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Download e-book for kindle: Applied Quaternionic Analysis by Kravchenko V. V.

By Kravchenko V. V.

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Lim K0 [f ](x) = P0 [f ]( ); 2 3x! 2 K0 [f ](x) = Q0 [f ]( ): Substituting this into the preceding equality we complete the proof. Theorem 8 implies some very nice properties of the operators P , Q and S . First of all we prove the following Theorem 9. 30) P2 = P ; Q2 = Q ; P Q = Q P = 0: Proof. 30) are equivalent, so that it is enough 36 2. 30). Let f 2 C 0; ( ; H(C)). K [f ] 2 ker D ( ) and due to the Cauchy integral formula we obtain that K [f ](x) = K [K [f ]](x): Now letting x ! 30). The second is proved in a similar way considering the exterior of .

Div ~j + ~j: satis…es the equation ! 6). This procedure of diagonalization can be written in matrix form 0 10 1 ! D i! E @ A@ A=B ! " 1 A and 1 0 @ B D 0 0 1 = D+ 0 1@ 2 1 0 1 ! E AB @ A ! " 1= 1= 1 A: In this way we obtain two decoupled equations for the unknown vectors !

That is, we have to calculate the following expression p ! q v ( (t)) 1 iq p + V = ( @t iD) 4 jx c 4 " 1 s( (t))j (1 M ( (t))) : 56 3. PHYSICAL MODELS REDUCING TO THE OPERATOR D We introduce the auxiliary functions 1 ; jx s( (t))j (1 M ( (t))) p ! 13) V= 1 ( @t c 1 ( @t c iD) [f ] ' iD) ['] f: It is easy to see that the scalar part of the expression on the right-hand side is zero. For this purpose we give the following formulas, which can be veri…ed by a straightforward computation. 00(t) = c jx 1 sj (1 grad = @t ' = 1 0= 1 M )3 (c2 M 2 + hs00; x M 1 c(1 00(t) + jx sj jx ; x M ) jx cM sj (1 2 si js0j2 ); s ; sj ; M )2 !

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Applied Quaternionic Analysis by Kravchenko V. V.


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