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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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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.
On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing.
 Proc. Roy. Soc. London A 452
"... We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every di ..."
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Cited by 11 (5 self)
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We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every direction in the complex plane at the singularity (Stokes directions not excepted) there exists a nonempty set of solutions whose difference from the "optimally" (i.e., near the least term) truncated asymptotic series is of the same (exponentially small) order of magnitude as the least term of the series. There is a family of generalized Borel summation formulas B which commute with the usual algebraic and analytic operations (addition, multiplication, differentiation, etc). We show that there is exactly one of them, B0 , such that for any formal series solution ~ f, B0 ( ~ f) differs from the optimal truncation of ~ f by at most the order of the least term of ~ f . We show in addition that the Berry (1989) smoothing phenomenon is universal within this class of differential systems. Whenever the terms "beyond all orders" change in crossing a Stokes line, these terms vary smoothly on the Berry scale arg(x) jxj \Gamma1=2 and the transition is always given by the error function; under the same conditions we show that Dingle's rule of signs for Stokes transitions holds. 1