Real integrand->complex result.

*To*: mathgroup at smc.vnet.net*Subject*: [mg20293] Real integrand->complex result.*From*: William Golz <wgolz at attglobal.net>*Date*: Tue, 12 Oct 1999 03:39:30 -0400*Sender*: owner-wri-mathgroup at wolfram.com

Synopsis of the problem: I get a complex result of the form f(t)=(a+bi)g(t) when evaluating a real integrand F(x,t)*phi(x) over the interval {x:0<x<L). All of the variables have been properly declared as real and the functions have been declared as real for real arguments (using the package "ReIm"). To see the problem more clearly, the code can be copied into Mathematica (3 or 4) and executed. (The Traditional or Standard Forms will not paste into a text document such as this) When evaluating the integral \!\(\[Integral]\_0\%\[ScriptL]\( simpF\[CurlyPhi]\_n\) \[DifferentialD]x\) over the interval (0,ScriptL) where ScriptL=1. The integrand \!\(simpF\[CurlyPhi]\_n\ is, for n=1, given by \!\(TraditionalForm \`\(-\(1\/6\)\)\ \[ExponentialE]\^\(\(\(-3\)\ \[Tau]\)/2\)\ \((\((3 + 2\ \[Pi]\^2)\)\ \(cos(\[Pi]\ x)\) - 6\ \[ExponentialE]\^\(x/2\) + 3)\)\ \((cos(3.66558239083868908`\ x) + sin(3.66558239083868908`\ x))\)\) and for n=2, \!\(TraditionalForm \`\(-\(1\/6\)\)\ \[ExponentialE]\^\(\(\(-3\)\ \[Tau]\)/2\)\ \((\((3 + 2\ \[Pi]\^2)\)\ \(cos(\[Pi]\ x)\) - 6\ \[ExponentialE]\^\(x/2\) + 3)\)\ \((cos(6.58302641111680308`\ x) + sin(6.58302641111680308`\ x))\)\) I get the result for n=1,2, respectively \!\(TraditionalForm \`\((\(-2.34710341268466837`\) + 0.`\ \[ImaginaryI])\)\ \[ExponentialE]\^\(\(\(-3\)\ \[Tau]\)/2\)\) \!\(TraditionalForm \`2\[InvisibleSpace]'' ''\[InvisibleSpace]\(\(( \(-1.26446964467229783`\) + 0.`\ \[ImaginaryI])\)\ \[ExponentialE]\^\(\(\(-3\)\ \[Tau]\)/2\)\)\) For simple functions F, the product F*phi can be integrated by parts, where the result contains no potential for complex coefficients when the variables are real valued. However, evaluating an identical integrand in Mathematica will give the result f(t)=(a+0.i)g(t). There does not seem to be any way to avert this, either by defining the variables to be real and the functions to be real valued (for real arguments). The evaluation of the integral is also quite slow, even for n=2, even when the expression is simplified as much as possible, and the final program will need to evaluate a larger number of terms. Thanks in advance, Bill