Integral of Bessel function, square root and gaussian

[vuellesse]

Hi! Does anyone know how to solve the following integral analitically?

\int^{1}_{0} dx \ e^{B x^{2}} J_{0}(i A \sqrt{1-x^{2}}), where A and B are real numbers.

Thanks!

 

[JDoolin]

Not really sure, but maybe if I take a stab at the question, someone else will answer better.

would it help to replace x=cos(θ)? then x^2 = cos^2(θ), dx =-sin(θ)dθ, \sqrt{1-x^2}=sin(\theta) and have your integral go from θ= ∏/2 to 0.

and you might try looking carefully at each of the "recurrence relations" for Bessel functions and see if they help.