# Power Series Expansion for Exponential Integral Function/Formulation 1

## 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

$\blacksquare$