[Date Index]
[Thread Index]
[Author Index]
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
Prev by Date:
**Re: Summary: Which[] as Textbook Input, Plot[] Questions**
Next by Date:
**Re: Summary: Which[] as Textbook Input, Plot[] Questions**
Previous by thread:
**Re: bugs in Mathematica 5.1**
Next by thread:
**Re: bugs in Mathematica 5.1**
| |