Re: Constant function Integrate Assumption

• To: mathgroup at smc.vnet.net
• Subject: [mg47218] Re: Constant function Integrate Assumption
• From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
• Date: Wed, 31 Mar 2004 02:57:42 -0500 (EST)
• References: <c4bdnf\$6tr\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```dsr14 at Cornell.edu (Daryl Reece) wrote:
> I am trying to use units within an Integration and would like
> Mathematica to understand that the units (Month) are not a function of
> the integration variable, but I have not had any luck with my
> attempts.
>
> Specifically
>
> Integrate[UnitStep[t Months], t]
>
> should yield
>
> t UnitStep[t]

Not exactly. I'll explain in a moment.

> but I get the unevaluated form, because Mathematica does not know how to
> assume that Month is not a function of t.

Yes, you get an unevaluated form, but that's not due to Mathematica
thinking that Month (or Months, whichever) might be a function of t.
Consider, for example,

In[1]:= Integrate[t Month, t]

Out[1]= (Month t^2)/2

showing that Mathematica assumes, just as you wish, that Month is
independent of t.

> I've tried Assumptions, SetAttribute, Replace...to no avail.  I've
> also hunted extensively online (Wolfram, this group and MathGroup)
> without any results, so any help would be greatly appreciated.

The problem with something like Integrate[UnitStep[t Month], t] seems to
be that we cannot convince Mathematica that Month is positive unless we
actually specify a numerical value for it. Note first that

In[2]:= Integrate[UnitStep[Pi t], t]

Out[2]= t UnitStep[t]

works as desired because Mathematica knows that Pi>0. We also see that

In[3]:= a=-3; Integrate[UnitStep[a t], t]

Out[3]= t-t UnitStep[t]

is correct because Mathematica knows that a<0.

Prior to your posting, I would have thought that the following would work:

In[4]:= Clear[a]; Assuming[a>0,Integrate[UnitStep[a t], t]]

Out[4]= Integrate[UnitStep[a t], t]

but we see, alas, that it doesn't work. So can someone suggest a way to
get Integrate[UnitStep[t Month], t] to evaluate to t UnitStep[t] without
having to assign a specific positive value to Month?

David Cantrell

```

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