Date: Aug 5, 2013 3:58 PM
Author: Nasser Abbasi
Subject: integration question: 1/(Sqrt[1 - x^2])*Exp[-(1 - x)^2]

Integration experts:

Looking at

1/(Sqrt[1 - x^2])*Exp[-(1 - x)^2]

and trying to integrate it from zero to infinity.

When replacing Exp[-(1 - x)^2] by Exp[-(x)^2] it works.
For example, on Mathematica:

------------------------
Integrate[1/(Sqrt[1 - x^2])*Exp[-(x)^2], {x, 0, Infinity}]
(* -((I*BesselK[0, -(1/2)])/(2*Sqrt[E])) *)
-------------------------

Notice:

Limit[1/(Sqrt[1 - x^2])*Exp[-(x^2)], x -> Infinity]

gives zero as expected. But

Limit[1/(Sqrt[1 - x^2])*Exp[-(1 - x^2)], x -> Infinity]

gives

DirectedInfinity[-I]

So the problem is that replacing Exp[-(x)^2] by Exp[-(1 - x)^2]
makes the integrand blow up.

And I am not sure I understand this part since as x->large value, then
exp(-(1-x)^2) will go to zero also just as fast as exp(-(0-x)^2)?


thanks,
--Nasser