Computer evaluation of the complementary error function

1975 ◽  
Vol 43 (9) ◽  
pp. 833-834 ◽  
Author(s):  
Daniel L. Decker
Keyword(s):  
Error Function ◽  
Computer Evaluation ◽  
Complementary Error Function

scholarly journals Saddle points of the complementary error function

1973 ◽  
Vol 27 (122) ◽  
pp. 409-409 ◽  
Author(s):  
Henry E. Fettis ◽  
James C. Caslin ◽  
Kenneth R. Cramer
Keyword(s):  
Error Function ◽  
Saddle Points ◽  
Complementary Error Function

Hyperasymptotics for integrals with finite endpoints

1992 ◽  
Vol 439 (1906) ◽  
pp. 373-396 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Complex Plane ◽  
Airy Function ◽  
Error Function ◽  
Steepest Descent ◽  
Similar Form ◽  
Complete Asymptotic Expansion ◽  
Complementary Error Function ◽  
Plane Passing ◽  
Method Of Steepest Descent

Berry & Howls (1991) (hereinafter called BH) refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp {– kf ( z )} along doubly infinite contours in the complex plane passing over saddlepoints of f ( z ). Here we derive analogous results for integrals with integrands of a similar form, but whose local expansions in powers of 1/ k are made about the finite endpoints of semi-infinite contours of integration. We treat endpoints where f ( z ) behaves locally linearly or quadratically. Generically, local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f ( z ). We derive ‘resurgence relations’ which express the original integral exactly as a truncated endpoint expansion plus a remainder, involving the global saddle structure of f ( z ) via integrals through certain ‘adjacent’ saddles. The saddles adjacent to the endpoint are determined by a topological rule. If the least term of the endpoint expansion is the N 0 ( k ) th, summing to here calculates the endpoint integral up to an error of approximately exp ( – N 0 ( k )). We develop a scheme, involving iteration of the new resurgence relations with a similar one derived in BH, which can reduce this error down to exp( – 2.386 N 0 ( k )). This ‘hyperasymptotic’ formalism parallels that of BH and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane, provided that it does not coincide with other saddles. We illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc( x ) and a constructed ‘incomplete’ Airy function.

Comparative Study of Hydrogen Diffusion in Hot-Wire and Glow-Discharge-Deposited a-Si:H

1999 ◽  
Vol 557 ◽  
Author(s):  
J. Shinar ◽  
R. Shinar ◽  
K. E. Junge ◽  
E. Iwaniczko ◽  
A. H. Mahan ◽  
...  
Keyword(s):  
Glow Discharge ◽  
Hydrogen Diffusion ◽  
Secondary Ion Mass Spectrometry ◽  
Error Function ◽  
Interface Layer ◽  
Hot Wire ◽  
Release Times ◽  
Multiple Trapping ◽  
Complementary Error Function ◽  
Secondary Ion

AbstractLong-range atomic H motion in hot-wire deposited (HW) a-Si:H is compared directly to that in glow-discharge deposited (GD) a-Si:H by monitoring the deuterium secondary ion mass spectrometry (DSLMS) profiles in [GD a-Si:H]/[GD a-Si:(H,D)]/[HW a-Si:H] multilayers vs annealing temperature and time. While the profiles in the GD layer are in excellent agreement with complementary error-function behavior and previous studies, the profiles in the HW layer suggest that the multiple-trapping motion of the H and D atoms is much slower, possibly due to an interface layer of defects. However, an exponential “tail” of D atoms extends deep into the HW layer, probably due to a long diffusion length of mobile D atoms, consistent with the established release times of H and D from the GD layer and H loss typical during growth of HW films. The results are also discussed in terms of the H exchange model and compared to previous NMIR studies of HW a-Si:H, which suggest that most of the hydrogen in the HW layer is concentrated in H-rich clusters dispersed in a network of very low H content.

Distributions of ground movements parallel to deep excavations in clay

2006 ◽  
Vol 43 (1) ◽  
pp. 43-58 ◽  
Author(s):  
Jill Roboski ◽  
Richard J Finno
Keyword(s):  
Error Function ◽  
Ground Movement ◽  
Deep Excavation ◽  
Direction Parallel ◽  
Ground Movements ◽  
Movement Data ◽  
Empirical Procedure ◽  
Complementary Error Function ◽  
Excavation Support ◽  
Support Wall

An empirical procedure for fitting a complementary error function (erfc) to settlement and lateral ground movement data in a direction parallel to an excavation support wall is proposed based on extensive optical survey data obtained around a 12.8 m deep excavation in Chicago. The maximum ground movement and the height and length of an excavation wall define the erfc fitting function. The erfc fit is shown to apply to three other excavation projects where substantial ground movement data were reported.Key words: excavations, clays, ground movements, performance data.

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