Re: GenerateConditions->False gives fine result!

*To*: mathgroup at smc.vnet.net*Subject*: [mg73976] Re: GenerateConditions->False gives fine result!*From*: "dimitris" <dimmechan at yahoo.com>*Date*: Sat, 3 Mar 2007 23:54:55 -0500 (EST)*References*: <es9238$3ig$1@smc.vnet.net>

Hi and thanks a lot regarding answering to my messages/questions apperared recently about Integrate. > This is because the integral does in fact diverge in the Riemann sense. T= aking the simpler example with {t->3, x->2}: Now I did plots of the integrand I understand where was my mistake. Thanks again! In fact what confused me (which should have confused!) was the fact that In[78]:= (1/(2*Pi*I))*Integrate[q/(E^(2*I*q)*Sqrt[q^2 + 9]), {q, -Infinity, Infinity}, GenerateConditions -> False] N[%] Out[78]= -((3*BesselK[1, 6])/Pi) Out[79]= -0.001283348797177626 In[82]:= Chop[(1/(2*Pi*I))*NIntegrate[q*(Exp[(-I)*q*x]/Sqrt[q^2 + t^2]) /. {t - > 3, x -> 2}, {q, -Infinity, Infinity}, Method -> Oscillatory]] SequenceLimit::seqlim: The general form of the sequence could not be \ determined, and the result may be incorrect. Out[82]= -0.0012833487971660698 I=2Ee. the two results are almost the same; this lead me to ignore the messages from SequenceLimit! Very amateurish mistake! Anyway...Integrate rules! I need more reading! Kind Regards Dimitris =CF/=C7 Bhuvanesh =DD=E3=F1=E1=F8=E5: > This is because the integral does in fact diverge in the Riemann sense. T= aking the simpler example with {t->3, x->2}: > > In[1]:= integrand = q/(E^(2*I*q)*Sqrt[q^2 + 9]); > > In[2]:= Limit[integrand, q->Infinity] > > Out[2]= (1 + I) Interval[{-1, 1}] > > GenerateConditions->False, in addition to checking convergence and lookin= g for singularities, also does Hadamard-type integrals. Here's another exam= ple: > > In[1]:= Integrate[1/(x^(1 + I/2)*(1 + x)), {x, 0, 1}] > > 1 > Integrate::idiv: Integral of ---------------- does not converge on {0, 1}. > 1 + I/2 > x (1 + x) > > 1 > Out[1]= Integrate[----------------, {x, 0, 1}] > 1 + I/2 > x (1 + x) > > In[2]:= Integrate[1/(x^(1 + I/2)*(1 + x)), {x, 0, 1}, GenerateCondition= s->False] //InputForm > > Out[2]//InputForm= (-PolyGamma[0, -I/4] + PolyGamma[0, 1/2 - I/4])/2 > > Bhuvanesh, > Wolfram Research.