Bounds and algorithms for the -Bessel function of imaginary order
Bounds and algorithms for the -Bessel function of imaginary order
复制标题
虚数阶 -Bessel 函数的界限和算法
DOI：
10.1112/s1461157013000028
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
H. Then
中科院分区：
文献类型：
--
作者：
A. Booker;Andreas Strömbergsson;H. Then
利用最速下降路径，证明了修正Bessel函数K ir（x）的精确有界和数值隐含常数
的虚阶及其关于阶的前两个导数。我们还证明了更一般的（混合）衍生物的精确渐近界，而无需计算出数值隐含常数。此外，我们还给出了计算${K}_{ir}（x）$的绝对快速收敛级数
及其导数，以及基于傅立叶插值的公式，用于计算$r$的许多值
.最后，我们在一个软件库中实现了这些特征的一个子集，用于快速和严格地计算${K}_{ir}（x）$
.
Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function ${K}_{ir} (x)$
of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of ${K}_{ir} (x)$
and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of $r$
. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of ${K}_{ir} (x)$
.