Does Mathematica Integral Score just display as well-formatted text?

Question

Trying the following exponential integral:

Integrate[ Exp[-2 A Sqrt[x^2 + a^2] + I ( x Subscript[k, x] + b )], {x, 0, Infinity} ] 

Am I returning something that is fairly printed but not priced? Does this mean that Mathematica cannot fulfill the integral and fail? I do not see messages about what happened.

To provide context for this question, this is a split up to one dimensional version of the integral that I did with paper and pencil:

 \int d^3 re^{ i \vec{k} \cdot \vec{r} } e^{ - 2 Z r/ a_0 } 

The point of mathematical mathematics was to check if I can check my result:

 16 \pi Z a_0^3/(4 Z^2 + k^2 a_0^2)^2 

Answer 1

That's right, when Mathematica pulls the integral back, usually this means that it does not know how to do it. Sometimes it can execute the integral only under certain assumptions (such as `a> 0 'and real), but it usually gives the answer as a giant if-statement with them.

This particular integral that you have is not trivial at all. And this does not match the form of any special functions that I am familiar with. Thus, it is possible that it cannot be expressed in closed form in terms of special functions that Mathematica has.

Also, what is the point of Subscript[k, x] ? I tried to remove it, and Mathematica still cannot complete the integral.

Answer 2

With a simple change of variables, the "simple" case Exp[-2 A Sqrt[a^2 + x^2] gives unlimited functions:

 Integrate[Exp[-2*A*Sqrt[a^2 + x^2]]*Dt[x] /. x -> Sqrt[t^2 - a^2] /. Dt -> (1 & ), {t, a, Infinity}, Assumptions -> a > 0 < A] 

To obtain

 a*(-BesselK[0, 2*a*A] + BesselK[1, 2*a*A])