复误差函数的小虚部近似
Accurate approximations for the complex error function with small imaginary argument
AI总结：
本文提出两种高效近似方法，用于计算复误差函数在虚部较小时的值，通过Dawson积分实现快速算法，第一种近似精度达10^-9，第二种达10^-13，后者在所需领域具有更高精度。
摘要：
本文提出两种高效近似方法，用于计算复误差函数$w(z)$在虚部较小时的值，范围为$0 \leq \operatorname{Re}[z] \leq 15$。这些近似方法基于实参数Dawson积分$F(x)$，使快速算法实现高效计算。通过随机输入数$x$和$y$的误差分析显示，在实部和虚部中，第一种近似精度超过$10^{-9}$和$10^{-14}$，而第二种近似精度超过$10^{-13}$和$10^{-14}$。第一种近似计算稍快，但第二种近似在所需领域提供卓越的高精度覆盖。
英文摘要：
In this paper we present two efficient approximations for the complex error function $w \left( {z} \right)$ with small imaginary argument $\operatorname{Im}{\left[ { z } \right]} < < 1$ over the range $0 \le \operatorname{Re}{\left[ { z } \right]} \le 15$ that is commonly considered difficult for highly accurate and rapid computation. These approximations are expressed in terms of the Dawson's integral $F\left( x \right)$ of real argument $x$ that enables their efficient implementation in a rapid algorithm. The error analysis we performed using the random input numbers $x$ and $y$ reveals that in the real and imaginary parts the average accuracy of the first approximation exceeds ${10^{ - 9}}$ and ${10^{ - 14}}$, while the average accuracy of the second approximation exceeds ${10^{ - 13}}$ and ${10^{ - 14}}$, respectively. The first approximation is slightly faster in computation. However, the second approximation provides excellent high-accuracy coverage over the required domain.
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