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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.
The Gamma Function
, 2000
"... 8> z 1 ds = 2 1 z Z 1 0 t 2z 1 e t 2 =2 dt: Artin [2] showed that (z) is the only smooth logconvex extension of the factorial function, so these integrals must coincide with the functions dened by Euler's limit (z) = lim n!1 1 2 3 n z(z + 1)(z + 2) (z + n) n z (2) The autho ..."
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8> z 1 ds = 2 1 z Z 1 0 t 2z 1 e t 2 =2 dt: Artin [2] showed that (z) is the only smooth logconvex extension of the factorial function, so these integrals must coincide with the functions dened by Euler's limit (z) = lim n!1 1 2 3 n z(z + 1)(z + 2) (z + n) n z (2) The author acknowledges gratefully the support of US NSF grant DMS9626829. 1 DRAFT and by Weierstrass' innite product 1=(z) = ze z 1 Y n=1 1 + z n<F18
2. Specification
"... This function evaluates the incomplete gamma functions in the normalised form P (a; x) = 1\Gamma (a) R x 0 t a\Gamma 1e\Gamma t dt Q(a; x) = 1\Gamma (a) R 1 x t a\Gamma 1e\Gamma t dt; ..."
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This function evaluates the incomplete gamma functions in the normalised form P (a; x) = 1\Gamma (a) R x 0 t a\Gamma 1e\Gamma t dt Q(a; x) = 1\Gamma (a) R 1 x t a\Gamma 1e\Gamma t dt;
GAMDIS CERN Program Library G106 Author(s) : K.S. K"olbig Library: MATHLIBSubmitter: Submitted: 01.05.1990Language: Fortran Revised: 15.03.1993
"... Gamma Distribution Function subprogram GAMDIS calculates the gamma distribution function (incomplete gamma function) P (x; a) = 1\Gamma (a) Z x 0 e \Gamma t ta\Gamma 1 dt for real arguments x * 0 and a? 0. Structure: FUNCTION subprogramUser Entry Name: ..."
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Gamma Distribution Function subprogram GAMDIS calculates the gamma distribution function (incomplete gamma function) P (x; a) = 1\Gamma (a) Z x 0 e \Gamma t ta\Gamma 1 dt for real arguments x * 0 and a? 0. Structure: FUNCTION subprogramUser Entry Name:
Getting Started..........................................................................................................................ix
"... For contact information, please visit www.vni.com/contact ..."
Q(a, x). 2. Specification #include #include
"... nag incomplete gamma (s14bac) nag incomplete gamma (s14bac) computes values for the incomplete gamma functions P (a, x) and ..."
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nag incomplete gamma (s14bac) nag incomplete gamma (s14bac) computes values for the incomplete gamma functions P (a, x) and
CM LIFTINGS OF SUPERSINGULAR ELLIPTIC CURVES
, 904
"... Abstract. Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by OD to supersingular elliptic curves in characteristic p. In the algorithm we first determin ..."
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Abstract. Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by OD to supersingular elliptic curves in characteristic p. In the algorithm we first determine an explicit constant Dp so that D > Dp implies that the map is necessarily surjective and then we compute explicitly the cases D  < Dp. Supposant vraie la conjecture de Riemann généralisée nous présentons un algorithme qui, donné un nombre prémier p, calcule l’ensemble des discriminants fondamentaux D < 0 tels que l’application de reduction modulo un premier aux dessus p des courbes elliptiques avec multiplication complexe par OD vers les courbes elliptiques supersingular en characteristique p est surjective. Dans l’algorithme, nous d’abord determinons une borne Dp explicite, tel que D > Dp implique que l’application est necessairement surjective et puis nous calculons explicitement les cas D  < Dp. 1.