Error function is odd

Theorem

The following formula holds: $$\mathrm{erf}(-z)=-\mathrm{erf}(z),$$ where $\mathrm{erf}$ denotes the error function (i.e. $\mathrm{erf}$ is an odd function).

Proof

From the definition, $$\mathrm{erf}(x)=\dfrac{2}{\sqrt{\pi}} \displaystyle\int_0^x e^{-\tau^2} \mathrm{d}\tau.$$ So, $$\begin{array}{ll} \mathrm{erf}(-x) &= \dfrac{2}{\sqrt{\pi}} \displaystyle\int_0^{-x} e^{-\tau^2} \mathrm{d}\tau \\ &\stackrel{u=-\tau}{=} -\dfrac{2}{\sqrt{\pi}} \displaystyle\int_0^{x} e^{-u^2} \mathrm{d}u, \end{array}$$ proving that $\mathrm{erf}$ is odd. $\blacksquare$

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 7.1.9