## Uniform bounds for the complementary incomplete gamma function, Preprint at http://locutus.cs.dal.ca:8088/archive/00000335

Citations: | 1 - 1 self |

### BibTeX

@MISC{Borwein_uniformbounds,

author = {Jonathan M. Borwein and O-yeat Chan},

title = {Uniform bounds for the complementary incomplete gamma function, Preprint at http://locutus.cs.dal.ca:8088/archive/00000335},

year = {}

}

### OpenURL

### Abstract

Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : |z |> c|a − 1|} with a real and z complex. Our results show that within the hyperboloid, |Γ(a, z) | is of order |z | a−1 e − Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z.

### Citations

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(Show Context)
Citation Context ...r explicit upper bounds on |Γ(a, z)| in large regions of the complex z-plane in order to give effective asymptotic formulas for Laguerre Polynomials [2]. While there are many asymptotic formulas [5], =-=[7]-=-, [8], [9], and inequalities [1], [4], [6] for Γ(a, z) in the literature, these results were 1 Research supported by NSERC and the Canada Research Chair program. 1s2 not applicable in the context of [... |

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Citation Context ... related to the generalized complementary error function erfcp(x) := 2 √ π � ∞ x e −tp dt = 2 p √ π Γ(1 p , xp ). (1.4) One can find an extended and highly readable overview on γ(a, z) and Γ(a, z) in =-=[3]-=-. Our present investigation was motivated by the need for explicit upper bounds on |Γ(a, z)| in large regions of the complex z-plane in order to give effective asymptotic formulas for Laguerre Polynom... |

15 |
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Citation Context ...rge regions of the complex z-plane in order to give effective asymptotic formulas for Laguerre Polynomials [2]. While there are many asymptotic formulas [5], [7], [8], [9], and inequalities [1], [4], =-=[6]-=- for Γ(a, z) in the literature, these results were 1 Research supported by NSERC and the Canada Research Chair program. 1s2 not applicable in the context of [2] either because the errors to the asympt... |

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Citation Context ... z)| in large regions of the complex z-plane in order to give effective asymptotic formulas for Laguerre Polynomials [2]. While there are many asymptotic formulas [5], [7], [8], [9], and inequalities =-=[1]-=-, [4], [6] for Γ(a, z) in the literature, these results were 1 Research supported by NSERC and the Canada Research Chair program. 1s2 not applicable in the context of [2] either because the errors to ... |

8 |
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(Show Context)
Citation Context ... upper bounds on |Γ(a, z)| in large regions of the complex z-plane in order to give effective asymptotic formulas for Laguerre Polynomials [2]. While there are many asymptotic formulas [5], [7], [8], =-=[9]-=-, and inequalities [1], [4], [6] for Γ(a, z) in the literature, these results were 1 Research supported by NSERC and the Canada Research Chair program. 1s2 not applicable in the context of [2] either ... |

5 |
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(Show Context)
Citation Context ...in large regions of the complex z-plane in order to give effective asymptotic formulas for Laguerre Polynomials [2]. While there are many asymptotic formulas [5], [7], [8], [9], and inequalities [1], =-=[4]-=-, [6] for Γ(a, z) in the literature, these results were 1 Research supported by NSERC and the Canada Research Chair program. 1s2 not applicable in the context of [2] either because the errors to the a... |

5 | Uniform asymptotics for the incomplete gamma functions, starting from negative values of the parameters
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(Show Context)
Citation Context ...licit upper bounds on |Γ(a, z)| in large regions of the complex z-plane in order to give effective asymptotic formulas for Laguerre Polynomials [2]. While there are many asymptotic formulas [5], [7], =-=[8]-=-, [9], and inequalities [1], [4], [6] for Γ(a, z) in the literature, these results were 1 Research supported by NSERC and the Canada Research Chair program. 1s2 not applicable in the context of [2] ei... |

2 |
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Citation Context ... present investigation was motivated by the need for explicit upper bounds on |Γ(a, z)| in large regions of the complex z-plane in order to give effective asymptotic formulas for Laguerre Polynomials =-=[2]-=-. While there are many asymptotic formulas [5], [7], [8], [9], and inequalities [1], [4], [6] for Γ(a, z) in the literature, these results were 1 Research supported by NSERC and the Canada Research Ch... |

2 |
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Citation Context ...ed for explicit upper bounds on |Γ(a, z)| in large regions of the complex z-plane in order to give effective asymptotic formulas for Laguerre Polynomials [2]. While there are many asymptotic formulas =-=[5]-=-, [7], [8], [9], and inequalities [1], [4], [6] for Γ(a, z) in the literature, these results were 1 Research supported by NSERC and the Canada Research Chair program. 1s2 not applicable in the context... |