Power Series Expansion for Error Function

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Theorem

$\displaystyle \map \erf x = \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }$

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