The error function $\mathrm{erf}$ is defined by $$\mathrm{erf}(x)=\dfrac{2}{\sqrt{\pi}}\displaystyle\int_0^x e^{-\tau^2} \mathrm{d}\tau.$$

Graph of $\mathrm{erf}$.
Domain coloring of $\mathrm{erf}$.

Properties

Taylor series for error function

Theorem

The following formula holds: $$\mathrm{erf}(z)=\dfrac{2}{\sqrt{\pi}}e^{-z^2}\displaystyle\sum_{k=0}^{\infty} \dfrac{2^k}{1 \cdot 3 \cdot \ldots \cdot (2k+1)} z^{2k+1},$$ where $\mathrm{erf}$ denotes the error function and $\pi$ denotes pi.

Proof

References

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

Theorem

The following formula holds: $$\mathrm{erf}(\overline{z}) = \overline{\mathrm{erf}(z)},$$ where $\mathrm{erf}$ denotes the error function and $\overline{z}$ denotes the complex conjugate.

Proof

References

Theorem: The following formula holds: $\dfrac{1}{2} \left( 1 + \mathrm{erf} \left( \dfrac{x-\mu}{\sqrt{2}\sigma} \right) \right)=\dfrac{1}{\sigma \sqrt{2 \pi}} \displaystyle\int_{-\infty}^x \exp \left( -\dfrac{(t-\mu)^2}{2\sigma^2} \right)\mathrm{d}t.$

Proof: █