       Unwanted definite-integral complex result from real integrand

• To: mathgroup at smc.vnet.net
• Subject: [mg21422] Unwanted definite-integral complex result from real integrand
• From: uhap023 at vms.rhbnc.ac.uk (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 wolfram.com

```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.

Integrate[%,{wa,-a,a}]

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?

Thanks.
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 rhbnc.ac.uk
SPAN:   19.875
Fax:    01784 472794

```

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