210
6 Appendices
U 22C = r
2
1 P 2(1)
⎧
⎨
⎩
ln[R O (1 + α 0 )]
θ,φ
P 2(2) dΩ 2
+
θ,φ
ln
1 +
α 2 P 2(2)
(1 + α 0 )
P 2(2) dΩ 2
⎫
⎬
⎭
.
(6.56)
The first integral in (6.56) vanishes by (6.32) because we can insert a factor of P 0(2)
in the integrand. For the second integral, if α 2 is small, then we have an integrand
of the form ln(1 + x) where x will be a small quantity. To deal with this, invoke the
expansion
ln(1 + x) ∼ x −
1
2
x
2
+ · · ·
(6.57)
This gives
U 22C = r
2
1 P 2(1)
α 2
(1 + α 0 )
θ,φ
P 2(2) P 2(2) dΩ 2
−
α
2
2
2(1 + α 0 )
2
θ,φ
P
3
2(2) dΩ 2
⎫
⎬
⎭
.
(6.58)
The first integral in (6.58) evaluates to 4π /5. The second one involves the cube of
P 2(2) . It turns out that we will not need this second term, but we will carry it along
for the time being with a compacting notation and write U 2CC in the form
U 22C = r
2
1 P 2(1)
4π
5
α 2
1 + α 0
−
πα
2
2
(1 + α 0 )
2
[2, 2, 2]
,
(6.59)
where
[2, 2, 2] =
θ,φ
P
3
2(2) dΩ 2 .
(6.60)
Gathering (6.53), (6.55), and (6.59) into (6.48) gives
U 2 = −
2 π
3
r
2
1 P 0(1) + 2π R
2
O P 0(1)
(1 + α 0 )
2
+
1
5
α
2
2
+r
2
1 P 2(1)
4π
5
α 2
1 + α 0
−
πα
2
2
(1 + α 0 )
2
[2, 2, 2]
.
(6.61)