Asymptotic Expansion for Exponential Integral Function

Theorem

Formulation 1

Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:

$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$

Then:

$\ds \map \Ei x \sim \frac {e^{-x} } x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {n!} {x^n}$

Formulation 2

Asymptotic Expansion for Exponential Integral Function/Formulation 2