Determinants and Their Applications in Mathematical Physics

300 6. Applications of Determinants in Mathematical Physics

Denote this particular solution byUr. Then,

tr=(−ω)

r

Ur,

where

Ur=

2 n

∑

j=1

(−1)

j− 1

Mj(c)fr(xj)

ε

∗

j

(6.10.54)

and the symbol∗denotes the complex conjugate. This function is of the

form (4.13.3), where

aj=

(−1)

j− 1

Mj(c)

ε

∗

j

(6.10.55)

andN=2n. These choices ofajandNmodify the functionkrdefined in

(4.13.5). Denote the modifiedkrbywr, which is given explicitly in (6.10.3).

Since the results of Section 4.13.2 are unaltered by replacingωby (−ω),

it follows from (4.13.22) and (4.13.23) withn→mthat

Pm=(−1)

m(m−1)/ 2

2

m

2

− 1

∣

∣

wi+j+wi+j− 2

∣

∣

m

,

Qm=(−1)

m(m−1)/ 2

2

(m−1)

2 ∣

∣w

i+j− 2

∣

∣

m

. (6.10.56)

Applying the theorem in Section 6.10.4,

Pm=2

m

2

− 1

ρ

−m(m−1)

{

V 2 n(c)

}m− 1

H

(m)

2 n(ε),

Qm=2

(m−1)

2

ρ

−m(m−1)

{

V 2 n(c)

}m− 1

H

(m)

2 n

(

1

ε

∗

)

. (6.10.57)

Hence,

Pn

Pn− 1

=2

2 n− 1

ρ

−2(n−1)

V 2 n(c)

H

(n)

2 n

(ε)

H

(n−1)

2 n (ε)

. (6.10.58)

Also, applying (6.10.32),

Qn+1

Qn

=2

2 n− 1

ρ

− 2 n

V 2 n(c)

H

(n+1)

2 n

(

1 /ε

∗

)

H

(n)

2 n

(

1 /ε

∗

)

=− 2

2 n− 1

ρ

− 2 n

V 2 n(c)

H

(n−1)

2 n

(ε

∗

)

H

(n)

2 n

(ε∗)

. (6.10.59)

Sinceτj=ρεj, (the third line of (6.10.26)), the functionsFandGdefined

in Section 6.2.8 are given by

F=H

(n−1)

2 n (ρε)=ρ

n− 1

H

(n−1)

2 n (ε),

G=H

(n)

2 n

(ρε)=ρ

n

H

(n)

2 n

(ε). (6.10.60)