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The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
Computing The Hilbert Transform Of The Generalized Laguerre And Hermite Weight Functions
, 2000
"... Explicit formulae are given for the Hilbert transform Z R \Gamma w(t)dt=(t \Gamma x), where w is either the generalized Laguerre weight function w(t) = 0 if t 0, w(t) = t ff e \Gammat if 0 ! t ! 1, and ff ? \Gamma1, x ? 0, or the Hermite weight function w(t) = e \Gammat 2 , \Gamma1 ! t ! ..."
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Explicit formulae are given for the Hilbert transform Z R \Gamma w(t)dt=(t \Gamma x), where w is either the generalized Laguerre weight function w(t) = 0 if t 0, w(t) = t ff e \Gammat if 0 ! t ! 1, and ff ? \Gamma1, x ? 0, or the Hermite weight function w(t) = e \Gammat 2 , \Gamma1 ! t ! 1, and \Gamma1 ! x ! 1. Furthermore, numerical methods of evaluation are discussed based on recursion, contour integration and saddlepoint asymptotics, and series expansions. We also study the numerical stability of the threeterm recurrence relation satisfied by the integrals Z R \Gamma n (t; w)w(t)dt=(t \Gamma x), n = 0; 1; 2; : : : , where n ( \Delta ; w) is the generalized Laguerre, resp. the Hermite, polynomial of degree n. AMS subject classification: 65D30, 65D32, 65R10. Key words: Hilbert transform, classical weight functions, computational methods. 1