誤差関数の定義と性質

応用数学入門　第13章 §13.1　★ $\operatorname{erf},\operatorname{erfc}$ の定義・特殊値。熱伝導の答えで使う。

$$\operatorname{erf}(z)=\frac{2}{\sqrt\pi}\int_0^z \mathrm{e}^{-\eta^2}\,\mathop{}\!\mathrm{d}\eta,\qquad \operatorname{erfc}(z)=1-\operatorname{erf}(z)=\frac{2}{\sqrt\pi}\int_z^\infty \mathrm{e}^{-\eta^2}\,\mathop{}\!\mathrm{d}\eta.$$

性質：$\operatorname{erf}(0)=0$、$\operatorname{erf}(\infty)=1$、$\operatorname{erf}(-z)=-\operatorname{erf}(z)$（奇関数）、$\dfrac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d} z}\operatorname{erf}(z)=\dfrac{2}{\sqrt\pi}\mathrm{e}^{-z^2}$。