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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 ! ..."
Abstract

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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
SUMMARY
, 706
"... A new generalization of Dawson’s integral function based on the link between a Riccati nonlinear differential equation and a secondorder ordinary differential equation is reported. The MacLaurin expansion of this generalized function is built up and to this end an explicit formula for a generic cof ..."
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A new generalization of Dawson’s integral function based on the link between a Riccati nonlinear differential equation and a secondorder ordinary differential equation is reported. The MacLaurin expansion of this generalized function is built up and to this end an explicit formula for a generic cofactor of a triangular matrix is deduced. 1.