How to evaluate this notoriously hard exponential integral

Solution 1:

$\textbf{Ansatz}$ I was able to make some progress in calculating the antiderivative $$I=\int \exp \left( -\beta \exp \left( t~\delta \right) \right) t^{\alpha }dt$$

where $\delta $ may set to $$\delta =1-2H\text{ or }\delta =2-2H$$ The integral may be expressed by the infinity sum

$$I\left( \alpha ,\beta ,\delta ,t\right) =\frac{1}{\delta ^{1+\alpha }}% \sum_{k=0}^{\infty }\frac{\beta ^{k}\left( -1\right) ^{k-\alpha }}{% k!k^{1+\alpha }}\Gamma \left( 1+\alpha ,-k~\delta ~t\right) +C$$ This can be simply proved by the differentiation. For some special values e.g. $\alpha=n>0$ explicit expressions can be obtained stocha. A general expression is in work. For that an intensity study of similar methods, published in the standard publication of [MITTAG-LEFFLER FUNCTIONS] (https://www.researchgate.net/publication/45870179_Mittag-Leffler_Functions_and_Their_Applications) has to be done. Note that $k=0$ is a problem, which has to be taken care of. In case of the integration limits $% [0~~1]$ the sum can be split. If e.g. $\beta =\delta =1$

$$I\left( \alpha ,t\right) =\sum_{k=1}^{\infty }\frac{\left( -1\right) ^{k-\alpha }}{k!k^{1+\alpha }}\Gamma \left( 1+\alpha ,-k~t\right) $$

$$\int_{0}^{1}\exp \left( -\exp \left( t\right) \right) t^{\alpha }dt=\frac{1}{% \alpha +1}+\left( I\left( 1\right) -I\left( 0\right) \right) $$

For other integration limits, one has to expand the integral e.g.

$$\lim_{s\rightarrow 0}\int \exp \left( t~s-\exp \left( t\right) \right) t^{\alpha }dt=\lim_{s\rightarrow 0}\sum_{k=0}^{\infty }\frac{\left( -1\right) ^{1+k}}{k!\left( s-k\right) ^{1+\alpha }}\Gamma \left( 1+\alpha ,\left( s-k\right) ~t\right) $$

Note: This is the Laplacetransform of:

$$L\left[ \exp \left( -\beta \exp \left( t~\delta \right) \right) t^{\alpha }% \right] $$

which may be calculated by the convolution theorem´tattvamasi.

$\textbf{Addendum}$ I finally managed to identify the infinity sum for $T=0$. In this case the solution may be expressed in terms of a $\lambda$-Generalized Hurwitz-Lerch- Zeta Function, which is defined e.g. in Srivastava.