Determinants and Their Applications in Mathematical Physics

264 6. Applications of Determinants in Mathematical Physics

where

A=|ars|n,

ars=δrser+

2

br+bs

=asr,

er= exp(−brx+b

3

rt+εr).

Theεrare arbitrary constants and thebrare constants such that thebr+

bs=0but are otherwise arbitrary.

Two independent proofs of this theorem are given in Sections 6.7.2 and

6.7.3. The method of Section 6.7.2 applies nonlinear differential recurrence

relations in a function of the cofactors ofA. The method of Section 6.7.3

involves partial derivatives with respect to the exponential functions which

appear in the elements ofA.

It is shown in Section 6.7.4 thatAis a simple multiple of a Wronskian and

Section 6.7.5 consists of an independent proof of the Wronskian solution.

6.7.2 The First Form of Solution..............

First Proof of Theorem 6.1.3.The proof begins by extracting a wealth

of information about the cofactors ofAby applying the double-sum rela-

tions (A)–(D) in Section 3.4 in different ways. Apply (A) and (B) with

f

′

interpreted first asfxand then asft. Apply (C) and (D) first with

fr=gr=br, then withfr=gr=b

3

r. Later, apply (D) withfr=−gr=b

2

r.

Appling (A) and (B),

v=Dx(logA)=−

∑

r

∑

s

δrsbrerA

rs

=−

∑

r

brerA

rr

, (6.7.3)

Dx(A

ij

)=

∑

r

brerA

ir

A

rj

. (6.7.4)

Applying (C) and (D) withfr=gr=br,

∑

r

∑

s

[

δrs(br+bs)er+2

]

A

rs

=2

∑

r

br,

which simplifies to

∑

r

brerA

rr

+

∑

r

∑

s

A

rs

=

∑

r

br, (6.7.5)

∑

r

brerA

ir

A

rj

+

∑

r

∑

s

A

is

A

rj

=

1

2

(bi+bj)A

ij

. (6.7.6)