Asymptotic Expansion for Exponential Integral Function
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Theorem
- $\displaystyle \map \Ei x \sim \frac {e^{-x} } x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {n!} {x^n}$
where:
- $\Ei$ is the exponential integral function
- $\sim$ 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 $\displaystyle \map \Ei x = \int_x^\infty \frac {e^{-u} } u \rd u$: $35.9$