By Vinogradov et al.

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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 [55]. 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 [32]).

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.

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Canonical Problems in Scattering and Potential Theory Part by Vinogradov et al.
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