Re: integration

*To*: mathgroup at smc.vnet.net*Subject*: [mg92599] Re: integration*From*: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>*Date*: Tue, 7 Oct 2008 07:06:51 -0400 (EDT)

Dear Experts, 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\)\) Regards, RG Hi, RG, your problem seems to be in Mathematics, rather than in Mathematica. The integral you need (as it is now) is too cumbersome. Make the integrand simpler by hiding and simplifying your notations, and the result will become more understandable. For example, below is your integral along with the result in which I denoted some terms by a, b and c to make it shorter: In[2]:= \!\( \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(s\)]\(Cos[\ t\ a + b + c*Log[S + r\ t]] \[DifferentialD]t\)\) Out[2]= If[(Re[S/(r s)] >= 0 && S/(r s) != 0) || Re[S/(r s)] <= -1 || Im[S/(r s)] != 0, -(1/(2 a)) S^(-\[ImaginaryI] c) (-((\[ImaginaryI] a S)/ r))^(-\[ImaginaryI] c) (S^(2 \[ImaginaryI] c) Gamma[1 + \[ImaginaryI] c, -((\[ImaginaryI] a S)/ r)] (-\[ImaginaryI] Cos[b - (a S)/r] + Sin[b - (a S)/r]) + (( a^2 S^2)/r^2)^(\[ImaginaryI] c) Gamma[1 - \[ImaginaryI] c, (\[ImaginaryI] a S)/ r] (\[ImaginaryI] Cos[b - (a S)/r] + Sin[b - (a S)/r])) + (1/( 2 a))(r s + S)^(-\[ImaginaryI] c) (-((\[ImaginaryI] a (r s + S))/ r))^(-\[ImaginaryI] c) ((r s + S)^(2 \[ImaginaryI] c) Gamma[1 + \[ImaginaryI] c, -((\[ImaginaryI] a (r s + S))/ r)] (-\[ImaginaryI] Cos[b - (a S)/r] + Sin[b - (a S)/r]) + (( a^2 (r s + S)^2)/r^2)^(\[ImaginaryI] c) Gamma[1 - \[ImaginaryI] c, (\[ImaginaryI] a (r s + S))/ r] (\[ImaginaryI] Cos[b - (a S)/r] + Sin[b - (a S)/r])), Integrate[Cos[b + a t + c Log[t r + S]], {t, 0, s}, Assumptions -> ! ((Re[S/(r s)] >= 0 && S/(r s) != 0) || Re[S/(r s)] <= -1 || Im[S/(r s)] != 0)]] This is your result after simplification: In[3]:= -(1/(2 a)) S^(-\[ImaginaryI] c) (-((\[ImaginaryI] a S)/ r))^(-\[ImaginaryI] c) (S^(2 \[ImaginaryI] c) Gamma[1 + \[ImaginaryI] c, -((\[ImaginaryI] a S)/ r)] (-\[ImaginaryI] Cos[b - (a S)/r] + Sin[b - (a S)/r]) + (( a^2 S^2)/r^2)^(\[ImaginaryI] c) Gamma[1 - \[ImaginaryI] c, (\[ImaginaryI] a S)/ r] (\[ImaginaryI] Cos[b - (a S)/r] + Sin[b - (a S)/r])) + 1/(2 a)(r s + S)^(-\[ImaginaryI] c) (-((\[ImaginaryI] a (r s + S))/ r))^(-\[ImaginaryI] c) ((r s + S)^(2 \[ImaginaryI] c) Gamma[1 + \[ImaginaryI] c, -((\[ImaginaryI] a (r s + S))/ r)] (-\[ImaginaryI] Cos[b - (a S)/r] + Sin[b - (a S)/r]) + (( a^2 (r s + S)^2)/r^2)^(\[ImaginaryI] c) Gamma[1 - \[ImaginaryI] c, (\[ImaginaryI] a (r s + S))/ r] (\[ImaginaryI] Cos[b - (a S)/r] + Sin[b - (a S)/r])) // FullSimplify Out[3]= (1/(2 r))\[ExponentialE]^(-((\[ImaginaryI] (b r + a S))/ r)) (\[ExponentialE]^(2 \[ImaginaryI] b) S^(1 + \[ImaginaryI] c) ExpIntegralE[-\[ImaginaryI] c, -((\[ImaginaryI] a S)/ r)] - \[ExponentialE]^(2 \[ImaginaryI] b) (r s + S)^( 1 + \[ImaginaryI] c) ExpIntegralE[-\[ImaginaryI] c, -((\[ImaginaryI] a (r s + S))/ r)] + \[ExponentialE]^((2 \[ImaginaryI] a S)/ r) (S^(1 - \[ImaginaryI] c) ExpIntegralE[\[ImaginaryI] c, (\[ImaginaryI] a S)/ r] - (r s + S)^(1 - \[ImaginaryI] c) ExpIntegralE[\[ImaginaryI] c, (\[ImaginaryI] a (r s + S))/r])) It is already comprehensible. Probably you may go still further by checking properties of the integral exponents. Have a look into the book: Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions with formulas, graphs and mathematical tables. (National Bureau of Standards, 1964). Another way I would go is to try to transform it first, and to integrate afterwards. Success, Alexei -- Alexei Boulbitch, Dr., Habil. Senior Scientist IEE S.A. ZAE Weiergewan 11, rue Edmond Reuter L-5326 Contern Luxembourg Phone: +352 2454 2566 Fax: +352 2454 3566 Website: www.iee.lu This e-mail may contain trade secrets or privileged, undisclosed or otherwise confidential information. If you are not the intended recipient and have received this e-mail in error, you are hereby notified that any review, copying or distribution of it is strictly prohibited. Please inform us immediately and destroy the original transmittal from your system. Thank you for your co-operation.