Re: integration
- To: mathgroup at smc.vnet.net
- Subject: [mg92606] Re: integration
- From: RG <gobithaasan at gmail.com>
- Date: Wed, 8 Oct 2008 06:24:17 -0400 (EDT)
- References: <gcchbh$s2c$1@smc.vnet.net> <gcfiaj$giu$1@smc.vnet.net>
On Oct 7, 7:48 pm, Jean-Marc Gulliet <jeanmarc.gull... at gmail.com> 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])]] > > HTH, > -- Jean-Marc Dear All, i think M6 would be able to give a simplified answer, which is more understandable without the appearance of imaginary numbers in the answer. The assumption of the integral should be: [1]{k1,k2,r,s,S} are real numbers [2] r > -1 [3] S > 0 [4] 0<= s<= S I tried doing with these assumption, but the imaginary part still exists.. Is there anyway to ask M6 to give the right assumption for imaginary-free answer? Thank you very much sirs...