Determinants and Their Applications in Mathematical Physics
326 Appendix ∂V ∂u =V(V+1−x) from which it follows thatSmsatisfies the nonlinear recurrence relation Sm+1=(1−x)Sm+ m ∑ r=0 ( m r ...
A.7 Symmetric Polynomials 327 Also, gnj(xi)=0,j=i. (A.7.5) Examples σ (3) 2 =x 1 x 2 +x 1 x 3 +x 2 x 3 , σ (n) r 0 =1, 1 ≤r≤n, ...
328 Appendix A.8 Differences Given a sequence{ur}, thenthh-difference ofu 0 is written as ∆ n h u 0 and is defined as ∆ n hu^0 = ...
A.8 Differences 329 where f(x)= n ∑ r=0 (−1) r ( n r ) x 2 r+1 2 r+1 , f ′ (x)= n ∑ r=0 (−1) r ( n r ) x 2 r =(1−x 2 ) n . f(x)= ...
330 Appendix =− 1 n+1 [ n+1 ∑ r=0 (−1) r ( n+1 r ) x 2 r − n+1 ∑ r=0 (−1) r ( n+1 r ) ] =− 1 n+1 [(1−x 2 ) n+1 −0] S(0) =− 1 n+1 ...
A.9 The Euler and Modified Euler Theorems on Homogeneous Functions 331 The first is due to Euler. The second is similar in natur ...
332 Appendix Illustration.The function f=Ax 0 x 2 x 4 x 6 + Bx 0 x 2 x 2 3 x 5 x 1 + Cx 2 0 x 1 x 5 3 +Dx 8 2 Ex 3 0 x^4 +Fx 4 1 ...
A.10 Formulas Related to the Function (x+ √ 1+x^2 ) 2 n 333 Proof. Replacexby−x − 1 in (A.10.1), multiply byx 2 n , and putx 2 = ...
334 Appendix λnr= n n+r ( n+r 2 r ) , 1 ≤r≤n, λn 0 =1,n≥0; μnr= 2 rλnr n , 1 ≤r≤n, μn 0 =0,n≥ 0. (A.10.8) Changing the sign ofxi ...
A.11 Solutions of a Pair of Coupled Equations 335 Proof of (a).Putx=shθ. Then, gn= 1 2 (e 2 nθ +e − 2 nθ ) =ch 2nθ, gm+n+gm−n= c ...
336 Appendix This solution can be particularized still further using Cauchy’s theorem. First, allowCto embraceαbut notβand then ...
A.12 B ̈acklund Transformations 337 A.12 B ̈acklund Transformations It is shown in Section 6.2.8 on brief historical notes on th ...
338 Appendix ∇ζ ′ + =− c ζ 2 + ∇ζ+, ∇ 2 ζ ′ +=− c ζ 2 + [ ∇ 2 ζ+− 2 ζ+ (∇ζ+) 2 ] . Hence, 1 2 (ζ ′ + +ζ ′ − )∇ 2 ζ ′ + −(∇ 2 ζ ′ ...
A.12 B ̈acklund Transformations 339 ∇· ( ∇ψ φ ) = 1 φ^2 (φ∇ 2 ψ−∇φ·∇ψ), (A.12.10) ∇· ( ∇ψ φ 2 ) = 1 φ 3 (φ∇ 2 ψ− 2 ∇φ·∇ψ), (A.12 ...
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A.13 Muir and Metzler, A Treatise on the Theory of Determinants 341 A.13 Muir and Metzler, A Treatise on the Theory of Determina ...
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Bibliography MR = Mathematical Reviews Zbl = Zentralblatt f ̈ur Mathematik PA = Physics Abstracts A. Abian, A direct proof of Ta ...
344 Bibliography T.H. Andres, W.D. Hoskins, R.G. Stanton, The determinant of a class of skew- symmetric Toeplitz matrices.Linear ...
Bibliography 345 B. Beckermann, G. Muhlbach, A general determinant identity of Sylvester type and some applications.Linear Alg. ...
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