Mathematica integral evaluation just shows up as nicely formatted text?

Question

Attempting the following exponential integral:

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

I get back something that is pretty printed, but not evaluated? Does this mean that Mathematica could not perform the integral and gave up? I don't see any messages saying that's what happened.

To provide context for this question, this is a stripped down to one dimension version of the integral I'd done with paper and pencil:

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

The point of trying mathematica was to see if I could verify my result:

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

Answer 1

Correct, when Mathematica spits the integral back out it usually means it doesn't know how to do it. Sometimes, it can only do the integral under certain assumptions (such as `a > 0' and real), but it will usually give the answer as a gigantic if-statement with those.

This particular integral you have is not trivial at all. And it doesn't fit the form of any of the special functions that I'm familiar with. So it's possible that it can't be expressed in closed form in terms of the special functions that Mathematica has.

Also, what's the point of the Subscript[k, x]? I tried removing it and Mathematica still can't do the integral.

Answer 2

With a simple change of variables the 'simple' case Exp[-2 A Sqrt[a^2 + x^2] yields besselfunctions:

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 get

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