Bounds for double exponential integrals I understand that the double-exponential integral
$$
F(a,b,C) := \int_{C}^\infty \exp(-a \exp(b x)) \, dx \quad \text{(with $a,b>0$ and $C \geq 0$)}
$$
can in general not be solved in closed-form.
I wonder wether there are 'simple expressions' in terms of $a,b,C$ as upper and lower bounds for $F(a,b,C)$ available?

Replace $x$ with $\frac{y}{b}$, then $y$ with $\log z$. Then you are left with an exponential integral, for which there are many well-known approximations, for instance the one given by the Gauss continued fraction.
