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18
Inequalities for zerobalanced hypergeometric functions
 Trans. Amer. Math. Soc
, 1995
"... Abstract. The authors study certain monotoneity and convexity properties of the Gaussian hypergeometric function and those of the Euler gamma function. 1. ..."
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Abstract. The authors study certain monotoneity and convexity properties of the Gaussian hypergeometric function and those of the Euler gamma function. 1.
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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Cited by 15 (1 self)
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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.
Counting Your Customers’ the Easy Way: An Alternative to the Pareto/NBD Model,” Working Paper, Wharton Marketing Department
, 2004
"... “Counting Your Customers ” the Easy Way: An Alternative to the Pareto/NBD Model Today’s managers are very interested in predicting the future purchasing patterns of their customers, which can then serve as an input into “lifetime value ” calculations. Among the models that provide such capabilities, ..."
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Cited by 10 (0 self)
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“Counting Your Customers ” the Easy Way: An Alternative to the Pareto/NBD Model Today’s managers are very interested in predicting the future purchasing patterns of their customers, which can then serve as an input into “lifetime value ” calculations. Among the models that provide such capabilities, the Pareto/NBD “Counting Your Customers ” framework proposed by Schmittlein, Morrison, and Colombo (1987) is highly regarded. But despite the respect it has earned, it has proven to be a difficult model to implement, particularly because of computational challenges associated with parameter estimation. We develop a new model, the betageometric/NBD (BG/NBD), which represents a slight variation in the behavioral “story ” associated with the Pareto/NBD, but it is vastly easier to implement. We show, for instance, how its parameters can be obtained quite easily in Microsoft Excel. The two models yield very similar results in a wide variety of purchasing environments, leading us to suggest that the BG/NBD could be viewed as an attractive alternative to the Pareto/NBD in most applications.
2000), Numerical and asymptotic aspects of parabolic cylinder functions
 J. Comp. Appl. Math
"... Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are mo ..."
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Cited by 7 (6 self)
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Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver’s results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered. 1991 Mathematics Subject Classification: 33C15, 41A60, 65D20.
Computing complex Airy functions by numerical quadrature
 Numer. Algorithms
, 2001
"... Integral representations are considered of solutions of the Airy differential equation Wtt  Z W  0 for computing Airy functions for complex values of Z. In a first method contour integral representations of the Airy functions are written as nonoscillating integrals for obtaining stable repr ..."
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Cited by 5 (3 self)
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Integral representations are considered of solutions of the Airy differential equation Wtt  Z W  0 for computing Airy functions for complex values of Z. In a first method contour integral representations of the Airy functions are written as nonoscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with wellknown algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.
Asymptotic approximations to truncation errors of series representations for special functions
 348 (SpringerVerlag
, 2007
"... Summary. Asymptotic approximations (n → ∞) to the truncation errors rn = ν=0 aν of infinite series ∑∞ ν=0 aν for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation ∆rn = an+1. ..."
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Cited by 4 (4 self)
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Summary. Asymptotic approximations (n → ∞) to the truncation errors rn = ν=0 aν of infinite series ∑∞ ν=0 aν for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation ∆rn = an+1. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding EulerMaclaurin formula. In the case of the other series considered – the Gaussian hypergeometric series 2F1(a, b; c; z) and the divergent asymptotic inverse power series for the exponential integral E1(z) – the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples. 1
On nonoscillating integrals for computing inhomogeneous Airy functions
 Math. Comput
, 2001
"... Abstract. Integral representations are considered of solutions of the inhomogeneous Airy differential equation w ′ ′ − zw = ±1/π. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function ..."
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Cited by 3 (3 self)
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Abstract. Integral representations are considered of solutions of the inhomogeneous Airy differential equation w ′ ′ − zw = ±1/π. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain nonoscillating integrals for complex values of z. In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results. 1.
Preconditioned GMRES for oscillatory integrals
, 2008
"... None of the existing methods for computing the oscillatory integral ∫ b a f(x)e iωg(x) dx achieve all of the following properties: high asymptotic order, stability, avoiding the computation of the path of steepest descent and insensitivity to oscillations in f. We present a new method that satisfie ..."
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Cited by 2 (2 self)
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None of the existing methods for computing the oscillatory integral ∫ b a f(x)e iωg(x) dx achieve all of the following properties: high asymptotic order, stability, avoiding the computation of the path of steepest descent and insensitivity to oscillations in f. We present a new method that satisfies these properties, based on applying the gmres algorithm to a preconditioned differential operator.
GEOMETRIC PROPERTIES OF QUASICONFORMAL MAPS AND SPECIAL FUNCTIONS
, 2008
"... Our goal is to provide a survey of some topics in quasiconformal analysis of current interest. We try to emphasize ideas and leave proofs and technicalities aside. Several easily stated open problems are given. Most of the results are joint work with several coauthors. In particular, we adopt result ..."
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Cited by 2 (1 self)
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Our goal is to provide a survey of some topics in quasiconformal analysis of current interest. We try to emphasize ideas and leave proofs and technicalities aside. Several easily stated open problems are given. Most of the results are joint work with several coauthors. In particular, we adopt results from the book authored by AndersonVamanamurthyVuorinen [AVV6]. Part 1. Quasiconformal maps and spheres Part 2. Conformal invariants and special functions Part 3. Recent results on special functions
Guaranteed Precision for Transcendental and Algebraic Computation made Easy
, 2006
"... Dedicated to the friends and families who blessed and supported me iv ..."
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Dedicated to the friends and families who blessed and supported me iv