March 7, 2017

Applied Quaternionic Analysis by Kravchenko V. V.

By Kravchenko V. V.

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1) where D := D + M ; 2 H(C). 1) we have to distinguish di¤erent cases depending on the algebraic properties of . The following observations will help us to understand the structure of the integral operators. 2 = 6 0. We introduce the following auxiliary p := 0 . 2 and ! so that Im 0. The complex quaternions + ! and (1) Let are zero divisors. ) 42 2. ELEMENTS OF QUATERNIONIC ANALYSIS are mutually complementary projection operators. ) + ( 0 + ! ) = 2 ( 0 ! 4) + commute with the oper- and D . 12).

U (t; x) = iq s( (t))j (1 M ( (t))) ; p 4 jx q! 12) reduces to a simple di¤erentiation. 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.

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 .

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