
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 singlevalued. 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^2y^2)/(x^2+y^2)^2, {y, 0, 1}, {x, 0, 1}] Out: Pi/4
In: Integrate[(x^2y^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%2C2Pi%2C2Pi%7D%2C%7By%2CPi%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 noninteger factors in the exponential. > >  > Murray Eisenberg murray@math.umass.edu > Mathematics & Statistics Dept. > Lederle Graduate Research Tower phone 413 5491020 (H) > University of Massachusetts 413 5 (W) > 710 North Pleasant Street fax 413 5451801 > Amherst, MA 010039305 > > > > > >

