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Re: integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92582] Re: integration
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 7 Oct 2008 07:03:43 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <gcchbh$s2c$1@smc.vnet.net>

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])]]

HTH,
-- Jean-Marc



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