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Topic: wrong result when computing a definite integral
Replies: 4   Last Post: Jan 14, 2013 12:01 AM

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 Murray Eisenberg Posts: 2,105 Registered: 12/6/04
Re: wrong result when computing a definite integral
Posted: Jan 12, 2013 9:51 PM

Unless there's some issue of branches of complex functions involved that
I'm missing, it should not matter here which order of integration you
use -- since the limits of integration are constants. However, if you
wrap each integrand in ComplexExpand,

a = Integrate[ComplexExpand[Exp[I*Sqrt[3]*y]], {x, -2*Pi, 2*Pi},
{y, -Pi, Pi}]
b = Integrate[ComplexExpand[Exp[I*Sqrt[3]*y]], {y, -Pi, Pi}, {x,
-2*Pi, 2*Pi}]

then you obtain the same result:

{a, b} // InputForm
{(8*Pi*Sin[Sqrt[3]*Pi])/Sqrt[3], (8*Pi*Sin[Sqrt[3]*Pi])/Sqrt[3]}
a == b
True

On Jan 11, 2013, at 10:22 PM, Alex Krasnov <akrasnov@eecs.berkeley.edu> wrote:

> Integrate takes the integration variables in prefix order, so perhaps you
> meant the following:
>
> In: Integrate[Exp[I*Sqrt[3]*y], {y, -Pi, Pi}, {x, -2*Pi, 2*Pi}]
> Out: (8*Pi*Sin[Sqrt[3]*Pi])/Sqrt[3]
>
> Thu, 10 Jan 2013, Dexter Filmore wrote:
>

>> i run into this problem today when giving a bunch of easy integrals to mathematica.
>> here's a wolfram alpha link to the problem:
>>

http://www.wolframalpha.com/input/?i=Integrate%5BExp%5BI+Sqrt%5B3%5Dy%5D%2C%7Bx%2C-2Pi%2C2Pi%7D%2C%7By%2C-Pi%2CPi%7D%5D#
>>
>> the integrand does not depend on the 'x' variable, the inner

integration should only result in a factor of 4Pi, and the correct
result is a real number, yet the below integral gives a complex number
which is far off from the correct value:
>> Integrate[Exp[I Sqrt[3] y], {x, -2 Pi, 2 Pi}, {y, -Pi, Pi}] -> -((4 I (-1 + E^(2 I Sqrt[3] Pi)) Pi)/Sqrt[3])
>>
>> from some trial and error it seems the result is also incorrect for non-integer factors in the exponential.

---
Murray Eisenberg murray@math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 5 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305

Date Subject Author
1/11/13 Alex Krasnov
1/12/13 Murray Eisenberg
1/14/13 Alex Krasnov