Power Series Expansion for Error Function
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Theorem
- $\ds \map \erf x = \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }$
\(\ds \map \erf x\) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \paren {x - \frac {x^3} {3 \times 1!} + \frac {x^5} {5 \times 2!} - \frac {x^7} {7 \times 3!} + \frac {x^9} {9 \times 2!} - \cdots}\) |
where:
- $\erf$ denotes the error function
- $x$ is a real number.
Proof
\(\ds \map \erf x\) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u\) | Definition of Error Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \int_0^x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {n!} } \rd u\) | Definition of Real Exponential Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {n!} \paren {\int_0^x u^{2 n} \rd u}\) | Power Series is Termwise Integrable within Radius of Convergence | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }\) | Primitive of Power |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Error Function $\ds \map \erf x = \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u$: $35.1$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error function