Determinants and Their Applications in Mathematical Physics
286 6. Applications of Determinants in Mathematical Physics Hence, the first term ofF is given by AxA ∗ x=(−1) n+1 Qn+1,n+2Qn+2, ...
6.10 The Einstein and Ernst Equations 287 Hence, referring to (6.9.28) and applying the Jacobi identity, (−1) n F= ∣ ∣ ∣ ∣ Qn+1, ...
288 6. Applications of Determinants in Mathematical Physics brs=ω s−r ars. (6.10.6) En=|ers|n=(−1) n A (n+1) 1 ,n+1 =(−1) n A (n ...
6.10 The Einstein and Ernst Equations 289 Proof. Multiply therth row ofAbyω −r ,1≤r≤nand thesth column byω s ,1≤s≤n. The effect ...
290 6. Applications of Determinants in Mathematical Physics Lemma 6.19. a. ∂epq ∂ρ +ω ∂apq ∂z = ( q−p ρ ) epq, b. ∂apq ∂ρ +ω ∂ep ...
6.10 The Einstein and Ernst Equations 291 =− ( E A ) 2 ∑ p ∑ q ∂apq ∂z E p 1 E nq . Hence, referring to Lemma 6.19, ∂E n 1 ∂ρ +ω ...
292 6. Applications of Determinants in Mathematical Physics which is equivalent to (b). This completes the proof of Lemma 6.20. ...
6.10 The Einstein and Ernst Equations 293 Proof. The proof is by induction and applies the B ̈acklund transforma- tion theorems ...
294 6. Applications of Determinants in Mathematical Physics Hence, the application of transformationγtoPngivesP ′ n . In order t ...
6.10 The Einstein and Ernst Equations 295 Exercise.The one-variable Hirota operatorsHxandHxxare defined in Section 5.7 and the d ...
296 6. Applications of Determinants in Mathematical Physics whereτj is a function which appears in the Neugebauer solution and i ...
6.10 The Einstein and Ernst Equations 297 Proof. Proof of (a).Denote the determinant on the left byWm. wi+j− 2 +wi+j= 2 n ∑ k=1 ...
298 6. Applications of Determinants in Mathematical Physics where Em= m ∏ r=1 εk r , K= m ∑ r=1 (kr−1). (6.10.39) Applying Theor ...
6.10 The Einstein and Ernst Equations 299 6.10.5 Physically Significant Solutions From the theorem in Section 6.10.2 on the inte ...
300 6. Applications of Determinants in Mathematical Physics Denote this particular solution byUr. Then, tr=(−ω) r Ur, where Ur= ...
6.10 The Einstein and Ernst Equations 301 Hence, Pn Pn− 1 = { 2 ρ } 2 n− 1 V 2 n(c) { G F } , Qn+1 Qn =− { 2 ρ } 2 n− 1 V 2 n(c) ...
302 6. Applications of Determinants in Mathematical Physics 6.10.6 The Ernst Equation The Ernst equation, namely (ξξ ∗ −1)∇ 2 ξ= ...
6.11 The Relativistic Toda Equation — A Brief Note 303 x=t √ 1 −a 2 = ct √ 1+c 2 . (6.11.4) Equations (6.11.1)–(6.11.3) are sati ...
Appendix A A.1 Miscellaneous Functions The Kronecker Delta Function δij= { 1 ,j=i 0 ,j=i. q ∑ j=p xjδjr= { 0 ,p≤r≤q, 0 , otherw ...
A.1 Miscellaneous Functions 305 δi,odd= { 1 ,iodd, 0 ,ieven. δi 1 i 2 ;j 1 j 2 = { 1 , (j 1 ,j 2 )=(i 1 ,i 2 ) 0 , otherwise. Th ...
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