Re: Bug in Integrate? (v. 5.0)
- To: mathgroup at smc.vnet.net
- Subject: [mg45284] Re: [mg45232] Bug in Integrate? (v. 5.0)
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Sat, 27 Dec 2003 05:00:21 -0500 (EST)
- References: <200312231013.FAA12219@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 23 Dec 2003, at 19:13, Antonio González wrote:
> Trying to find the Fourier series of
>
> f[t_]=Sinh[a Cos[t]]
>
> I tried the integral
>
> J = Integrate[4 Cos[t] Sinh[a Cos[t]]/Pi,{t,0,Pi/2}]
>
> whose result should be
>
> J = 2 BesselI[1,a]
>
> and, in fact, this is the answer in Mathematica 4.1.
> In Mathematica 5.0, I got the wrong result
>
> J = 0
>
> Can anybody explain why?
>
> Antonio
>
>
>
>
First transform the integrand so that Sinh is expressed asa sum of
exponentials. You can do it as follows:
Apart[4*Cos[t]*(Sinh[a*Cos[t]]/Pi) /.
Sinh[x_] -> TrigToExp[Sinh[x]]]
-((2*Cos[t])/(E^(a*Cos[t])*Pi)) + (2*E^(a*Cos[t])*Cos[t])/
Pi
Now let's take the two terms separately and evaluate first in version
5.0 and then in 4.2. Version 5 gives:
Integrate[(2*E^(a*Cos[t])*Cos[t])/Pi, {t, 0, Pi/2}]
-((a*BesselI[1, a])/Sqrt[a^2]) + StruveL[-1, a]
and
Integrate[(2*Cos[t])/(E^(a*Cos[t])*Pi), {t, 0, Pi/2}]
-((a*BesselI[1, a])/Sqrt[a^2]) + StruveL[-1, a]
Note that we got the same answers, so they cancel giving the (wrong)
answer 0.
In Mathematica 4.2 on the other hand the answers are:
Integrate[(2*E^(a*Cos[t])*Cos[t])/Pi, {t, 0, Pi/2}]
BesselI[1, a] + StruveL[-1, a]
Integrate[(2*Cos[t])/(E^(a*Cos[t])*Pi), {t, 0, Pi/2}]
-BesselI[1, a] + StruveL[-1, a]
Hence subtracting the answers we get the correct value 2 BesselI[1, a] .
To conclude: the wrong answer given by Mathematica 5.0 appears to be
caused by the following:
J=Integrate[(2*E^(a*Cos[t])*Cos[t])/Pi, {t, 0, Pi/2}]
-((a*BesselI[1, a])/Sqrt[a^2]) + StruveL[-1, a]
which is correct only when a<0
FullSimplify[J, a < 0]
BesselI[1, a] + StruveL[-1, a]
but wrong when a>0:
FullSimplify[J, a > 0]
-BesselI[1, a] + StruveL[-1, a]
Andrzej Kozlowski
Chiba, Japan
http://www.mimuw.edu.pl/~akoz/
- References:
- Bug in Integrate? (v. 5.0)
- From: "Antonio González" <gonferREMOVE@esi.us.es>
- Bug in Integrate? (v. 5.0)