Re: Problem with definite integrals having symbolic limits

• To: mathgroup at smc.vnet.net
• Subject: [mg21853] Re: [mg21850] Problem with definite integrals having symbolic limits
• From: Bojan Bistrovic <bojanb at physics.odu.edu>
• Date: Wed, 2 Feb 2000 22:54:13 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

> Dear MathGroup,
>
> I discovered what I think is an inconsistency and would like to check my
> opinion and offer a suggestion to Wolfram.
> When you do a "definite" integration with symbolic limits (instead of
> concrete numbers), the result you get may not be unconditionally true. The
> necessary conditions are not stated by Mathematica, although
> "GenerateConditions" is set to True.
>
> The notebook added demonstrates this problem. I suggest that a warning
> message should be given, that describes the conditions for the stated
> result.
>
> With best regards,
> Wolter Kaper
> dept. of Chemistry
> Univ. of Amsterdam
>
The "problem" is the folowing: you use Set to define you function

In[1]:= Int[xend_] = Integrate[1/x^2, {x, -2, xend},
GenerateConditions -> True]

The right hand side is evaluated AT THIS TIME, so, since "xend" is just a
symbol, the result is calculated and assigned to your function:

Out[1]= -1/2 - 1/xend

In[2]:= ?Int
Out[2]= Global`Int
Int[xend_] = -1/2 - xend^(-1)

So when you type Int[2], you get the wrong result. What you should do is

In[3]:= Int[xend_] := Integrate[1/x^2, {x, -2, xend},
GenerateConditions -> True]

In[4]:= ?Int
Out[4]= Global`Int
Int[xend_] := Integrate[1/x^2, {x, -2, xend}, GenerateConditions -> True]

In[5]:= Int[2]
Out[5]= Infinity

Now, SHOULD the integrate return the result like:

If[xend > -2 && xend < 0, -(2 + xend)/(2*xend), Integrate[a/x^2, {x, -2, xend}]]

Well, if you try

Integrate[a/x^2, {x, -2, xend}, GenerateConditions -> True, Assumptions->{a>0}]

you get

If[xend > -2 && xend < 0, -(a*(2 + xend))/(2*xend), Integrate[a/x^2, {x, -2,
xend}]]

It looks bizzare :-). SetDelayed will solve your problem for the time, and as
for the question about inconsistency, it looks like one to me.

Bye, Bojan

--
-------------------------------------------------------------
Bojan Bistrovic,                       bojanb at physics.odu.edu
Old Dominion University, Physics Department,      Norfolk, VA
-------------------------------------------------------------

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