Manipulating $\int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$

Question

Is there a way to express the integral

$I(x_{0}, t) = \int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$,

where $x_{0} \neq 0$ and $t \ge 0$, in terms of more well-known functions? In particular, I am interested in a form where $t$ appears in the limits and not in the integrand, analogous to how something like the integral of $sin (x t) \, \text{e}^{-x^{2}}$ may be expressed in terms of error functions. My attempts thus far have only yielded

$I(x_{0}, t) = \text{e}^{-t^{2} / 4} \Big(\int_{x_{0} - i t/2}^{\infty} \frac{\text{e}^{-x^{2}} (z - i t / 2)}{z^{2} + t^{2} / 4} \, dz + \int_{x_{0} + i t/2}^{\infty} \frac{\text{e}^{-x^{2}} (z + i t / 2)}{z^{2} + t^{2} / 4} \, dz \Big)$,

from which I cannot find a way to proceed further. My next step would be to split the limits, introducing integrals to and from $0$, although this introduces singularities which naturally cancel but are nonetheless undesirable.

Any analyses of the integral are welcome.

For completeness (although not relevant to the actual question itself), I have an incrementally increasing variable $t$ with fixed $x_{0}$, for which I intend to compute this integral. For large $t$, it becomes highly oscillatory.

Answer 1

This is a partial answer. By using the definition of the $\operatorname{Ei}$ exponential integral function we can get for all $a>0$: $$ I(a,0)=-\frac{\operatorname{Ei}(-a^2)}{2}. $$

Answer 2

OP here - I hadn't logged on StackExchange in a while and forgot I had an account when I originally posted.

I've since discovered one way to get $t$ out of the integrand, and that's by differentiating under the integral sign:

$\frac{d I}{d t} = - \int_{x_{0}}^{\infty} \sin (x t) \, \text{e}^{-x^{2}} \, dx = - \frac{1}{2 i} \, \text{e}^{-t^{2}/4} \int_{x_{0} - i t / 2}^{x_{0} + i t / 2} \text{e}^{-z^{2}} \, dz$,

$I (x_{0}, t) = \frac{1}{2} \int_{x_{0}^{2}}^{\infty} \frac{1}{x} \, \text{e}^{-x} \, d x - \frac{1}{2} \int_{0}^{t} \text{e}^{-x^{2} / 4} \, \Big(\int_{0}^{x / 2 - i x_{0}} \, \text{e}^{z^{2}} \, d z + \int_{0}^{x / 2 + i x_{0}} \, \text{e}^{z^{2}} \, d z \Big) \,d x$.

I would still be interested whether there is a better way, though.