Power Series Expansion for Exponential Integral Function

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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:

\(\ds \map \Ei x\) \(=\) \(\ds -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {x^n} {n \times n!}\)
\(\ds \) \(=\) \(\ds -\gamma - \ln x + \frac x {1 \times 1!} - \frac {x^2} {2 \times 2!} + \frac {x^3} {3 \times 3!} - \dots\)


Formulation 2

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

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

Then:

\(\ds \map \Ei x\) \(=\) \(\ds \gamma + \ln x + \sum_{n \mathop = 1}^\infty \frac {x^n} {n \times n!}\)
\(\ds \) \(=\) \(\ds \gamma + \ln x + \frac x {1 \times 1!} + \frac {x^2} {2 \times 2!} + \frac {x^3} {3 \times 3!} + \dots\)