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