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

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?

-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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