Consider the following contour integral:

$$\oint_C \frac{dz}{z} e^{-a \sqrt{z} + b/z} e^{t z} $$

where $C$ is the contour defined here. We apply Cauchy's theorem and set the integral to zero. As the outer radius $\to \infty$, we find that

$$\int_{c-i \infty}^{c+i \infty} \frac{ds}{s} e^{-a \sqrt{s} + b/s} e^{s t} + \int_{-\infty}^0 \frac{dx}{x} e^{-i a \sqrt{x}-b/x} e^{-t x} + i \int_{\pi}^{-\pi} d\phi \ \, e^{-a \sqrt{\epsilon} e^{i \phi/2}} e^{b/\epsilon e^{-i \phi}} e^{t \epsilon e^{i \phi}} \\ + \int_0^{\infty} \frac{dx}{x} e^{i a \sqrt{x}-b/x} e^{-t x} = 0$$

The third term on the LHS is actually quite tricky. Through a substitution, I get

$$-i 2 \int_{-\pi/2}^{\pi/2} d\phi \, e^{-a \sqrt{\epsilon} e^{i \phi}} e^{b/\epsilon e^{i 2 \phi}} e^{t \epsilon e^{i 2 \phi}} $$

which we can express as a contour integral over a half-circle connected by a line segment about the imaginary axis with a small detour about the origin. As $\epsilon \to 0$, the integral about that line segment goes to zero and the integral is simply $i \pi$ times the sum of the residues at the origin of

$$(2/z) e^{-a \sqrt{\epsilon} z} e^{b/(\epsilon z^2)} e^{-t \epsilon z^2}$$

Thus, leaving out all of the fun details of expanding the above in a Laurent series, I get that

$$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} \frac{ds}{s} e^{-a \sqrt{s} + b/s} e^{s t} = I_0\left ( 2 \sqrt{b t} \right ) + \sum_{k=1}^{\infty} \frac{(a^2 b)^k}{k! (2 k)!}-\frac1{\pi} \int_{-\infty}^{\infty} dx \, \frac{\sin{a x}}{x} e^{-b/x^2} e^{-t x^2}$$

You would think that the sum in the middle would be some well-known special function like a Bessel or something, but no dice. (Mathematica returns $\, _0F_2\left(;\frac{1}{2},1;\frac{a^2 b}{4}\right)$ which does not ring a bell.) I imagine I could sit around and play with the properties of this function but I would prefer not to reinvent the wheel if I do not need to.

The integral looks like a train wreck, i.e., I cannot think of a way to evaluate it yet. Of course you all will be the first to know if I get any flashes of brilliance.