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Integrate E^(I x) Bug?




Is this a bug?
Note:  I is the Sqrt[-1]

In[49]:=

et01=E^(I x)
Integrate[et01,{x,0,Infinity}]

et02=ExpToTrig[et01]
Integrate[et02,{x,0,Infinity}]

et03=Integrate[et01,{x,0,a}]
Limit[et03,a->Infinity]


Out[49]=
\!\(E\^\(I\ x\)\)

Out[50]=
I

^^^  What?

Out[51]=
Cos[x]+I Sin[x]

Integrate::"idiv": 
    "Integral of \!\(\(Cos[x]\) + \(I\\ \(Sin[x]\)\)\) does not converge
on \ \!\({0, \*InterpretationBox[\"\\[Infinity]\",
DirectedInfinity[1]]}\)." Out[52]=
\!\(\*
  RowBox[{
    SubsuperscriptBox["", "0", 
      InterpretationBox["",
        DirectedInfinity[ 1]]], 
    \(\((Cos[x] + I\ Sin[x])\) \[DifferentialD]x\)}]\)

^^^ the Cos[x]+I Sin[x] form of E^(I x) does not integrate!

Out[53]=
\!\(I - I\ E\^\(I\ a\)\)

Out[54]=
\!\(\*
  RowBox[{"Limit", "[", 
    RowBox[{\(I - I\ E\^\(I\ a\)\), ",", 
      RowBox[{"a", "\[Rule]", 
        InterpretationBox["",
          DirectedInfinity[ 1]]}]}], "]"}]\)


I know forms like E^( (I - a) x ) should converge if a>0 on (0,Infinity)
because you can visualise this as  E^(I x) E^(- a x) and the E^(- a x)
damps out the E^(I x) term as x->Infinity

Mathematica Version 3.0.1.1x running on a Macintosh PowerPC 7100/66

Thanks for the Help.



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