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A bug in Integrate?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg22810] A bug in Integrate?
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Fri, 31 Mar 2000 01:01:13 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

While replying to another posting (Re: [mg22760] Got a trouble with the
Limit[]) I was trying to think of a way to get Mathematica to effectively
tell us that Lim[a^-x,x->Infinity] is 0 if a>1. One indirect way to do this
is to make Mathematica evaluate

Integrate[Log[a]*a^x, {x, 1, Infinity}, Assumptions -> {Im[a] == 0}] +
Integrate[Log[a]*a^x, x] /. x -> 1

I was expecting to get an answer of the form If[a>1,...]  but this is what
happened:

In[1]:=
Integrate[Log[a]*a^x, {x, 1, Infinity}, Assumptions -> {Im[a] == 0}] +
    Integrate[Log[a]*a^x, x] /. x -> 1

Out[1]=
0

This looks like a bug to me (rather than Mathematica deliberately choosing
only to consider the case when the integral exists)  particularly if you
compare it with:

In[2]:=
Integrate[x^a, {x, 1, Infinity}, Assumptions -> {Im[a] == 0}]

Out[2]=
             1                a
If[a < -1, ------, Integrate[x , {x, 1, Infinity}]]
           -a - 1
--
Andrzej Kozlowski
Toyama International University
Toyama, Japan
http://sigma.tuins.ac.jp/




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