Power Series Expansion for Exponential Integral Function plus Logarithm

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Theorem

$\displaystyle \map \Ei x = -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {x^n} {n \times n!}$

where:

$\Ei$ denotes the exponential integral function
$\gamma$ denotes the Euler-Mascheroni constant
$x$ is a real number with $x > 0$.


Proof

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

$\blacksquare$


Sources