虚误差函数
贡献者： 待更新
虚误差函数的定义为
\begin{equation}
\operatorname{erfi} (x) = - \mathrm{i} \operatorname{erf} ( \mathrm{i} x) = \frac{1}{\sqrt{\pi}} \int_{-x}^x \mathrm{e} ^{t^2} \,\mathrm{d}{t}
= \frac{2}{\sqrt{\pi}} \int_0^x \mathrm{e} ^{t^2} \,\mathrm{d}{t} ~.
\end{equation}
推导如下，使用换元法 $t = \mathrm{i} \tau$ 有
\begin{equation}
\operatorname{erfi} (x) = \frac{- \mathrm{i} }{\sqrt{\pi}} \int_{- \mathrm{i} x}^{ \mathrm{i} x} \mathrm{e} ^{t^2} \,\mathrm{d}{t}
= \frac{- \mathrm{i} }{\sqrt{\pi}} \int_{-x}^{x} \mathrm{e} ^{-( \mathrm{i} \tau)^2} \,\mathrm{d}{( \mathrm{i} \tau)}
= \frac{1}{\sqrt{\pi}} \int_{-x}^{x} \mathrm{e} ^{\tau^2} \,\mathrm{d}{\tau} ~.
\end{equation}
同理可得
\begin{equation}
\operatorname{erfi} ( \mathrm{i} x) = \mathrm{i} \operatorname{erf} (x)~,
\end{equation}
其导函数为
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}{x}} \operatorname{erfi} (x) = \frac{2}{\sqrt{\pi}} \mathrm{e} ^{x^2}~,
\end{equation}
所以
\begin{equation}
\int \mathrm{e} ^{x^2} \,\mathrm{d}{x} = \frac{\sqrt{\pi}}{2} \operatorname{erfi} (x) + C~.
\end{equation}
与误差函数的级数展开同理，$ \operatorname{erfi} (x)$ 的级数展开为
\begin{equation}
\operatorname{erfi} (x) = \frac{2}{\sqrt{\pi}} \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)n!}
= \frac{2}{\sqrt{\pi}} \left(x + \frac{x^3}{3} + \frac{x^5}{10} + \frac{x^7}{42} + \frac{x^9}{216} \dots \right) ~.
\end{equation}