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Re: integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92584] Re: integration
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 7 Oct 2008 07:04:05 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <gcchbh$s2c$1@smc.vnet.net> <48EA5916.2050008@gmail.com>

Jean-Marc Gulliet wrote:

> RG wrote:
> 
>> I have been trying to simplify(integrate) the following function, but
>> M6 seems to give a complex answer which i cann't understand.. please
>> help.
>>
>> x[s_]=\!\(
>> \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(s\)]\(Cos[
>> \*FractionBox[\(r\ t\ \((\(-\[Kappa]0\) + \[Kappa]1 + r\ \[Kappa]1)\)
>> + \((1 + r)\)\ S\ \((\[Kappa]0 - \[Kappa]1)\)\ \((\(-Log[S]\) + Log[S
>> + r\ t])\)\),
>> SuperscriptBox[\(r\), \(2\)]]] \[DifferentialD]t\)\)
> 
> First, notice that if we use the *InputForm* of the above expression, we 
> can easily add assumptions on the parameters of the integral (or we 
> could use *Assuming*), for instance that S, r, and s are real and r != 0 
> or s > 0.
> 
> However, it seems that the above integral has no solution if the 
> parameter S is positive. On the other hand, ff we allow S to be negative 
> (or complex) then the integral has a symbolic complex solution.
> 
> In[49]:= Integrate[
>  Cos[(r*t*(-\[Kappa]0 + \[Kappa]1 + r*\[Kappa]1) + (1 + r)*
>       S*(\[Kappa]0 - \[Kappa]1)*
>             (-Log[S] + Log[S + r*t]))/r^2], {t, 0, s},
>    Assumptions -> S > 0]
> 
> Out[49]= Integrate[
>  Cos[(r t (-\[Kappa]0 + \[Kappa]1 + r \[Kappa]1) + (1 +
>       r) S (\[Kappa]0 - \[Kappa]1) (-Log[S] + Log[S + r t]))/r^2], {t,
>    0, s}, Assumptions -> S > 0]
> 
> In[46]:= Integrate[
>  Cos[(r*t*(-\[Kappa]0 + \[Kappa]1 + r*\[Kappa]1) + (1 + r)*
>       S*(\[Kappa]0 - \[Kappa]1)*
>             (-Log[S] + Log[S + r*t]))/r^2], {t, 0, s},
>    Assumptions -> {Element[{S, r, s}, Reals], r != 0, s > 0}]
> 
> Out[46]= If[r S > 0 || s + S/r <= 0, (1/(
>  2 (\[Kappa]0 - \[Kappa]1 - r \[Kappa]1)))
>  r S^(-((I (1 + r) S (\[Kappa]0 - \[Kappa]1))/r^2)) (r s + S)^(-((
> 
> [... output partially deleted ...]
> 
>        r^2)] Sin[(S \[Kappa]0)/r^2 - (S \[Kappa]1)/r^2 - (
>        S \[Kappa]1)/r]),
>  Integrate[
>   Cos[(r t (-\[Kappa]0 + \[Kappa]1 + r \[Kappa]1) + (1 +
>        r) S (\[Kappa]0 - \[Kappa]1) (-Log[S] + Log[S + r t]))/
>    r^2], {t, 0, s},
>   Assumptions ->
>    r != 0 && s > 0 && r S <= 0 && r (r s + S) > 0 &&
>     S \[Element] Reals]]
> 
> 
> You can manipulate further the integral thanks to *FullSimplify* and 
> some assumptions on the parameters.
> 
> Assuming[r S > 0 || s + S/r <= 0,
>  FullSimplify[
>   1/(2 (\[Kappa]0 - \[Kappa]1 - r \[Kappa]1))
>    r S^(-((I (1 + r) S (\[Kappa]0 - \[Kappa]1))/r^2)) (r s + S)^(-((
> 
>         [... input partially deleted ...]
> 
>         S \[Kappa]1)/r])]]

It took a long time, but the last expression returned the following 
result (which is valid only for r S > 0 || s + S/r <= 0):

(1/(2 r))E^(-((I S (-\[Kappa]0 + \[Kappa]1 + r \[Kappa]1))/
   r^2)) (S ExpIntegralE[-((I (1 + r) S (\[Kappa]0 - \[Kappa]1))/
       r^2), -((I S (-\[Kappa]0 + \[Kappa]1 + r \[Kappa]1))/r^2)] -
    S^(-((I (1 + r) S (\[Kappa]0 - \[Kappa]1))/r^2)) (r s + S)^(
     1 + (I (1 + r) S (\[Kappa]0 - \[Kappa]1))/r^2)
      ExpIntegralE[-((I (1 + r) S (\[Kappa]0 - \[Kappa]1))/r^2), -((
       I (r s + S) (-\[Kappa]0 + \[Kappa]1 + r \[Kappa]1))/r^2)] +
    E^((2 I S (-\[Kappa]0 + \[Kappa]1 + r \[Kappa]1))/
     r^2) (S ExpIntegralE[(I (1 + r) S (\[Kappa]0 - \[Kappa]1))/r^2, (
         I S (-\[Kappa]0 + \[Kappa]1 + r \[Kappa]1))/r^2] - (S^3)^((
        I (1 + r) S (\[Kappa]0 - \[Kappa]1))/r^2) (r s + S)^(
        1 - (I (1 + r) S (\[Kappa]0 - \[Kappa]1))/
         r^2) ((\[Kappa]0 - (1 + r) \[Kappa]1)^2)^((
        I (1 + r) S (\[Kappa]0 - \[Kappa]1))/
        r^2) (S^2 (r s + S)^2 (\[Kappa]0 - (1 + r) \[Kappa]1)^2)^(-((
         I (1 + r) S (\[Kappa]0 - \[Kappa]1))/
         r^2)) ((r s + S) (S + r Conjugate[s]))^((
        I (1 + r) S (\[Kappa]0 - \[Kappa]1))/r^2)
         ExpIntegralE[(I (1 + r) S (\[Kappa]0 - \[Kappa]1))/r^2, (
         I (r s + S) (-\[Kappa]0 + \[Kappa]1 + r \[Kappa]1))/r^2]))

Regards,
-- Jean-Marc


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