UnitStep warning: 2nd post

*To*: mathgroup at smc.vnet.net*Subject*: [mg53289] UnitStep warning: 2nd post*From*: jackgold at umich.edu*Date*: Tue, 4 Jan 2005 03:13:54 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Hello Everyone, I have posted earlier that UnitStep followed by Function Expand is faulty. Example 1: UnitStep[x^2]//FunctionExpand returns UnitStep[-x] + UnitStep[x] which is false at x - 0. As one might expect, Example 2: UnitStep[-x^2]//FunctionExpand returns 0 which is false at x = 0. Now the 2nd edition: Example 3: UnitStep[x^2]//FullSimplify returns 1 which is correct! However, Example 4: UnitStep[-x^2]//FullSimplify returns 0 which is incorrect. Example 5: UnitStep[1 - x^2]//FunctionExpand returns -UnitStep[-1 + x] + UnitStep[1 + x] which is incorrect at x = 1 and is correct at x = -1. Example 6: UnitStep[1-x^2]//FullSimplify performs no simplification. I repeat an earlier remark. Those responsible for these "bugs" and inconsistencies, have neglected the subtle behavior of UnitStep at points of its discontinuity. A correct examination will show the need for introducing DiscreteDelta as part of the expansion of UnitStep["polynomial"]. The first step in this is to realize that UnitStep[-x]//FunctionExpand = 1 - UnitStep[x] + DiscreteDelta[x] similarly with UnitStep[-x]//FullSimplify Jack Goldberg