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Extra info for Canonical Problems in Scattering and Potential Theory Part I: Canonical problems in scattering and potential theory

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181) and (1. 182). These basic integral representations of Abel kind will be extensively exploited in later chapters. Other useful relationships can be found in . 7 Dual equations and single- or double-layer surface potentials Let S0 be an open surface, which is a portion of a larger closed surface S; let S1 be the complementary part of S0 in S (thus S = S0 ∪S1 ) so that S1 may be regarded as an “aperture” in S. Given S0 , the choice of S (and hence S1 ) may be made arbitrarily, but we shall require that it satisfies the hypotheses for the application of Green’s theorem (see ).

1. 160) where the notation for the Pochhammer symbol def def (a)k = a (a + 1) . . (a + k − 1) ; (a)0 = 1 (1. 161) → has been used; the upper parameters − a = (a1 , . . , ap ) are unrestricted, → − whereas the lower parameters b = (b1 , . . , bq ) are restricted so that no bj is zero or a negative integer. Note that when a is neither zero nor a negative integer, Γ (a + k) . (1. 2). If the one of upper parameters is equal to zero or a negative integer, then the series terminates and is a hypergeometric polynomial.

215), as desired.