zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A uniform asymptotic expansion for the incomplete gamma function. (English) Zbl 1013.33002
A new uniform asymptotic expansion is given for the incomplete gamma function $\Gamma(a,z)$ valid for large $z$. The expansion is compared with one developed by the reviewer [SIAM J. Math. Anal. 10, 757-766 (1979; Zbl 0412.33001)]. The coefficients of the new expansion are simpler to evaluate near the transition point $a=z$, the domain of validity for the complex parameters $a$ and $z$ is smaller. In the new expansion the error function again gives the main approximation, higher terms are related with the error function (and with parabolic cylinder functions). Numerical examples are given to illustrate the accuracy of the expansion.

MSC:
33B20Incomplete beta and gamma functions
WorldCat.org
Full Text: DOI
References:
[1] B.C. Berndt, Ramanujan’s Notebooks Part I, Springer, New York, 1985, p. 64. · Zbl 0555.10001
[2] Didonato, A. R.; Morris, A. H.: Computation of the incomplete gamma functions. ACM trans. Math. software 12, 377-393 (1986) · Zbl 0623.65016
[3] Dingle, R. B.: Asymptotic expansions: their derivation and interpretation. (1973) · Zbl 0279.41030
[4] H.P. Dopper, Asymptotische Ontwikkelingen van de Onvolledige Gammafuncties. Dissertation, University of Groningen, 1942.
[5] Erdélyi, A.; Wyman, M.: The asymptotic evaluation of certain integrals. Arch. rational mech. Anal. 14, 217-260 (1963) · Zbl 0168.37903
[6] Gautschi, W.: Exponential integral $\int 1 \infty $e-xtt-ndt for large values of n. J. res. Nat. bur. Standards 62, 123-125 (1959) · Zbl 0118.32604
[7] Gradshteyn, I. S.; Ryzhik, I. M.: Tables of integrals, series, and products. (1980) · Zbl 0521.33001
[8] Kowalenko, V.; Frankel, N. E.: Asymptotics for the Kummer function of Bose plasmas. J. math. Phys. 35, 6179-6198 (1994) · Zbl 0815.33013
[9] Mahler, K.: Ueber die nullstellen der unvollstaendigen gammafunktionen. Rend. circ. Mat. Palermo 54, 1-41 (1930) · Zbl 56.0310.01
[10] Olver, F. W. J.: The generalized exponential integral. Internat. ser. Num. math. 119, 497-510 (1994) · Zbl 0821.33001
[11] Paris, R. B.: New asymptotic formulas for the Riemann zeta function on the critical line. Proceedings of the international workshop on special functions, 21--25 June 1999, Hong Kong (2000)
[12] Paris, R. B.; Cang, S.: An asymptotic representation for ${\zeta}$( 1 2+it). Methods appl. Anal. 4, 449-470 (1997) · Zbl 0913.11033
[13] Schell, H. -J.: Asymptotische entwicklungen für die unvollständige gammafunktion. Wiss. schr. Tech. univ. Karl-marx-stadt 22, 477-485 (1980) · Zbl 0531.33003
[14] Temme, N. M.: The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. anal. 10, 757-766 (1979) · Zbl 0412.33001
[15] Temme, N. M.: Computational aspects of incomplete gamma functions with large complex parameters. Internat. ser. Num. math. 119, 551-562 (1994) · Zbl 0821.65006
[16] Temme, N. M.: Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters. Methods appl. Anal. 3, 335-344 (1996) · Zbl 0863.33002
[17] Temme, N. M.: Special functions: an introduction to the classical functions of mathematical physics. (1996) · Zbl 0856.33001
[18] Tricomi, F. G.: Asymptotische eigenschaften der unvollständigen gammafunktion. Math. Z. 53, 136-148 (1950) · Zbl 0038.22105
[19] Whittaker, E. T.; Watson, G. N.: Modern analysis. (1965) · Zbl 0108.26903
[20] Wong, R.: On uniform asymptotic expansion of definite integrals. J. approx. Theory 7, 76-86 (1973) · Zbl 0251.41011