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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
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