Re: wrong result when computing a definite integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg129427] Re: wrong result when computing a definite integral*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Sat, 12 Jan 2013 21:51:50 -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>

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>