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Incomplete gamma functions for large values of their variables. (English) Zbl 1076.33001
The authors derive simple asymptotic expansions of the incomplete gamma functions $\Gamma(a,z)$ and $\gamma(a,z)$ for large $a$ and $z$. They write $\Gamma(a,z)$ or $\gamma(a,z)$ as an exponential factor times another factor. This second factor is expanded at the asymptotically relevant point of the exponential factor (saddle point or end point). They require four different expansions to cover the region $(a,z)\in C\times (C-R^{-})$ depending on the value of $a-z$. Three of them valid away from the transition point $a=z$ in the regions (i) $R(a)<0$, (ii) $R(a)>-1$, $R(a)>R(z)$ and (iii) $R(a)>-1$, $R(a)<R(z)$ correspondingly and the fourth one valids around the transition point $a=z$.

##### MSC:
 33B20 Incomplete beta and gamma functions 33F05 Numerical approximation and evaluation of special functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
##### Keywords:
incomplete gamma functions; asymptotic expansions
Full Text:
##### References:
 [1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions. (1970) · Zbl 0171.38503 [2] Gautschi, W.: Exponential integral $\int 1\infty e$ - xtt - ndt for large values of n. J. res. Natl. bur. Standards 62, 123-125 (1959) · Zbl 0118.32604 [3] López, J. L.; Temme, N. M.: Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. appl. Math. 103, 241-258 (1999) · Zbl 1006.41003 [4] López, J. L.; Temme, N. M.: The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis. J. comput. Appl. math. 133, 623-633 (2001) · Zbl 0990.33010 [5] J.L. López, N.M. Temme, Convergent asymptotic expansions of Charlier, Laguerre and Jacobi polynomials, Proc. Roy. Soc. Edinburgh Sect. A, in press [6] Mahler, K.: Ueber die nullstellen der unvollstaendigen gammafunktionen. Rend. circ. Mat. Palermo 54, 1-41 (1930) · Zbl 56.0310.01 [7] Olver, F. W. J.: Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms. SIAM J. Math. anal. 22, 1475-1489 (1991) · Zbl 0738.41030 [8] Paris, R. B.: A uniform asymptotic expansion for the incomplete gamma function. J. comput. Appl. math. 148, 323-339 (2002) · Zbl 1013.33002 [9] Temme, N. M.: The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. anal. 10, 757-766 (1979) · Zbl 0412.33001 [10] 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 [11] Temme, N. M.: Special functions: an introduction to the classical functions of mathematical physics. (1996) · Zbl 0856.33001 [12] Tricomi, F. G.: Asymptotische eigenschaften der unvollständigen gammafunktion. Math. Z. 53, 136-148 (1950) · Zbl 0038.22105