Calculation of modified Bessel functions in the complex domain

Evaluation of the Modified Bessel Function of the Third Kind of Imaginary Orders

“…Series expansions are expected to work better as x becomes smaller and as a becomes larger; this fact can be understood from Eq. (9).…”

“…In the literature several approaches to the evaluation of K ia (x) can be found ( [2,3,7,[9][10][11]), which are mainly based on integral representations. In [16] the τ -method is used for computing the related function K 1/2+iβ (x), with x, β ∈ (0, 10].…”

Bounds and algorithms for the -Bessel function of imaginary order

“…In particular, [3] gives precise bounds on the error terms and one could in principle follow [29] to get quite precise numerical bounds on the error in the asymptotic expansions of K ir (x) and its derivative with respect to x. Besides, a whole range of methods have been employed to bring the numerical integration forward [15,23], for instance, deforming the contour of integration [25], rearranging the oscillatory integrand [24], using Fourier transform methods [9], using the method of steepest descent [19], [20, pp. 117(bottom)-123]. Moreover, a generalized Simpson rule for numerical quadrature of oscillatory integrals was developed [13], a variety of series and continued fraction expansions have been utilized [10,11,41], and hyperasymptotic expansions established [32].…”