Asymptotic Expansion for Exponential Integral Function/Formulation 1

Theorem

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:

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

where $\sim$ denotes asymptotic equivalence as $x \to \infty$.

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Exponential Integral $\ds \map \Ei x = \int_x^\infty \frac {e^{-u} } u \rd u$: $35.9$