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Unwanted definite-integral complex result from real integrand

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  • Subject: [mg21422] Unwanted definite-integral complex result from real integrand
  • From: uhap023 at (Tom Crane)
  • Date: Thu, 6 Jan 2000 01:46:37 -0500 (EST)
  • Organization: Dept. Physics, Royal Holloway, University of London
  • Sender: owner-wri-mathgroup at

Dear All,
	Can anybody help me with the following problem? I am trying to do a 
definite integral on a real expression but Mathematica always returns a complex 
result. I need a real result and believe the result should be real. 
Moveover, I want to produce a Fortran function of the result and even if I 
were to use Fortran's complex datatype, terms of the form CosIntegal(x+Iy) 
in the result are a problem since I want to use external functions (eg. NAG 
library) to evaluate these special functions.

My integrand is,

(g*t1*(t - t1 + t*t1^2*(wa + wb)^2 + t1^3*(wa + wb)^2))/
  (1 + t1^2*(wa + wb)^2)^2 - 
 (g*t1^2*((-1 + t1^2*(wa + wb)^2)*Cos[t*(wa + wb)] + 
    2*t1*(wa + wb)*Sin[t*(wa + wb)]))/(E^(t/t1)*(1 + t1^2*(wa + wb)^2)^2)

and the integral is, eg.


By expanding the integrand into partial fractions I can partially
understand what going on w.r.t. the trig containing terms - they comprise,
something like, eg. Sin[t*(wa+wb)]/<a polynomial in wa>. I imagine that
Mathematica then tries to shoehorn this expression into the Sine Integral, the
complex terms arising from the manipulations/solutions of the polynomial
in the denominator?? None of this gets me anywhere and in any case
imaginary terms appear elsewhere in the integrated result. 

I have browsed MathGroup and tried a few thing but to no avail. Some of
the things I have tried/are trying, are; 

(1) Loading Calculus`Limits` before doing the integral.
(2) Using ComplexExpand[%,TargetFunctions->{Re,Im}] on the complex result of 
the Integrate command.
(3) Using Assumptions with Integrate to tell it that all of the variables 
in the integrand are always +ve.
(4) Running FullSimplify on the integrated result but I strongly doubt
that it will remove all the imaginary terms. 

Essentially, the thing I need to tell Mathematica is: Do the Integral, feel free 
to use your knowledge of special functions etc. to make a more useful
result, but *don't* introduce any complex arithmetic. How can I persuade
it to do this? 

Tom Crane.

Ps. I'm using Mathematica 4.0 on Win98.

Tom Crane, Dept. Physics, Royal Holloway, University of London, Egham Hill,
Egham, Surrey, TW20 0EX, England. 
Email:	T.Crane at
SPAN:   19.875
Fax:    01784 472794

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