Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences · 1955 · 4.3K citations · 3 references
Natural SciencesMathematicsPhysicsDifferential EquationsStochastic ProcessesTheoretical PhysicsIntegral EquationsSpecial FunctionsAsymptotic ExpansionsBessel FunctionsHypergeometric Distribution
The paper focuses on evaluating and tabulating integrals of Lipschitz–Hankel type involving products of Bessel functions. The authors aim to derive a formula expressing these integrals via a hypergeometric function. They evaluate the integral in common cases, derive approximate expansions for small \(b/a\), develop recurrence relations between integrals, and provide tables of values. The resulting tables enable evaluation of several specific integrals when \(b/a\) and \(c/a\) are small.
This paper is concerned with the evaluation and tabulation of certain integrals of the type (* 00 I(p, v; A) = J J fa t) ) e~cttxdt. In part I of this paper, a formula is derived for the integrals in terms of an integral of a hypergeometric function. This new integral is evaluated in the particular cases which are of most frequent use in mathematical physics. By means of these results, approximate expansions are obtained for cases in which the ratio b/a is small or in which b~a and is small. In part II, recurrence relations are developed between integrals with integral values of the parameters pt, v and A. Tables are given by means of which 7(0, 0; 1), 7(0, 1; 1), 7(1, 0; 1), 7(1,1; 1), 7(0, 0 ;0), 7(1, 0;'0), 7(0, 1; 0), 7(1, 1; 0), 7(0,1; - 1 ), 7(1,0; - 1 ) and 7(1,1; - 1 ) may be evaluated for 0 < b/a ^ 2, 0 ^ c/a ^ 2.
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