Re: Integration Problems

• To: mathgroup at smc.vnet.net
• Subject: [mg13933] Re: [mg13926] Integration Problems
• From: Wouter Meeussen <w.meeussen.vdmcc at vandemoortele.be>
• Date: Tue, 8 Sep 1998 02:52:31 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```dear Georgios,

(509.399 - 2.92295 10^-5 I) etc, there *would* be a problem, but
(509.399 - 2.92295 10^-15 I) etc..
*
is clearly a floating point 'residue' caused by your giving a numerical
parameter 4.73 (with no accuracy attached to it). Somewhere Mathematica
must have swithed to the NIntegrate routine.

look for instance at the indefinite Integrate[W2[x,t],x] : Out[5]=
(-1.653497140625001 + 0.*I)*
(32./E^(9.46*x) - 0.002493010912484319*E^(9.46*x) -
308.0638986263043*x +
((0. + 0.*I)*Cos[4.73*x])/E^(4.73*x) -
(1.129788293085026 - 1.77635683940025*^-15*I)*
E^(4.73*x)*Cos[4.73*x] - 32.*Cos[9.46*x] -
((128. + 0.*I)*Sin[4.73*x])/E^(4.73*x) -
(2.664535259100375*^-15 + 0.*I)*E^(4.73*x)*
Sin[4.73*x] + 0.*I*Sin[9.46*x])*q[2, 1][t]^2

the imaginary parts are already in there!

But in the indefinite integral with symbolic parameter a in place of
4.73 :

Out[8]=
-((E^(2*a*x - 2*(-(a*(1 - x)) - a*x))*
(a^2/E^(a*(1 - x)) + a^2/E^(a*x) + a^2*Cos[a*x] -
a^2*Sin[a*x])^2*
(-E^(-2*a) + E^(-4*a*x) - (2*a*x)/E^(2*a*x) -
4*a*E^(-a - 2*a*x)*x - 4*E^(-a - a*x)*Cos[a*x] -
Cos[2*a*x]/E^(2*a*x) - (4*Sin[a*x])/E^(3*a*x))*
q[2, 1][t]^2)/
(2*a*(E^(a*(1 - x)) + E^(a*x) +
E^(a*(1 - x) + a*x)*Cos[a*x] -
E^(a*(1 - x) + a*x)*Sin[a*x])^2))

you will find no imaginary 'residues', so the integration technique must
have been different.

fill in the upper & lower bound,
In[10]:=(int/. x->1) - (int /. x->0) and do a FullSimplify on that:
Out[11]=
(a^3*(1 + 2*a - 2/E^(2*a) + 4*Cos[a] + Cos[2*a] +
(4*(-1 + a + Sin[a]))/E^a)*q[2, 1][t]^2)/2

only then would I introduce  %/. a->4.73 Out[12]=
509.3985730174712*q[2, 1][t]^2

I hope this puts you onto some nifty ideas (;-))

wouter.

At 01:22 07.09.98 -0400, you wrote:
>Dear Sirs ,
>
>while performing the following evaluation, i received an unexpected
>result:
>
>1.) W1[x_,t_]:=
>    q[2,1][t] (Sin[4.73 x] - Cos[4.73 x] + Exp[-4.73 x] + Exp[-4.73
>(1-x)]);
>
>2.) W2[x_,t_]:=Evaluate[(D[W1[x,t],{x,2} ])^2]
>
>3.) Integrate[W2[x,t],{x,0,1}]
>
>
>(509.399 -2.922295 X 10^-5 I) q[2,1][t]^2
>
>Could you please tell me, why does an imaginary part arise though the
>function
>that is being integrated is real ?
>
>Thank you very much for your time.
>
>Georgios
>
>

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vdmcc at vandemoortele.be

```

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