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Re: Why two results of same eq. are different?

• To: mathgroup at smc.vnet.net
• Subject: [mg44753] Re: Why two results of same eq. are different?
• From: "Tanel Telliskivi" <maria.08.6689847 at telia.com>
• Date: Tue, 25 Nov 2003 00:45:21 -0500 (EST)
• References: <200311200816.DAA01508@smc.vnet.net> <bpkpd7$ce1$1@smc.vnet.net>
• Reply-to: "Tanel Telliskivi" <maria.08.6689847 at telia.com>
• Sender: owner-wri-mathgroup at wolfram.com

Hello,
"4.1 for Microsoft Windows (December 11, 2000)"

In[2]:=
f[a_]=Integrate[Log[2,1+10*x]*Exp[-x]*x,{x,0,a},Assumptions\[Rule]a>0]

Out[2]=
-((1/(20*Log[2]))*((20 - 20*E^a + 18*E^(1/10 + a)*
      ExpIntegralEi[-(1/10)] + 18*E^(1/10 + a)*Gamma[0, 1/10 + a] +
     9*E^(1/10 + a)*Log[1/(-1 - 10*a)] - 9*E^(1/10 + a)*
      Log[-1 - 10*a] + 20*Log[1 + 10*a] + 20*a*Log[1 + 10*a] +
     18*E^(1/10 + a)*Log[1 + 10*a])/E^a))

Out[4]=
4.05855803419172 + 0.*I

/Tanel Telliskivi




"Andrzej Kozlowski" <akoz at mimuw.edu.pl> skrev i meddelandet
news:bpkpd7$ce1$1 at smc.vnet.net...
> On 20 Nov 2003, at 17:16, Sung Jin Kim wrote:
>
> > Dear all,
> >
> > I got very extraordinary results today from below two same integrals
> > except one is symbolic one and the other is numeric one:
> > A. In[1]= N[Integrate[ Log[2, 1 + 10*x]*Exp[-x]*x, {x, 0, Infinity}]]
> >     Out[1]= -3.77002
> > B. In[2]= NIntegrate[ Log[2, 1 + 10*x]*Exp[-x]*x, {x, 0, Infinity}]
> >     Out[2]= 4.05856
> >
> > Why did I got the different results of these, surprisingly?
> >
> > Thank you in advance!
> > ---
> > Sung Jin Kim
> > A member of MCL in SNU: kimsj at mobile.snu.ac.kr,
> > A MTS of i-Networking Lab in SAIT: communication at samsung.com
> >
> >
> >
> The answer given by Integrate is clearly wrong and seems to be due to
> Mathematica's failure to deal with a difficult limit.
>
> Not that if we set
>
>
> f[a_] = Integrate[Log[2, 1 + 10*x]*Exp[-x]*x, {x, 0, a}, Assumptions ->
> a > 0]
>
>
> (1/Log[1024])*(Log[10*a + 1]/E^a - 10*E^(1/10)*Gamma[2, a +
> 1/10]*Log[10*a + 1] +
>     E^(1/10)*MeijerG[{{}, {1}}, {{0, 0}, {}}, a + 1/10] -
>     10*E^(1/10)*MeijerG[{{}, {1, 1}}, {{0, 0, 2}, {}}, a + 1/10] +
>     10*E^(1/10)*MeijerG[{{}, {1, 1}}, {{0, 0, 2}, {}}, 1/10] -
>     E^(1/10)*MeijerG[{{}, {1}}, {{0, 0}, {}}, 1/10])
>
> Then this agrees with NIntegrate, e.g.
>
>
> NIntegrate[Log[2,1+10*x]*Exp[-x]*x,{x,0,20}]
>
>
> 4.05856
>
>
> N[f[20]]
>
>
> 4.05856
>
> (Note also that using N for large a's will produce wrong answers unless
> you use more digits. For example, compare N[f[40]] with N[f[40],20]).
>
> In any case, the answer given for Integrate is almost certainly right,
> but Mathematica is unable to correctly find the limit of the expression
> as a->Infinity. It seems to preform some numerical checks that it is
> unable to carry out.
>
> Andrzej Kozlowski
> Chiba, Japan
> http://www.mimuw.edu.pl/~akoz/
>

• References:
  • [Integrate] Why two results of same eq. are different?
    • From: "Sung Jin Kim" <kimsj@mobile.snu.ac.kr>