Power Series Expansion for Exponential Integral Function/Formulation 1

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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:

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

where $\gamma$ denotes the Euler-Mascheroni constant.


Proof

\(\ds \map \Ei x\) \(=\) \(\ds -\gamma - \ln x + \int_0^x \frac {1 - e^{-u} } u \rd u\) Characterization of Exponential Integral Function
\(\ds \) \(=\) \(\ds -\gamma - \ln x + \int_0^x \frac 1 u \paren {1 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^n} {n!} } \rd u\) Definition of Real Exponential Function
\(\ds \) \(=\) \(\ds -\gamma - \ln x + \int_0^x \frac 1 u \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {u^n} {n!} } \rd u\)
\(\ds \) \(=\) \(\ds -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n!} \paren {\int_0^x u^{n - 1} \rd u}\) Power Series is Termwise Integrable within Radius of Convergence
\(\ds \) \(=\) \(\ds -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {x^n} {n \times n!}\) Primitive of Power

$\blacksquare$


Sources

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