       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

```

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