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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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 Alex Krasnov Posts: 15 Registered: 10/3/12
Re: wrong result when computing a definite integral
Posted: Jan 14, 2013 12:01 AM

I should have probably read the problem before posting. There should be no
issue with branches, since Exp is single-valued. Interestingly, the
incorrect result differs by a phase factor of (-2+Sqrt[3])*Pi.

I also noticed that the documentation states that Integrate computes
multiple integrals. It actually computes interated integrals in prefix
notation:

Integrate[f, y, x] <=> Integrate[dy*Integrate[dx*f]]

This is clear from the following example:

In: Integrate[(x^2-y^2)/(x^2+y^2)^2, {y, 0, 1}, {x, 0, 1}]
Out: -Pi/4

In: Integrate[(x^2-y^2)/(x^2+y^2)^2, {x, 0, 1}, {y, 0, 1}]
Out: Pi/4

Since multiple and iterated integrals are equal only through Fubini's
theorem and similar results, perhaps the documentation should be
corrected.

Alex

On Sat, 12 Jan 2013, Murray Eisenberg wrote:

> 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