Re: troubling simple integral
- To: mathgroup at smc.vnet.net
- Subject: [mg91346] Re: troubling simple integral
- From: Alberto Verga <Alberto.Verga at laposte.net>
- Date: Sun, 17 Aug 2008 06:38:35 -0400 (EDT)
- References: <g86844$hr7$1@smc.vnet.net>
On Aug 16, 11:54 am, did <didier.o... at hotmail.com> wrote: > Mathematica 6.0.3.0 has trouble computing the quite simple integral > > In[2]:= FullSimplify[ Integrate[ (b + k*x) / k^2 * Exp[-k*x] * > Sin[k*a] * Sin[k*y] , {k, 0,Infinity}, > Assumptions -> x > 0 && Im[a] == 0 && Im[b] == 0 && Im[y] == = 0 ]] > > Out[2]= -(1/8) I (b + 2 I x Log[1 + (4 a y)/(x^2 + (a - y)^2)]) > > which is obviously wrong. Repeating the computation, I get different > results. > > What should I do to get a correct answer? > (This one I can do by hand, but I have more complicated similar ones > to evaluate.) The integral proportional to b (the one proportional to x poses no problem) gives a strange result if you specifies Assumptions-> Element[{a,y},Reals] In[11]:= Integrate[1/k^2*Exp[-k]*Sin[a k] Sin[y k], {k, 0, Infinity}, Assumptions -> Element[{a, y}, Reals]] Out[11]= I/8 However if you replace the parameters by some constants (here Catalan and EulerGamma) you get a result In[7]:= ii1 = Integrate[ 1/k^2*Exp[-k]*Sin[Catalan k] Sin[EulerGamma k], {k, 0, Infinity}] /. {Catalan -> a, EulerGamma -> y} Out[7]= 1/4 (-2 a (ArcTan[a - y] - ArcTan[a + y]) + 2 y (ArcTan[a - y] + ArcTan[a + y]) + Log[1 + a^2 - 2 a y + y^2] - Log[1 + a^2 + 2 a y + y^2]) The same result can be obtained using the option GenerateConditions -> False: In[9]:= ii2 = Integrate[1/k^2*Exp[-k]*Sin[a k] Sin[y k], {k, 0, Infinity}, GenerateConditions -> False] Out[9]= 1/4 (2 (-a + y) ArcTan[a - y] + 2 (a + y) ArcTan[a + y] + Log[1 + I a - I y] + Log[1 - I a + I y] - Log[I (-I + a + y)] - Log[-I (I + a + y)]) In[10]:= Table[ii1 - ii2, {a, .1, 1., .2}, {y, .1, 1., .2}] // Chop Out[10]= {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}} I agree with you, the output I/8 is difficult to understand... Alberto Verga