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MathGroup Archive 2007

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Re: integrate issue (5.2)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg77136] Re: integrate issue (5.2)
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Mon, 4 Jun 2007 03:58:30 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <f3u3u4$2kr$1@smc.vnet.net>

dimitris wrote:
> This is quite trivial so I am sure it has been pointed
> out a long time ago.
> In fact I believe that mathematica 6 does not
> show this buggy behavior.
> 
> Anyway...
> 
> In[32]:=
> $Version
> 
> Out[32]=
> "5.2 for Microsoft Windows (June 20, 2005)"
> 
> In[40]:=
> LaplaceTransform[Log[t], t, o]
> Out[40]=
> -((EulerGamma + Log[o])/o)
> 
> which is correct.
> 
> However
> 
> In[42]:=
> Integrate[Log[t]*Exp[(-s)*t], {t, 0, Infinity}, GenerateConditions ->
> True]
> 
> returns divergence message which is not in general truth. Try for
> example
> 
> In[44]:=
> Integrate[(Log[t]*Exp[(-#1)*t] & ) /@ {1, 4, 10}, {t, 0, Infinity}]
> Out[44]=
> {-EulerGamma, -(EulerGamma/4) - Log[2]/2, (1/10)*(-EulerGamma -
> Log[10])}
> 
> Being more specifying,
> 
> In[47]:=
> Integrate[Log[t]*Exp[(-s)*t], {t, 0, Infinity}, Assumptions -> s > 0]
> Out[47]=
> -((EulerGamma + Log[s])/s)
> 
> The integral converges for Re[s]>0, but Mathematica (5.2) fails to
> detect this.

Hi Dimitris,

Although this does not solve your problem, you are right in assuming 
that version 6.0 correctly handles the different cases.

In[1]:= Integrate[Log[t]*Exp[(-s)*t], {t, 0, Infinity},
    GenerateConditions -> True]

Out[1]= If[Re[s] > 0, -((EulerGamma + Log[s])/s),
  Integrate[Log[t]/E^(s*t),
      {t, 0, Infinity}, Assumptions -> Re[s] <= 0]]

In[2]:= Integrate[Log[t]*Exp[(-s)*t], {t, 0, Infinity}]

Out[2]= If[Re[s] > 0, -((EulerGamma + Log[s])/s),
  Integrate[Log[t]/E^(s*t),
      {t, 0, Infinity}, Assumptions -> Re[s] <= 0]]

In[3]:= Integrate[(Log[t]*Exp[(-#1)*t] & ) /@ {1, 4, 10}, {t, 0,
   Infinity}]

Out[3]= {-EulerGamma, -(EulerGamma/4) - Log[2]/2,
    (1/10)*(-EulerGamma - Log[10])}

In[4]:= Integrate[Log[t]*Exp[(-s)*t], {t, 0, Infinity},
  Assumptions -> s > 0]

Out[4]= -((EulerGamma + Log[s])/s)

Regards,
Jean-Marc


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