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Re: wrong result when computing a definite integral
*To*: mathgroup at smc.vnet.net
*Subject*: [mg129439] Re: wrong result when computing a definite integral
*From*: Alex Krasnov <akrasnov at eecs.berkeley.edu>
*Date*: Mon, 14 Jan 2013 00:02:50 -0500 (EST)
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*References*: <20130111023916.C722668DD@smc.vnet.net> <20130112032244.54DBC691E@smc.vnet.net>
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 at 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 at 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
>
>
>
>
>
>
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