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Exp[I x] vs. Cos[x] + I Sin[x] in Integrate
*To*: mathgroup at christensen.cybernetics.net
*Subject*: [mg2035] Exp[I x] vs. Cos[x] + I Sin[x] in Integrate
*From*: "Brian J. Albright" <albright at physics.ucla.edu>
*Date*: Sat, 16 Sep 1995 01:41:10 -0400
*Organization*: UCLA Department of Physics and Astronomy
Greetings.
A friend of mine and I have recently run into some apparent
peculiarities with the MMa Integrate function. If we enter
the following
In[1]:= Integrate[ Exp[I x] / (2 + Sin[x]), {x,0,2 Pi}]
then we get
Out[1]:= 0
Unfortunately, this is not the right answer. The correct answer,
obtainable through contour integration, is 2 Pi I (1 - 2/Sqrt[3]).
Interestingly, if we rewrite the integrand, changing "Exp[I x]"
into "Cos[x] + I Sin[x]":
In[2]:= Integrate[ (Cos[x] + I Sin[x]) / (2 + Sin[x]), {x,0,2 Pi}]
then we get
4 I Pi
Out[2]:= 2 I Pi - -------
Sqrt[3]
which matches our paper-and-pencil result. Does anyone know why
MMa gives different answers for the two?
Thanks in advance.
-Brian
ps. Incidentally, if I write the "Sin[x]" in In[1] as
"(Exp[I x] - Exp[-I x]) / (2 I)", I get
In[3]:= Integrate[ Exp[I x] / (2 + (Exp[I x] - Exp[-I x])/(2 I) ),
{x,0,2 Pi}]
Out[3]:= 0
Also, if I use NIntegrate rather than Integrate, I get
In[4]:= Chop[ NIntegrate[ Exp[I x] / (2 + Sin[x]), {x,0,2 Pi}] ]
Out[4]:= -0.972012 I
compare with
In[5]:= Out[2]//N
Out[5]:= -0.972012
--
Brian J. Albright |
Department of Physics and Astronomy, UCLA | To err is human...
albright at physics.ucla.edu | to err really big
http://bohm.physics.ucla.edu/~albright | is government.
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