Re: bugs in Mathematica 5.1

*To*: mathgroup at smc.vnet.net*Subject*: [mg54068] Re: bugs in Mathematica 5.1*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Wed, 9 Feb 2005 09:27:51 -0500 (EST)*Organization*: The University of Western Australia*References*: <cua579$hgs$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <cua579$hgs$1 at smc.vnet.net>, "Gennady Stupakov" <stupakov at yahoo.com> wrote: > Second is a more complicated integral that I recently encounted in my > research. > > In[2]:=Integrate[E^(a*Cos[x] - b*Cos[2*x]), {x, 0, 2*Pi}, > GenerateConditions -> True] > Out[2]=If[Re[a] < Re[b], 2*Pi*BesselI[0, -a + b], Integrate[E^(a*Cos[x] - > b*Cos[2*x]), {x, 0, > 2*Pi},Assumptions -> Re[a] >= Re[b]]] > > Let us check this result comparing it with numerical integration for, say, > b=2 and a=1: > > In[3]:= > b = 2.; > a = 1.; > {Integrate[E^(a*Cos[x] - b*Cos[2*x]), {x, 0, 2*Pi}], NIntegrate[E^(a*Cos[x] > + b*Cos[2*x]), > {x, 0, 2*Pi}]} > Out[5]={7.95493, 20.8711} > > Again, the analytical result is wrong. Actually, you have changed the sign of part of the integrand: Integrate[E^(a*Cos[x] - b*Cos[2*x]), {x, 0, 2*Pi}], NIntegrate[E^(a*Cos[x] + b*Cos[2*x]), {x, 0, 2*Pi}] Nevertheless, the analytic solution is indeed incorrect. To evaluate such integrals in general you can use http://functions.wolfram.com/03.02.23.0007.01 followed by straighforward trig (Fourier) integrals. For example, Integrate[E^(a Cos[x] + b Cos[2 x]), {x, 0, 2 Pi}] == 2 Pi Sum[BesselI[2 m, a] BesselI[m, b], {m, -Infinity, Infinity}] Although from http://functions.wolfram.com/03.02.16.0007.01, BesselI[0,a+b]==Sum[BesselI[m,a] BesselI[m,b], {m,-Infinity,Infinity}] as far as I am aware, there is no simpler closed-form for the sum involving BesselI[2 m, a]. This (doubly) infinite sum is rapidly convergent. For example 2 Pi Sum[BesselI[2 m, a] BesselI[m, b], {m, -4, 4}] /. {b -> 2., a -> 1.} agrees with the result from NIntegrate to better than 1 part in 10^10. Cheers, Paul -- Paul Abbott Phone: +61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul