Abels Equation And the Cauchy Integral Equation of the Second Kind

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Abels Equation And the Cauchy Integral Equation of the Second Kind
Arthur S Peters
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, / 4^^ ^, ^ / i-il^^l^ (1-^) 1 {i-?)^|^:MiM -X '^f (x) TT cot 77r + r; / g 7- Therefore by using this transformation, which we will denote by T, equation (2. L6) can be expressed as (2. 17) ], \.^]> [5]) the solution of (3-1) can be reduced to the solution of a Hilbert-Riemann boundary value problem. The reduction, as well as the solution of the subsidiary problem, is based on the use of an ingenious idea due to Carleman [6], namely the intro- duction of the function F(w) = f ii^i^ L 13 where ...w is unrestricted; and the use of the Plemelj formulas. This method for finding the solution of (3-1) is very elegantly effective and it Is perhaps Indispensable for finding the Integral representation for the solution of a Cauchy Integral equation of the third kind. For these reasons and others, the method Is likely to remain In Its position of preeminence. However, apart from an analysis of the properties of ^{z) which will Insure the existence of the Cauchy principal value for all or almost all values of C, it could be claimed that the Ideas mentioned above require the development of a function theoretic apparatus which is unnecessarily advanced and powerful for the solution of the real equation (2.

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