       # Re: definite integral bug (3.0)

• To: mathgroup@smc.vnet.net
• Subject: [mg12104] Re: definite integral bug (3.0)
• From: Bill Bertram <wkb@ansto.gov.au>
• Date: Sat, 25 Apr 1998 01:30:35 -0400
• Organization: ANSTO
• References: <6hjslb\$2pd@smc.vnet.net>

```Richard Easther wrote:
>
> Hi - I recently tried to persuade Mathematica to do the following
> definite integral,
>
>    Integrate[ Exp[-x^2] 1/(1 + Exp[  x]),{x,0,Infinity}]
>
> and it gives the answer
>
>                          Log[x]                 1 + Log[x]
>         PolyGamma[0, 1 + ------] - PolyGamma[0, ----------]
>                            2                        2
>        ---------------------------------------------------
>                                  2
>

This is most disturbing! Whereas for a one off integral of this kind one
could use NIntegate rather than Integrate, if one needed to define a
function of z as,

f[z_]:=Integrate[ Exp[-( z t)^2] /(1 + Exp[t]),{t,0,Infinity}]

to evaluate this for different values of z, then there is a problem. For
example evaluation of f yields,

1/2*(-PolyGamma[0, 1/2 + 2*Log[t]] +
PolyGamma[0, 1 + 2*Log[t]])

This is clearly rubbish and must indicate a bug in this version of
Mathematica.

On the other hand the simpler function,

g[z_]:=Integrate[ Exp[-( z t)^2] ,{t,0,Infinity}]

is OK, eg g -> Sqrt[Pi}/6.

Cheers,
Bill

```

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