Category:Power Series Expansion for Exponential Integral Function
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This category contains pages concerning Power Series Expansion for Exponential Integral Function:
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:
\(\ds \map \Ei x\) | \(=\) | \(\ds \gamma + \ln x + \sum_{n \mathop = 1}^\infty \frac {x^n} {n \times n!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \gamma + \ln x + \frac x {1 \times 1!} + \frac {x^2} {2 \times 2!} + \frac {x^3} {3 \times 3!} + \dots\) |
Pages in category "Power Series Expansion for Exponential Integral Function"
The following 3 pages are in this category, out of 3 total.