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version 1.3, 2005/10/20 11:22:27

version 1.4, 2012/06/11 05:23:52

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% $OpenXM: OpenXM/src/kan96xx/Doc/slope.sm1,v 1.2 2001/06/18 03:12:21 takayama Exp $

% $OpenXM: OpenXM/src/kan96xx/Doc/slope.sm1,v 1.3 2005/10/20 11:22:27 takayama Exp $

(oxasir.sm1.loaded) boundp not {

[(parse) (oxasir.sm1) pushfile] extension

} { } ifelse

Line 582 $Imported commands: slope $ message

(When slope.geometric is one, it outputs the geometric slopes.)

(As to the algorithm, see A.Assi, F.J.Castro-Jimenez and J.M.Granger)

( How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996)

(Note that the signs of the slopes are negative, but the absolute values)

(Note that the signs of the slopes s' are negative, but the absolute values -s')

(of the slopes are returned.)

$In other words, when pF+qV is the gap, -s'=q/p is returned.$

$Note that s=1-1/s' is called the slope in recent literatures. Solutions belongs to O(s).$

$The number s satisfies 1<= s.$

$We have r=s-1=-1/s', and kappa=1/r=-s',$

$which is used 1/Gamma(1+m*r) factor and exp(-tau^kappa)$

$in the Borel and Laplace transformations respectively.$

$Example 1: [ [(x^4 Dx + 3)] (x) [0 1] [-1 1]] slope :: $

$ The solution is exp(x^(-3)). $

$Example 2: [ [(x^3 Dx^2 + (x + x^2) Dx + 1)] [(x)] $