Applied Quaternionic Analysis by Kravchenko V. V.

By Kravchenko V. V.

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It is clear that for “reasonably good” functions (such as those satisfying the conditions of the above mentioned theorems) the integral K f is a continuous function (even more, its derivatives of any order exist), but on the boundary the kernel of the integral, the function K (x y), has a singularity which leaves no hope for the continuity of K f in the points of . The following fundamental fact, which is an analogue 34 2. ELEMENTS OF QUATERNIONIC ANALYSIS of the Plemelj-Sokhotski formulas from complex analysis, makes this completely clear.

In other words each function u 2 ker( + 2 ) can be repre- sented in a unique way as the following sum u = f + g; where f 2 ker D and g 2 ker D : For a given u the corresponding functions f and g are found easily. Namely, f= 1 D 2 u and g= 1 D u: 2 This fact is true also for scalar solutions u0 of the Helmholtz equation. 5. 27), we obtain the following asymptotic equality for u0 : ix ix f+ g+o jxj jxj u0 = ix jxj = + ix jxj 1 grad u0 2 = 1 u0 + 2 1 1 grad u0 + u0 2 2 = 1 jxj +o i x grad u0 + o jxj 1 jxj 1 jxj = ; which gives us the Sommerfeld radiation condition: i u0 (x) x ; grad u0 (x) jxj =o 1 jxj ; when jxj !

The next step is to obtain the Cauchy integral formula for null-solutions of the operator D in the exterior domain R3 n . Let us …rst discuss some heuristic arguments leading to the notion of a radiation condition at the in…nity. 19) ( + 2 )u(x) = (x); x 2 R3 ; which mathematically de…nes the fundamental solution of the Helmholtz operator. 19) has a physical sense we must remember that it describes a monochromatic wave generated by a point source situated in the origin. It is physically reasonable to require that u decrease at in…nity, which ensures …nite energy of the propagation process.

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