Re: Solving an integral in the limit.

*To*: mathgroup at smc.vnet.net*Subject*: [mg62423] Re: [mg62389] Solving an integral in the limit.*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Thu, 24 Nov 2005 06:33:21 -0500 (EST)*References*: <200511230612.BAA14091@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Josef Karthauser wrote: > I'm having trouble solving a complicated integral using mathematica, > and I'm looking for some wisdom on the matter. > > The problem can be summarised as follows. Mathematica can determine > the solution to, > > Integrate[E^(I*x^2)/ Sqrt[1 + x^2], {x, 0, Infinity}] > > but if I replace the upper bound with a free variable and take the limit > as it goes to Infinity mathematica doesn't manage it, > > Limit[Integrate[E^(I*x^2)/ Sqrt[1 + x^2], {x, 0, a}], a -> Infinity] > > Surely it should be able to determine that the answer is the same as in > the previous case. Is there anyway to pursuade it? > > Many thanks, > Joe One method might be to use a rule that pulls the limit point of approach into the bound of the integral. The rule below will do this for the upper bound of integration. In[1]:= Unprotect[Integrate]; In[2]:= Integrate /: Limit[Integrate[f_,{x_,a_,b_},opts___], b_->c_] := Integrate[f,{x,a,c},opts] In[3]:= Limit[Integrate[E^(I*x^2)/ Sqrt[1 + x^2], {x, 0, a}], a -> Infinity] Out[3]=//InputForm= (MeijerG[{{1/4, 3/4}, {}}, {{0, 0, 1/2}, {1/2}}, 1/4] + I*MeijerG[{{1/4, 3/4}, {}}, {{0, 1/2, 1/2}, {0}}, 1/4])/(4*Sqrt[2]*Pi) This relies on the good will of Integrate not to ever attempt, for example, an improper integral as a limit of a proper one. In such a scenario you'd get into infinite recursion whenever the proper one returned unevaluated (as happens with the example you give; fortunately the improper one is NOT done as a limit but instead handled directly). I doubt this will help you with your actual problem, by the way. It's just a way to force recognition of a limiting case that can be handled directly, when the "proper" case cannot. Daniel Lichtblau Wolfram Research

**References**:**Solving an integral in the limit.***From:*Josef Karthauser <joe@tao.org.uk>