a bug in Integrate (fixing!)
- To: mathgroup at smc.vnet.net
- Subject: [mg78732] a bug in Integrate (fixing!)
- From: dimitris <dimmechan at yahoo.com>
- Date: Sun, 8 Jul 2007 06:06:23 -0400 (EDT)
A (not so trivial!) workaround...
Noting that
In[1157]:=
Integrate[Exp[-z]*ArcTan[Sin[z]/(1 + Cos[z])], {z, 0, Infinity}]
(Collect[#1, Pi] & )[Simplify[Integrate[Exp[-z]*ArcTan[Sin[z]/(1 +
Cos[z])], {z, 0, Pi, Infinity}]]]
(Collect[#1, Pi] & )[Simplify[Integrate[Exp[-z]*ArcTan[Sin[z]/(1 +
Cos[z])], {z, 0, Pi, 3*Pi, Infinity}]]]
(Collect[#1, Pi] & )[Simplify[Integrate[Exp[-z]*ArcTan[Sin[z]/(1 +
Cos[z])], {z, 0, Pi, 3*Pi, 5*Pi, Infinity}]]]
(Collect[#1, Pi] & )[Simplify[Integrate[Exp[-z]*ArcTan[Sin[z]/(1 +
Cos[z])], {z, 0, Pi, 3*Pi, 5*Pi, 7*Pi, Infinity}]]]
Out[1157]=
1/2
Out[1158]=
1/2 - Pi/E^Pi
Out[1159]=
1/2 + (-E^(-3*Pi) - E^(-Pi))*Pi
Out[1160]=
1/2 + (-E^(-5*Pi) - E^(-3*Pi) - E^(-Pi))*Pi
Out[1161]=
1/2 + (-E^(-7*Pi) - E^(-5*Pi) - E^(-3*Pi) - E^(-Pi))*Pi
the requested definite integral is equal to
In[1163]:=
res=HoldForm[1/2 - Pi*Sum[Exp[(-(2*n + 1))*Pi], {n, 0, Infinity}]]
Indeed
In[1165]:=
ReleaseHold[res]
Simplify[ExpToTrig[%]]
Out[1165]=
1/2 - (E^Pi*Pi)/(-1 + E^(2*Pi))
Out[1166]=
(1/2)*(1 - Pi*Csch[Pi])
(*check*)
In[1167]:=
N[%]
Out[1167]=
0.3639854725089334
Cheers
Dimitris