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?

  • 1
    $\begingroup$ I think the Ei(x) function helps you. $\endgroup$
    – mick
    Jul 3 '15 at 14:18
  • $\begingroup$ you may use the fact that the integral is clearly dominated from the values of $x$ near $C$ $\endgroup$
    – tired
    Jul 3 '15 at 15:18

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.


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