Applied Quaternionic Analysis by Kravchenko V. V.

By Kravchenko V. V.

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1) ! t ! 2) ); 58 3. PHYSICAL MODELS REDUCING TO THE OPERATOR D ! t . ! E and H are complex vectors called the complex amplitudes of the electromagnetic …eld; ! is the frequency of oscillations. 1)! 4) we obtain the equations for the complex amplitudes E and H (the quantities and j characterizing the sources are also assumed to ! t )): ! 3) ! 4) ! rot E = i! 5) ! 6) ! 3) we …nd the relation between ! j: ! 7) and i! 1 the medium is supposed to be homogeneous. Very often " and are considered to be complex quantities.

Problem 1. (The interior Dirichlet problem for the operator D ) Given a complex quaternionic function g 2 C 0; ( ; H(C)), …nd a function f such that + D f (x) = 0; x2 f (x) = g(x); x2 . and Problem 2. 24) at in…nity. Let us analyse Problem 1 (Problem 2 can be analysed in a similar way). From Theorem 10 we see immediately that the solution of Problem 1 does not always exist because not all functions g are -extendable into + . 33). If this is the case then the solution of Problem 1, according to the Cauchy integral formula, is obtained from the Cauchy integral of g: f = K g.

1: It can be veri…ed immediately that this condition is ful…lled by u+ but not by u . We observe a similar situation in the case of the operator D . 22) ( x x 1 ) K(x) = o( ); 2 +i jxj jxj jxj when jxj ! 1: 28 2. ELEMENTS OF QUATERNIONIC ANALYSIS Let us see what happens with the function K+ . 22) is ful…lled by K+ . 22). Note that K+ is precisely the fundamental solution K used already on the preceding pages. Now we are ready to prove the Cauchy integral formula for the exterior domain. Theorem 6.

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