weird bugs in Integrate

*To*: mathgroup at smc.vnet.net*Subject*: [mg75303] weird bugs in Integrate*From*: dimitris <dimmechan at yahoo.com>*Date*: Tue, 24 Apr 2007 03:29:10 -0400 (EDT)

I don't know if it worths two years after Mathematica 5.2 was released to talk about bugs in Integrate but since: a) I couldn't find anything relevant in the archives and b) the integrals are rather trivial and the bugs very clear and weird, here we go... Consider the definite integral of the functions cot and tan in [a,b]. In[30]:= ints = HoldForm[Integrate[{Cot[x], Tan[x]}, {x, a, b}, GenerateConditions -> True]] (*I use the setting GenerateConditions->True because as I was informed does search harder for convergence/divergence. The default setting GenerateConditions->Automatic does not change my findings*) Here we taste Integrate In[31]:= ReleaseHold[ints /. {a -> Pi/3, b -> 2*(Pi/5)}] {N[%], ReleaseHold[ints /. Integrate[x__, GenerateConditions -> True] :> NIntegrate[x] /. {a -> Pi/3, b -> 2*(Pi/5)}]} Out[31]= {(1/2)*Log[(1/6)*(5 + Sqrt[5])], ArcCsch[2]} Out[31]= {{0.0936592,0.481212},{0.0936592,0.481212}} Indeed correct results. Let's make the things a little harder. Obviously, tan(x) has poles at the vanishing points of cos(x) whereas cot(x) has poles at the vanishing points of sin(x). So for example in the range [0,2Pi]. In[32]:= ({#1, Reduce[#1[x] == 0 && 0 <= x <= 2*Pi, x]} & ) /@ {Cos, Sin} Out[32]= {{Cos, x == Pi/2 || x == (3*Pi)/2}, {Sin, x == 0 || x == Pi= || x == 2*Pi}} In[33]:= MapIndexed[Plot[#1[x], {x, 0, 10*Pi}, PlotStyle -> Hue[#2[[1]]/3], PlotLabel -> ToString[#1]] & , {Tan, Cot}]; It is very obvious that the integrals does not converge e.g. in the range [0,4Pi] and [1,4Pi]. And Mathematica understands this. In[34]:= ReleaseHold[ints /. {a -> 0, b -> 4*Pi}] Integrate::idiv: Integral of Cot[x] does not converge on {0, 4\=F0}. Integrate::idiv: Integral of Tan[x] does not converge on {0, 4\=F0}. Out[34]= {Integrate[Cot[x], {x, 0, 4*Pi}, GenerateConditions -> True], Integrate[Tan[x], {x, 0, 4*Pi}, GenerateConditions -> True]} In[35]:= ReleaseHold[ints /. {a -> 1, b -> 4*Pi}] Integrate::idiv: Integral of Cot[x] does not converge on {1, 4\=F0}. Integrate::idiv: Integral of Tan[x] does not converge on {1, 4\=F0}. Out[35]= {Integrate[Cot[x], {x, 1, 4*Pi}, GenerateConditions -> True], Integrate[Tan[x], {x, 1, 4*Pi}, GenerateConditions -> True]} To my surprize however, In[39]:= ReleaseHold[ints /. {a -> 0, b -> Infinity}] Integrate::idiv : Integral of Cot[x] does not converge on {0,Infinity}. Out[39]= {Integrate[Cot[x], {x, 0, Infinity}, GenerateConditions -> True], - Log[Interval[{-1, 1}]]} I=2Ee. Mathematica fails to detect that the integral of Tan[x] does not converge on [0,Infinity). Even worse behavior appeared is the following examples In[42]:= ReleaseHold[ints /. {a -> 1, b -> Infinity}] Out[42]= {Log[Interval[{-Csc[1],Csc[1]}]],Log[Cos[1]]-Log[Interval[{-1,1}]]} In[43]:= ReleaseHold[ints /. {a -> 1000, b -> Infinity}] Out[43]= {Log[Interval[{-Csc[1000], Csc[1000]}]], Log[Cos[1000]] - Log[Interval[{-1, 1}]]} where for both Tan[x] and Cot[x] Mathematica fails to detect the divergence. Dimitris