Modified Bessel Functions of Imaginary Order and Positive Argument

Title: Modified Bessel Functions of Imaginary Order and Positive Argument

Abstract: This research article presents Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order. These functions, denoted as Kia(x) and Lia(x), are independent solutions of the modified Bessel equation of imaginary order. The article discusses the algorithm's accuracy and regions of application, which are valid in all regions of the (x, a) plane except close to the transition line a = x. The algorithm combines different methods of evaluation in different regions, ensuring high accuracy and efficiency. The article also provides a comparison between the different methods, which contributes to the algorithm's reliability.

Research Question: How can we efficiently and accurately compute the modified Bessel functions of purely imaginary order and positive argument?

Methodology: The article proposes an algorithm that combines different methods of evaluation in different regions. This approach ensures high accuracy and efficiency. The algorithm uses the Wronskian relation to validate the methods and ensure their consistency. The article also provides a comparison between the different methods, which contributes to the algorithm's reliability.

Results: The article presents Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order. These programs compute the functions Kia(x) and Lia(x) and their derivatives for real a and positive x. The algorithm's accuracy is better than 10−13 in the (x, a) region D1 = (0, 200) × (−200, 200), better than 5 × 10−13 in region D2 = (0, 500) × (−500, 500), and close to 10−12 in region D3 = (0, 1500) × (−1500, 1500). The article also discusses the regions of applicability of each method, which are valid in all regions of the (x, a) plane except close to the transition line a = x.

Implications: The article's findings have significant implications for the field of numerical analysis. The presented algorithm provides an efficient and accurate method for computing modified Bessel functions of purely imaginary order and positive argument. This will be particularly useful for researchers and practitioners working in areas such as physics, engineering, and mathematics. The algorithm's high accuracy and efficiency make it a valuable tool for solving problems that require the computation of these functions.

Link to Article: https://arxiv.org/abs/0401008v1 Authors: arXiv ID: 0401008v1