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