Determinants and Their Applications in Mathematical Physics

6.4 The Kay–Moses Equation 251

=

∑

i

biciφi[(ci+cj)A

ij

−φiφj]

=

∑

r

brcrφr[(cr+cj)A

rj

−φrφj],

∑

r

brcrA

rj

φ

′

r=

∑

r

brcrφr[cjA

rj

−φrφj], (6.4.12)

∑

r

brcrA

rj

(φ

′

r−φr)=

∑

r

brcrφr(cj−1)A

rj

−

∑

r

brcrφ

2

rφj.

Multiply bye

cju

/(cj−1), sum overj, and refer to (6.4.7):

∑

j,r

brcrA

rj

e

cju

(φ

′

r

−φr)

cj− 1

=

∑

r

brcrφ

2

r−

∑

r

brcrφ

2

r

∑

j

e

cju

φj

cj− 1

=F

∑

r

brcrφ

2

r

=

1

2

F(logA)

′′

, (6.4.13)

where

F=1−

∑

j

e

cju

φj

cj− 1

=1−

∑

i,j

e

(ci+cj)u

A

ij

cj− 1

. (6.4.14)

Differentiate and refer to (6.4.9):

F

′

=− 2

∑

r

brcr

∑

j

e

cju

A

rj

cj− 1

∑

i

e

ciu

A

ir

=− 2

∑

r

brcr

∑

j

φre

cju

A

rj

cj− 1

. (6.4.15)

Differentiate again and refer to (6.4.8):

F

′′

=2

∑

r

brcr

∑

j

e

cju

cj− 1

[

φ

2

rφj−cjφrA

rj

−φ

′

rA

rj

]

=P−Q−R, (6.4.16)

where

P=2

∑

j

e

cju

φj

cj− 1

∑

r

brcrφ

2

r

=(1−F)(logA)

′′

(6.4.17)

Q=2

∑

j,r

brcrcjφre

cju

A

rj

cj− 1