## (2003)

### Abstract

The inverse of the cumulative standard normal probability function.

### Citations

3572 |
The pricing of options and corporate liabilities
- Black, Scholes
- 1973
(Show Context)
Citation Context ...theory, the theory of errors, heat conduction, biology, economics, physics, neural networks [9], etc. It plays a fundamental role in the financial mathematics, being part of the Black-Scholes formula =-=[2]-=-, and its inverse is used in computing the implied volatility of an option [8]. Yet, little is known about the properties of the inverse function, e.g., series expansions, asymptotic behavior, integra... |

429 |
Special functions and their applications
- Lebedev
- 1965
(Show Context)
Citation Context ... [ e 1 2 x2 D (n−1) (e x2 2 ) ] x2 d dx [D(n) (e x2 2 )]+ ] 2 ) Summary 14 We conclude this section by comparing the properties of Pn(x) with the well known formulas for the Hermite polynomials Hn(x) =-=[7]-=-. Since the Hn are deeply related with the function N(x), we would expect to see some similarities between the Hn and the Pn. 7Pn(x) Hn(x) n − Pn(x) = e 2 x2D (n)(e x2 2 ) Hn(x) = (−1) nex2 d n dx ∑ ... |

127 | On the Lambert W function
- Corless, Gonnet, et al.
- 1996
(Show Context)
Citation Context ...= 2k, k ≥ 1 i=1 ∫∞ −∞ = 1 √ 2π = 0, n = 2k + 1, k ≥ 0 z n N ′ (z)dz ∫∞ −∞ z n 1 − e 2 z2 dz k∏ (2i + 1), n = 2k, k ≥ 1 i=1 2 95 Asymptotics Definition 20 We’ll denote by LW(x) the function Lambert W =-=[4]-=-, This function has the series representation [5] the derivative LW(x)e LW(x) = x (8) LW(x) = ∑ (−n) n−1 x n! n , n≥1 d LW(x) LW = dx x[1 + LW(x)] and it has the asymptotic behavior if x ̸= 0, LW(x) ∼... |

21 | A Sequence of Series for the Lambert W Function
- Corless, Jeffrey, et al.
- 1997
(Show Context)
Citation Context ... ≥ 0 z n N ′ (z)dz ∫∞ −∞ z n 1 − e 2 z2 dz k∏ (2i + 1), n = 2k, k ≥ 1 i=1 2 95 Asymptotics Definition 20 We’ll denote by LW(x) the function Lambert W [4], This function has the series representation =-=[5]-=- the derivative LW(x)e LW(x) = x (8) LW(x) = ∑ (−n) n−1 x n! n , n≥1 d LW(x) LW = dx x[1 + LW(x)] and it has the asymptotic behavior if x ̸= 0, LW(x) ∼ ln(x) − ln[ln(x)] x → ∞. Proposition 21 √ ( 1 S(... |

11 |
On the Calculation of the Inverse of the Error Function
- Strecok
- 1968
(Show Context)
Citation Context ...607-7045, USA (ddomin1@uic.edu) 1and some expressions for the derivatives and integrals. Carlitz [3], studied the arithmetic properties of the coefficients in the power series of inverfc(x). Strecok =-=[11]-=- computed the first 200 terms in the series of inverfc(x), and some expansions in series of Chebyshev polynomials. Finally, Fettis [6] studied inverfc(x) for small x, using an iterative sequence of lo... |

10 |
Rational Chebyshev approximations for the inverse of the error function
- Blair, Edwards, et al.
- 1976
(Show Context)
Citation Context ...ties of the inverse function, e.g., series expansions, asymptotic behavior, integral representations. The major work done has been in computing fast and accurate algorithms for numerical calculations =-=[1]-=-. Over the years a few articles have appeared with analytical studies of the closely related error function and its complement erf(x) = 2 √ π erfc(x) = 2 √ π ∫x 0 ∫ x ∞ e −t2 dt e −t2 dt . Philip [10]... |

8 |
The inverse of the error function
- Carlitz
- 1963
(Show Context)
Citation Context ...computer Science, University of Illinois at Chicago (m/c 249), 851 South Morgan Street, Chicago, IL 60607-7045, USA (ddomin1@uic.edu) 1and some expressions for the derivatives and integrals. Carlitz =-=[3]-=-, studied the arithmetic properties of the coefficients in the power series of inverfc(x). Strecok [11] computed the first 200 terms in the series of inverfc(x), and some expansions in series of Cheby... |

8 |
A stable algorithm for computing the inverse error function in the “tail-end” region
- Fettis
- 1974
(Show Context)
Citation Context ...s of the coefficients in the power series of inverfc(x). Strecok [11] computed the first 200 terms in the series of inverfc(x), and some expansions in series of Chebyshev polynomials. Finally, Fettis =-=[6]-=- studied inverfc(x) for small x, using an iterative sequence of logarithms. The purpose of this paper is to present some new results on the derivatives, integrals, and asymptotics of the inverse of th... |

4 |
The function inverfc θ
- Philip
- 1960
(Show Context)
Citation Context ... [1]. Over the years a few articles have appeared with analytical studies of the closely related error function and its complement erf(x) = 2 √ π erfc(x) = 2 √ π ∫x 0 ∫ x ∞ e −t2 dt e −t2 dt . Philip =-=[10]-=- introduced the notation “inverfc(x)” to denote the inverse of the complementary error function. He gave the first terms in the power series for inverfc(x), asymptotic formulas for small x in terms of... |

3 | Characterization of a class of sigmoid functions with applications to neural networks
- Menon, Mehrotra, et al.
- 1996
(Show Context)
Citation Context ...uss) distribution. It finds widespread application in almost every scientific discipline, e.g., probability theory, the theory of errors, heat conduction, biology, economics, physics, neural networks =-=[9]-=-, etc. It plays a fundamental role in the financial mathematics, being part of the Black-Scholes formula [2], and its inverse is used in computing the implied volatility of an option [8]. Yet, little ... |