroscopy are sometimes analysed in terms of second moments. The
second moment of a signal centred at angular frequencyω
0
is
whereg(ω)is a shape function for the signal. Evaluate the integral for the gaussian curve
Section 6.6
Evaluate the indefinite integrals:
- 65.Z
()
()( )()
xx
xx x
dx
2
33
123
−+
++ +
Z
()
()()
x
xx
dx
++
2
34
Z
dx
()()xx
21 3−+gTT()ωω=−−( ω)
21
2
2
0
2
π
exp
Z
ωωω ω ω
00
2
∞
()()− gd
vvvv
2
0
2
=Z
∞
fd()
v2
vvvv=Z
0
∞
fd()
f
m
kT
e
mkT()vv
v=
−
4
2
32
22
2π
π
Z
0
32
2∞
re dr
−rZ
0
22
2∞
re dr
−rZ
0
2
2∞
re dr
−rZ
0
5
π 25
sin xxdxcos.
ZZ
00
2
1
ππ 22sin cos sin cos
mn m nd
n
mn
θθθ= θ θθd
−
−
Zsin cos
54
xxdx.
ZZsin cos
sin cos
sin
mnmnmd
mn
n
mn
θθθ
θθ
=
−
+− 11
1
θθθθcos
nd
− 2
Zsin
nxdx,
6.8 Exercises 189