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MathGroup Archive 2000

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Re: Unwanted definite-integral complex result from real integrand

  • To: mathgroup at smc.vnet.net
  • Subject: [mg21497] Re: Unwanted definite-integral complex result from real integrand
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Wed, 12 Jan 2000 08:35:41 -0500 (EST)
  • Organization: University of Western Australia
  • References: <851esl$4bp@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Tom Crane wrote:

> 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.

Why do you need a real result? It is not always the simplest representation.

> 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.

Why? For many problems, Mathematica's numerics should be fast enough. What exactly
do you want to do with the result?

> 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.

To better see what is going on, consider integrating (a Sin[a])/(a^2 + 1), which
is, essentially, one term in your integrand:

In[1]:= Simplify[Integrate[(a Sin[a])/(a^2 + 1), a]]
Out[1]= (I*CosIntegral[-I + a]*Sinh[1] - I*CosIntegral[I + a]*
   Sinh[1] + Cosh[1]*(SinIntegral[-I + a] +
    SinIntegral[I + a]))/2

In[2]:= Collect[%, {Sinh[_], Cosh[_]}]
Out[2]= ((I*CosIntegral[-I + a] - I*CosIntegral[I + a])*Sinh[1])/2 +
 (Cosh[1]*(SinIntegral[-I + a] + SinIntegral[I + a]))/2

This result is indeed real for real a. You can explicitly show this by employing
the Symmetry Relations for Si and Ci (see Abramowitz and Stegun 5.2.19-20). E.g.,
Si(a+I)+Si(a-I) is real because Si(a-I) is the complex conjugate of Si(a+I) for
real a.

However, there is no simple expression (i.e., no other built-in special function)
for the real and imaginary parts of Si(a+b I) etc., and hence (without the
introduction of new special functions) there is no way to write Si(a+I)+Si(a-I) as
an explicit real function of a.

> 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?

And as you have effectively discovered, this is not always (mathematically)
possible!

Cheers,
    Paul



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