Page 315 - Determinants and Their Applications in Mathematical Physics

300 6. Applications of Determinants in Mathematical Physics

Denote this particular solution by U r . Then,
t r =(−ω) U r ,
r
where
(−1) M j (c)f r (x j )
2n j−1

U r = (6.10.54)
ε ∗
j=1 j
and the symbol ∗ denotes the complex conjugate. This function is of the
form (4.13.3), where
(−1) j−1 M j (c)
a j = (6.10.55)
ε ∗
j
and N =2n. These choices of a j and N modify the function k r defined in
(4.13.5). Denote the modified k r by w r , which is given explicitly in (6.10.3).
Since the results of Section 4.13.2 are unaltered by replacing ω by (−ω),
it follows from (4.13.22) and (4.13.23) with n → m that
2
2
P m =(−1) m(m−1)/2 m −1 w i+j + w i+j−2 ,
m
2
Q m =(−1) m(m−1)/2 (m−1) w i+j−2 . (6.10.56)
2
m
Applying the theorem in Section 6.10.4,
2 - . m−1 (m)
m −1 −m(m−1)
P m =2 ρ V 2n (c) H (ε),
2n

2 −m(m−1) - . m−1 (m) 1
(m−1)
Q m =2 ρ V 2n (c) H . (6.10.57)
ε
2n ∗
Hence,
(n)
H (ε)
2n−1 −2(n−1)
=2 ρ V 2n (c) 2n . (6.10.58)
P n
H (ε)
P n−1 (n−1)
2n
Also, applying (6.10.32),
(n+1)
H 1/ε ∗
Q n+1
=2 2n−1 −2n V 2n (c) 2n (n)
ρ
H 1/ε ∗
Q n
2n
(n−1)
∗
H (ε )
= −2 2n−1 −2n V 2n (c) 2n (n) . (6.10.59)
ρ
H (ε )
∗
2n
Since τ j = ρε j , (the third line of (6.10.26)), the functions F and G defined
in Section 6.2.8 are given by
(n−1) n−1 (n−1)
F = H (ρε)= ρ H (ε),
2n 2n
(n) (n)
G = H (ρε)= ρ H (ε). (6.10.60)
n
2n 2n