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Applied Quaternionic Analysis by Kravchenko V. V.

By Kravchenko V. V.

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F ( ) and according to the Plemelj-Sokhotski formulas K [f ](x) ! P [f ]( ). 31). 31) hold. Let us consider F (x) := K [f ](x); x 2 + . Then F 2 ker D ( + ) and again by the Plemelj- Sokhotski formulas, F j = P [f ] = f . The …rst part of the theorem is proved. The proof of the second part is analogous. Remark 5. 24). The function f itself is extendable in this sense into + or i¤ Q [f ] 0 or P [f ] 0 on 38 2. ELEMENTS OF QUATERNIONIC ANALYSIS respectively. In these cases we call the function -extendable into or + respectively.

5) and = ! (that is = 0). In this second 0 case we can consider both situations ! 2 S and ! 2 = S together. First of all let us notice that in both cases the operator K! admits the following representation K! 9) Z +#(x f#(x y) x jx y yj2 y)! gd y; i (x y) jx yj ! n (y)f (y)+ x 2 R3 n : 48 2. ELEMENTS OF QUATERNIONIC ANALYSIS Here #(x) = exp(i jxj) : 4 jxj = ! 2 H(C) can Moreover, the radiation condition for any be written also in a uni…ed form as can be seen from the following statement, which is nothing but the Cauchy integral formula for the operator D!

E AB @ A ! " 1= 1= 1 A: In this way we obtain two decoupled equations for the unknown vectors !

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