Ei(x)=-Integral from -x to infinity of e^(-t)/t dt
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Theorem
The following formula holds for $x>0$: $$\mathrm{Ei}(x) = \mathrm{PV} -\displaystyle\int_{-x}^{\infty} \dfrac{e^{-t}}{t} \mathrm{d}t,$$ where $\mathrm{Ei}$ denotes the exponential integral Ei, $\mathrm{PV}$ denotes the Cauchy principal value, and $e^{-t}$ denotes the exponential.
Proof
References
- James Whitbread Lee Glaisher: On certain definite integrals involving the exponential-integral (1881)... (previous)... (next)
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $5.1.2$ (note: this reference writes this formula with $\mathrm{Ei}(x)$ instead of $\mathrm{Ei}(-x)$)
