## Some (2007)

### BibTeX

@MISC{Dominici07some,

author = {Diego Dominici},

title = {Some},

year = {2007}

}

### OpenURL

### Abstract

properties of the inverse error function

### Citations

20 |
Uniform asymptotic smoothing of Stokes’s discontinuities
- Berry
(Show Context)
Citation Context ...atical physics, e.g., probability and statistics [Wal50], data analysis [Her88], heat conduction [Jae46], etc. It plays a fundamental role in asymptotic expansions [Olv97] and exponential asymptotics =-=[Ber89]-=-. Its inverse, which we will denote by inverf (z), inverf (z) = erf −1 (z), appears in multiple areas of mathematics and the natural sciences. A few examples include concentration-dependent diffusion ... |

10 |
Numerical solution of equations of the diffusion type with diffusivity concentration-dependent
- Philip
- 1955
(Show Context)
Citation Context ...erse, which we will denote by inverf (z), inverf (z) = erf −1 (z), appears in multiple areas of mathematics and the natural sciences. A few examples include concentration-dependent diffusion problems =-=[Phi55]-=-, [Sha73], solutions to Einstein’s scalar-field equations [LW95], chemical potentials [TM96], the distribution of lifetimes in coherent-noise models [WM99], diffusion rates in tree-ring chemistry [BKS... |

9 |
Rigorous WKB for finite-order linear recurrence relations with smooth coefficients
- Costin, Costin
- 1996
(Show Context)
Citation Context ...ew techniques in the computation of inverf (z) [Loz96]. In this paper, we analyze the asymptotic behavior of the derivatives dn dzn inverf (z) ∣ z=0 for large values of n, using a discrete WKB method =-=[CC96]-=-. In Section 2 we present 1991 Mathematics Subject Classification. Primary 33B20; Secondary 30B10, 34K25. Key words and phrases. Inverse error function, asymptotic analysis, discrete ray method, diffe... |

8 |
The inverse of the error function
- Carlitz
- 1963
(Show Context)
Citation Context ...nverf (z) and review our previous work on nested derivatives. In Section 3 we study a family of polynomials Pn(x) associated with the derivatives of inverf (z), which were introduced by L. Carlitz in =-=[Car63]-=-. Theorem 3.3 contains our main result on the asymptotic analysis of Pn(x). In Section 4 we give asymptotic approximations for dn dzn inverf (z) ∣ and some numerical z=0 results testing the accuracy o... |

8 |
A stable algorithm for computing the inverse error function in the “tail-end” region
- Fettis
- 1974
(Show Context)
Citation Context ...udied the function inverf (z) (see [Dom03b] and references therein), little is known about its analytic properties, the major work having been done in developing algorithms for numerical calculations =-=[Fet74]-=-. Dan Lozier, remarked the need for new techniques in the computation of inverf (z) [Loz96]. In this paper, we analyze the asymptotic behavior of the derivatives dn dzn inverf (z) ∣ z=0 for large valu... |

7 | Software needs in special functions
- Lozier
- 1996
(Show Context)
Citation Context ...its analytic properties, the major work having been done in developing algorithms for numerical calculations [Fet74]. Dan Lozier, remarked the need for new techniques in the computation of inverf (z) =-=[Loz96]-=-. In this paper, we analyze the asymptotic behavior of the derivatives dn dzn inverf (z) ∣ z=0 for large values of n, using a discrete WKB method [CC96]. In Section 2 we present 1991 Mathematics Subje... |

6 |
The inverse of the cumulative standard normal probability function
- Dominici
(Show Context)
Citation Context ...imes in coherent-noise models [WM99], diffusion rates in tree-ring chemistry [BKSH99] and 3D freehand ultrasound imaging [SJEMFAL+ 03]. Although some authors have studied the function inverf (z) (see =-=[Dom03b]-=- and references therein), little is known about its analytic properties, the major work having been done in developing algorithms for numerical calculations [Fet74]. Dan Lozier, remarked the need for ... |

4 |
Intraring variability of Cr, As, Cd, and Pb in red oak revealed by secondary ion mass spectrometry: Implications for environmental biomonitoring
- Brabander, Keon, et al.
- 1999
(Show Context)
Citation Context ...i55], [Sha73], solutions to Einstein’s scalar-field equations [LW95], chemical potentials [TM96], the distribution of lifetimes in coherent-noise models [WM99], diffusion rates in tree-ring chemistry =-=[BKSH99]-=- and 3D freehand ultrasound imaging [SJEMFAL+ 03]. Although some authors have studied the function inverf (z) (see [Dom03b] and references therein), little is known about its analytic properties, the ... |

4 | Nested derivatives: A simple method for computing series expansions of inverse functions
- Dominici
- 2003
(Show Context)
Citation Context ...gher derivatives of inverf (z), the nonlinearity of the recurrence makes it hard to analyze the asymptotic behavior of dn as n → ∞. Instead, we shall use an alternative technique that we developed in =-=[Dom03a]-=- and we called the method of ”nested derivatives”. The following theorem contains the main result presented in [Dom03a]. Theorem 2.4. Let Then, H(x) = h −1 (x), f(x) = 1 h ′ (x) , z0 = h(x0), |f(x0)| ... |

4 |
Asymptotics and special functions (AKP Classics), 2nd edition
- Olver
- 1997
(Show Context)
Citation Context ...ch of applied mathematics and mathematical physics, e.g., probability and statistics [Wal50], data analysis [Her88], heat conduction [Jae46], etc. It plays a fundamental role in asymptotic expansions =-=[Olv97]-=- and exponential asymptotics [Ber89]. Its inverse, which we will denote by inverf (z), inverf (z) = erf −1 (z), appears in multiple areas of mathematics and the natural sciences. A few examples includ... |

2 |
The role of error-functions in order to obtain relatively optimal classification. In Classification and related methods of data analysis
- Herden
- 1987
(Show Context)
Citation Context ...on erf(z), defined by erf(z) = 2 ∫z √ exp π ( −t 2) dt, 0 occurs widely in almost every branch of applied mathematics and mathematical physics, e.g., probability and statistics [Wal50], data analysis =-=[Her88]-=-, heat conduction [Jae46], etc. It plays a fundamental role in asymptotic expansions [Olv97] and exponential asymptotics [Ber89]. Its inverse, which we will denote by inverf (z), inverf (z) = erf −1 (... |

2 |
Some applications of the repeated integrals of the error function
- Jaeger
- 1946
(Show Context)
Citation Context ...(z) = 2 ∫z √ exp π ( −t 2) dt, 0 occurs widely in almost every branch of applied mathematics and mathematical physics, e.g., probability and statistics [Wal50], data analysis [Her88], heat conduction =-=[Jae46]-=-, etc. It plays a fundamental role in asymptotic expansions [Olv97] and exponential asymptotics [Ber89]. Its inverse, which we will denote by inverf (z), inverf (z) = erf −1 (z), appears in multiple a... |

2 |
Andante regime of scalar field dynamics in the chaotic inflationary universe. Phys
- Lidsey, Waga
- 1995
(Show Context)
Citation Context ..., appears in multiple areas of mathematics and the natural sciences. A few examples include concentration-dependent diffusion problems [Phi55], [Sha73], solutions to Einstein’s scalar-field equations =-=[LW95]-=-, chemical potentials [TM96], the distribution of lifetimes in coherent-noise models [WM99], diffusion rates in tree-ring chemistry [BKSH99] and 3D freehand ultrasound imaging [SJEMFAL+ 03]. Although ... |

2 |
Exact solutions for concentration dependent diffusion and the inverse complementary error function
- Shampine
- 1973
(Show Context)
Citation Context ...ch we will denote by inverf (z), inverf (z) = erf −1 (z), appears in multiple areas of mathematics and the natural sciences. A few examples include concentration-dependent diffusion problems [Phi55], =-=[Sha73]-=-, solutions to Einstein’s scalar-field equations [LW95], chemical potentials [TM96], the distribution of lifetimes in coherent-noise models [WM99], diffusion rates in tree-ring chemistry [BKSH99] and ... |

2 |
Transient response and the central limit theorem of probability
- Wallman
- 1950
(Show Context)
Citation Context ...uction The error function erf(z), defined by erf(z) = 2 ∫z √ exp π ( −t 2) dt, 0 occurs widely in almost every branch of applied mathematics and mathematical physics, e.g., probability and statistics =-=[Wal50]-=-, data analysis [Her88], heat conduction [Jae46], etc. It plays a fundamental role in asymptotic expansions [Olv97] and exponential asymptotics [Ber89]. Its inverse, which we will denote by inverf (z)... |

2 |
Lifetimes of agents under external stress. Phys
- Wilke, Martinetz
- 1999
(Show Context)
Citation Context ... concentration-dependent diffusion problems [Phi55], [Sha73], solutions to Einstein’s scalar-field equations [LW95], chemical potentials [TM96], the distribution of lifetimes in coherent-noise models =-=[WM99]-=-, diffusion rates in tree-ring chemistry [BKSH99] and 3D freehand ultrasound imaging [SJEMFAL+ 03]. Although some authors have studied the function inverf (z) (see [Dom03b] and references therein), li... |

1 |
Consideration on the potential-composition relationships observed with amorphous intercalation systems such as lixwo3, Solid State Ionics 84
- Kudo, Hibino
- 1996
(Show Context)
Citation Context ...of mathematics and the natural sciences. A few examples include concentration-dependent diffusion problems [Phi55], [Sha73], solutions to Einstein’s scalar-field equations [LW95], chemical potentials =-=[TM96]-=-, the distribution of lifetimes in coherent-noise models [WM99], diffusion rates in tree-ring chemistry [BKSH99] and 3D freehand ultrasound imaging [SJEMFAL+ 03]. Although some authors have studied th... |