## LOCATING THE ZEROS OF PARTIAL SUMS OF e z WITH RIEMANN-HILBERT METHODS (709)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Kriecherbauer709locatingthe,

author = {T. Kriecherbauer and A. B. J. Kuijlaars and K. D. T-r Mclaughlin and P. D. Miller},

title = {LOCATING THE ZEROS OF PARTIAL SUMS OF e z WITH RIEMANN-HILBERT METHODS},

year = {709}

}

### OpenURL

### Abstract

Dedicated to Percy Deift with gratitude and admiration. Abstract. In this paper we derive uniform asymptotic expansions for the partial sums of the exponential series. We indicate how this information will be used in a later publication to obtain full and explicitly computable asymptotic expansions with error bounds for all zeros of the Taylor polynomials pn−1(z) = Pn−1 k=0 zk / k!. Our proof is based on a representation of pn−1(nz) in terms of an integral of the form R e γ nφ(s) ds. We demonstrate how to derive uniform s−z expansions for such integrals using a Riemann-Hilbert approach. A comparison with classical steepest descent analysis shows the advantages of the Riemann-Hilbert analysis in particular for points z that are close to the critical points of φ. 1.

### Citations

173 | A steepest descent method for oscillatory Riemann–Hilbert problems
- Deift, Zhou
- 1993
(Show Context)
Citation Context ...antages of the RiemannHilbert analysis in particular for points z that are close to the critical points of φ. 1. Introduction During the past fifteen years and largely due to the ground breaking work =-=[5]-=-, [6] of Deift and Zhou, Riemann-Hilbert problems have become a powerful tool in asymptotic analysis with applications in many fields such as inverse scattering theory, integrable PDE’s, orthogonal po... |

47 |
On the Solvability of Painlevé
- Fokas, Zhou
- 1992
(Show Context)
Citation Context ...es of the RiemannHilbert analysis in particular for points z that are close to the critical points of φ. 1. Introduction During the past fifteen years and largely due to the ground breaking work [5], =-=[6]-=- of Deift and Zhou, Riemann-Hilbert problems have become a powerful tool in asymptotic analysis with applications in many fields such as inverse scattering theory, integrable PDE’s, orthogonal polynom... |

24 |
Uber eine Eigenschaft der Exponentialreihe
- Szego
- 1924
(Show Context)
Citation Context ... pn(z) := 1 + z + · · · + zn n! the partial sums of the exponential series. The problem to describe the asymptotic distribution of the zeros of pn was posed and solved in the classical paper of Szegő =-=[11]-=-. He proved that the zeros of pn, divided by n, converge in the limit n → ∞ to some curve D∞, now called Szegő curve, which consists of all complex numbers |z| ≤ 1 that satisfy the equation |ze1−z| = ... |

9 |
Scienti computation on mathematical problems and conjectures
- Varga
- 1990
(Show Context)
Citation Context ... of size O(1/ √ n) the rescaled polynomials pn(n + w √ n) can asymptotically be expressed in terms of the complementary error function erfc. Carpenter, Varga, and Waldvogel [4] (see also [13] and see =-=[12]-=- for an interesting discussion of zeros of the partial sums of ez ) then provided an asymptotic expansion of pn(nz) in compact subsets of C \ {z0}. These results were used in [4] to obtain lower and u... |

8 |
Complex zeros of the error function and of the complementary error function
- Fettis, Caslin, et al.
- 1973
(Show Context)
Citation Context ... s ∈ U0 ∩ γ . Pn is thus a local solution of the Riemann-Hilbert problem (RHP)1 in U0. Moreover, Pn is of a rather explicit nature, since h is related to the well studied complementary error function =-=[7]-=- (see also (4.5) below) and the Taylor coefficients of λ can8 T. KRIECHERBAUER, A. B. J. KUIJLAARS, K. D. T-R MCLAUGHLIN, AND P. D. MILLER be computed explicitly at ξ = 0 to all orders from the defin... |

8 |
Applied Asymptotic Analysis
- Miller
- 2006
(Show Context)
Citation Context ...n Theorems 4.1, 4.2 we state our results for the zeros of pn−1 in the upper half plane. 2. Classical steepest descent analysis In order to remind ourselves of the method of steepest descent (see also =-=[8]-=- for an elementary exposition) we first determine the large n asymptotics of a quantity that is related to Fn defined in (1.1) but is somewhat simpler to analyze. Let (2.1) Gn := 1 2πi ∫ γ e nφ(s) ds ... |

7 |
Asymptotics for the zeros of the partial sums of e z
- Carpenter, Varga, et al.
- 1991
(Show Context)
Citation Context ...that in neighborhoods of z0 of size O(1/ √ n) the rescaled polynomials pn(n + w √ n) can asymptotically be expressed in terms of the complementary error function erfc. Carpenter, Varga, and Waldvogel =-=[4]-=- (see also [13] and see [12] for an interesting discussion of zeros of the partial sums of ez ) then provided an asymptotic expansion of pn(nz) in compact subsets of C \ {z0}. These results were used ... |

5 |
A characterization of the exponential series
- Buckholtz
(Show Context)
Citation Context ...N-HILBERT 3 limiting distribution of the rescaled zeros on D∞. A first result presenting error bounds on the distance between the Szegő curve and the zeros of pn(nz) has been established by Buckholtz =-=[3]-=- who showed that they are located in the exterior of D∞ at a distance of at most 2e/ √ n. For each zero of pn(nz) its distance from D∞ is measured by the minimal distance between the zero and all the ... |

4 |
The zeros of the partial sums of the exponential function
- Newman, Rivlin
- 1972
(Show Context)
Citation Context ... Szegő curve. Subsequently more detailed asymptotics of pn(nz) have been derived. It turns out that z0 = 1 is a critical point where the asymptotic behavior changes. It was shown by Newman and Rivlin =-=[9]-=- that in neighborhoods of z0 of size O(1/ √ n) the rescaled polynomials pn(n + w √ n) can asymptotically be expressed in terms of the complementary error function erfc. Carpenter, Varga, and Waldvogel... |

4 |
On zero distribution of sections and tails of power series
- Ostrovskii
(Show Context)
Citation Context ...os of pn . Related results on the zeros of the partial sums of cos, sin and of more general sums of exponential functions can be found in [2] and in references therein. Finally, we mention the review =-=[10]-=- on the behavior of the zeros of more general sections and tails of power series. The main novel result proved in the present paper is Theorem 3.3 which provides an explicitly computable asymptotic ex... |

3 |
Zeros of the partial sums of cos(z) and sin(z
- Varga, Carpenter
(Show Context)
Citation Context ...orhoods of z0 of size O(1/ √ n) the rescaled polynomials pn(n + w √ n) can asymptotically be expressed in terms of the complementary error function erfc. Carpenter, Varga, and Waldvogel [4] (see also =-=[13]-=- and see [12] for an interesting discussion of zeros of the partial sums of ez ) then provided an asymptotic expansion of pn(nz) in compact subsets of C \ {z0}. These results were used in [4] to obtai... |

2 | Zeros of sections of exponential sums
- Bleher, Mallison
(Show Context)
Citation Context ...compact subsets of C \ {z0}. These results were used in [4] to obtain lower and upper bounds on the distance of the zeros from the Szegő curve. Note that up to the recent paper by Bleher and Mallison =-=[2]-=- no uniform asymptotics for pn(nz) and its zeros were available in a fixed size neighborhood of the critical point z0. In a different direction and using methods from logarithmic potential theory Andr... |

1 |
Angular distribution of zeros of the partial sums of e z via the solution of inverse logarithmic potential problem
- Andrievskii, Carpenter, et al.
(Show Context)
Citation Context ...n(nz) and its zeros were available in a fixed size neighborhood of the critical point z0. In a different direction and using methods from logarithmic potential theory Andrievskii, Carpenter and Varga =-=[1]-=- recently extended the results of Szegő [11] on the angular distribution of the zeros by proving uniform error bounds for regions including z0. We refer the reader to the recent review [14] for a desc... |

1 |
On the zeroes of the N- th partial sum of the exponential series, The American Mathematical Monthly 112 (2005), no. 10, 891–909. Fakultät für Mathematik, Universität Bochum, Universitätsstr
- Zemyan
(Show Context)
Citation Context ...r and Varga [1] recently extended the results of Szegő [11] on the angular distribution of the zeros by proving uniform error bounds for regions including z0. We refer the reader to the recent review =-=[14]-=- for a description of more results on the zeros of pn . Related results on the zeros of the partial sums of cos, sin and of more general sums of exponential functions can be found in [2] and in refere... |