       Re: Integration Bug

• To: mathgroup at smc.vnet.net
• Subject: [mg65279] Re: Integration Bug
• From: "Chris H. Fleming" <chris_h_fleming at yahoo.com>
• Date: Thu, 23 Mar 2006 06:58:48 -0500 (EST)
• References: <dvj4s8\$1j\$1@smc.vnet.net><dvrcbt\$a66\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Valeri Astanoff wrote:
> Hi Chris,
>
> Seems to me that if you convert to Ei form,
> care has to be taken of the argument sign :
>
> In:=toEiForm = {CosIntegral[x_ /; x \[Element] Reals] ->
>      (1/2)*ExpIntegralEi[(-I)*x] + (1/2)*ExpIntegralEi[I*x] +
>       I*Pi*(1 - UnitStep[x]), SinIntegral[x_ /; x \[Element] Reals] ->
>      (1/2)*I*(ExpIntegralEi[(-I)*x] - ExpIntegralEi[I*x] +
> I*Pi*Sign[x])};
>
> In:={CosIntegral,CosIntegral /. toEiForm}
> Out={CosIntegral, ExpIntegralEi[-I]/2 + ExpIntegralEi[I]/2}
>
> In:=%//N
> Out={0.337404, 0.337404 + 0.*I}
>
> In:={CosIntegral[-1],CosIntegral[-1] /. toEiForm}
> Out={CosIntegral[-1], I*Pi + ExpIntegralEi[-I]/2 +
> ExpIntegralEi[I]/2}
>
> In:=%//N
> Out={0.337404 + 3.14159*I, 0.337404 + 3.14159*I}
>
>
> hth
>
> Valeri Astanoff

Oh I agree. E1 has a branch cut singularity exactly like log, and Ei
opposite of that. I am pretty sure that Mathematica gives incorrect
integrals because it is integrating across branch cuts in a
discontinuous fashion.

But one can carefully convert from Si, Ci to E1 and I have done so to
get the answer in the form I want.

The problem is that I have more complicated integrals that I have to do
by hand because Mathematica wont attempt them unless I cast them as
purely exponential and thus give an incorrect answer.

```

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