Integrate E^(I x) Bug?
- To: mathgroup@smc.vnet.net
- Subject: [mg11469] Integrate E^(I x) Bug?
- From: gts@mindspring.com (Garrett Tim Sos)
- Date: Thu, 12 Mar 1998 01:34:47 -0500
- Organization: MindSpring Enterprises
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.