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