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By Kravchenko V. V.

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Notation Physical quantity Units in the mks system volume charge density coulomb/m3 j current density amper/m2 E electric …eld intensity volt/m H magnetic …eld intensity amper/m D electric induction vector coulomb/m2 B magnetic induction vector tesla=volt sec/m2 51 52 3. PHYSICAL MODELS REDUCING TO THE OPERATOR D The Maxwell equations are considered together with the so-called constitutive relations which describe the relations between the induction vectors and the …eld vectors. In general they can be written as follows D = D(E; H); B = B(E; H): The simplest interpretation of these relations is that, for instance, the electric induction D(!

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. Let us consider another boundary value problem for the operator D , the so-called jump problem. 5. THE OPERATOR D + I 39 Problem 3. 35) is ful…lled: f+ f = K [g]+ K [g] = P [g] + Q [g] = g: A much more di¢ cult problem is the analogue of the famous Riemann boundary value problem.

19). 19) in the case Im = 0 is to impose a radiation condition at in…nity. For the Helmholtz equation, this was proposed by Sommerfeld and has the following form. 21) i u(x) = o( 1 ); jxj when jxj ! 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+ .

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