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)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*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 > > > > > >

**References**:**wrong result when computing a definite integral***From:*Dexter Filmore <liquid.phynix@gmail.com>

**Re: wrong result when computing a definite integral***From:*Alex Krasnov <akrasnov@eecs.berkeley.edu>