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MathGroup Archive 1999

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Re: RE: ExpIntegralEi

  • To: mathgroup at smc.vnet.net
  • Subject: [mg18580] Re: [mg18491] RE: [mg18463] ExpIntegralEi
  • From: "Andrzej Kozlowski" <andrzej at tuins.ac.jp>
  • Date: Tue, 13 Jul 1999 01:01:21 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

Exp is an entire function (analytic in the enitre complex plane) and it has
no branch cuts. But there may well be some branches taken during the
integration process. One can see that something like this is going on if we
look at the following:



In[1]:=
c2 = -0.05018627683354541 - 0.153047656745338 I;
In[2]:=
c3 = -0.7828709924214918 + 0.2780791279205129 I;

In[2]:=
Table[Integrate[
    Exp[beta*c2 + s*(c3 + beta*c5)], {s, 0, 1}, {beta, 0,
      1}], {c5, -0.1, -0.7, -0.1}]

Out[2]=
{0.658685 + 0.0294099 I, 0.645025 + 0.0284488 I,

  0.631904 + 0.0275333 I, -4.42244 + 18.6302 I, -2.50325 + 14.4967 I,

  -1.51119 + 11.8412 I, -0.935727 + 9.99846 I}

Look at the sudden jump at c3=-0.4. By contrast NIntegrate gives:

In[3]:=
Table[NIntegrate[
    Exp[beta*c2 + s*(c3 + beta*c5)], {s, 0, 1}, {beta, 0,
      1}], {c5, -0.1, -0.7, -0.1}]

Out[3]=
{0.658685 + 0.0294099 I, 0.645025 + 0.0284488 I,

  0.631904 + 0.0275333 I, 0.619293 + 0.0266608 I,

  0.607166 + 0.0258288 I, 0.5955 + 0.0250351 I,

  0.584271 + 0.0242777 I}


So it does look liek some sort of singularity that mathematica encounters
during the integration process, though not being an expert at this sort of
thing I can't imagine what it might be.
--
Andrzej Kozlowski
Toyama International University
JAPAN
http://sigma.tuins.ac.jp
http://eri2.tuins.ac.jp


----------
>From: "Ersek, Ted R" <ErsekTR at navair.navy.mil>
To: mathgroup at smc.vnet.net
>To: mathgroup at smc.vnet.net
>Subject: [mg18580] [mg18491] RE: [mg18463] ExpIntegralEi
>Date: Thu, Jul 8, 1999, 12:08 PM
>

> Lionel ARNAUD reported the following with version 3.0.
> -------------------------
>
> c2 = -0.05018627683354541 - 0.153047656745338 I;
> c3 = -0.7828709924214918 + 0.2780791279205129 I;
> c5 = -0.6758555487562639 - 0.04753624179417532 I;
>
> Integrate[Exp[beta*c2+s*(c3+beta*c5)], {s,0,1},{beta,0,1}]
>
> NIntegrate[Exp[beta*c2+s*(c3+beta*c5)], {s,0,1},{beta,0,1}]
>
> The results given are:
> -0.228103 + 10.5644 I
>  0.587252 +  0.0191685 I
>
> Not the same !
> ----------------------
> Lionel,
>
> Version 4.0 gives the same result.  I wonder if the difference has to due
> with branch cuts along the path of integration. It may be that the two
> commands are integrating along different paths in the complex plane.
>
> I think Exp[z] has a branch cut where Im[z]=(Pi+2*n*Pi) (n is an integer).
> However it looks like the integration path doesn't cross a branch cut, so I
> am puzzled.
>
> Regards,
> Ted Ersek


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