       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:=
> \$Version
>
> Out=
> "5.2 for Microsoft Windows (June 20, 2005)"
>
> In:=
> LaplaceTransform[Log[t], t, o]
> Out=
> -((EulerGamma + Log[o])/o)
>
> which is correct.
>
> However
>
> In:=
> Integrate[Log[t]*Exp[(-s)*t], {t, 0, Infinity}, GenerateConditions ->
> True]
>
> returns divergence message which is not in general truth. Try for
> example
>
> In:=
> Integrate[(Log[t]*Exp[(-#1)*t] & ) /@ {1, 4, 10}, {t, 0, Infinity}]
> Out=
> {-EulerGamma, -(EulerGamma/4) - Log/2, (1/10)*(-EulerGamma -
> Log)}
>
> Being more specifying,
>
> In:=
> Integrate[Log[t]*Exp[(-s)*t], {t, 0, Infinity}, Assumptions -> s > 0]
> Out=
> -((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:= Integrate[Log[t]*Exp[(-s)*t], {t, 0, Infinity},
GenerateConditions -> True]

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

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

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

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

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

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

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

Regards,
Jean-Marc

```

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