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