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Hyperdeterminants. I. (English) Zbl 0866.65011
Hyperasymptotic expansions are given in terms of certain multiple integrals, the so-called hyperdeterminants. In the first section, the author gives the definition of the hyperdeterminants, and obtains several simple properties. Next section deals with an illustration of the role of the hyperdeterminants in global hyperasymptotic expansions. The main section contains a new integral representations for the hyperdeterminants. With these integral representations, the author is able to obtain convergent series expansions for the hyperdeterminants. These convergent expansions can be used to compute the hyperdeterminants to arbitrary precision.

##### MSC:
 65D20 Computation of special functions, construction of tables 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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##### References:
 [1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Nat. bur. Standards appl. Math. series 55 (1964) · Zbl 0171.38503 [2] Berry, M. V.; Howls, C. J.: Hyperasymptotics. Proc. roy. Soc. London ser. A 430, 653-668 (1990) [3] Berry, M. V.; Howls, C. J.: Hyperasymptotics for integrals with saddles. Proc. roy. Soc. London, ser. A 434, 657-675 (1991) · Zbl 0764.30031 [4] Howls, C. J.: Hyperasymptotics for integrals with finite endpoints. Proc. roy. Soc. London ser. A 439, 373-396 (1992) · Zbl 0773.30040 [5] Daalhuis, A. B. Olde: Hyperasymptotic expansions of confluent hypergeometric functions. IMA J. Appl. math. 49, 203-216 (1992) · Zbl 0781.33002 [6] Daalhuis, A. B. Olde: Hyperasymptotics and the Stokes phenomenon. Proc. roy. Soc. Edinburgh 123A, 731-743 (1993) · Zbl 0786.33004 [7] Daalhuis, A. B. Olde: Hyperasymptotic solutions of second-order linear differential equations II. Methods appl. Anal. 2, 198-211 (1995) · Zbl 0847.34061 [8] Daalhuis, A. B. Olde; Olver, F. W. J.: Hyperasymptotic solutions of second-order linear differential equations I. Methods appl. Anal. 2, 173-197 (1995) · Zbl 0847.34060 [9] Temme, N. M.: The numerical computation of the confluent hypergeometric function $U(a, b, z)$. Numer. math. 41, 63-82 (1983) · Zbl 0489.33001 [10] Temme, N. M.: Special functions: an introduction to the classical functions of mathematical physics. (1996) · Zbl 0856.33001 [11] N.M. Temme, Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters, Methods Appl. Anal., submitted. · Zbl 0863.33002 [12] Titchmarsh, E. C.: The theory of functions. (1939) · Zbl 0022.14602