Error Function is Odd

Theorem

$\map \erf {-x} = -\map \erf x$

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

\(\ds \)

\(=\)

\(\ds -\frac 2 {\sqrt \pi} \int_0^{-\paren {-x} } e^{-\paren {-u}^2} \rd u\)

substituting $u \mapsto -u$

\(\ds \)

\(=\)

\(\ds -\frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u\)

\(\ds \)

\(=\)

\(\ds -\map \erf x\)

Definition of Error Function

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Error Function $\ds \map \erf x = \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u$: $35.3$