
integration question: 1/(Sqrt[1  x^2])*Exp[(1  x)^2]
Posted:
Aug 5, 2013 3:58 PM


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((1x)^2) will go to zero also just as fast as exp((0x)^2)?
thanks, Nasser

