When is the Modified Bessel Function Equal to Its Derivative?

SIAM Review ◽  
1977 ◽  
Vol 19 (4) ◽  
pp. 737-737
Author(s):  
I. Nåsell
Keyword(s):  
Bessel Function ◽  
Modified Bessel Function

scholarly journals Bounds and algorithms for the -Bessel function of imaginary order

2013 ◽  
Vol 16 ◽  
pp. 78-108 ◽  
Author(s):  
Andrew R. Booker ◽  
Andreas Strömbergsson ◽  
Holger Then
Keyword(s):  
Bessel Function ◽  
Convergent Series ◽  
Steepest Descent ◽  
Modified Bessel Function ◽  
Software Library ◽  
Asymptotic Bounds ◽  
Mixed Derivatives ◽  
Rigorous Computation ◽  
Working Out ◽  
Precise Asymptotic

AbstractUsing 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${K}_{ir} (x)$.

scholarly journals An Integral involving a Modified Bessel Function of the Second Kind and an E-function

1953 ◽  
Vol 1 (3) ◽  
pp. 119-120 ◽  
Author(s):  
Fouad M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Modified Bessel Function ◽  
Image Position

§ 1. Introductory. The formula to be established iswhere m is a positive integer,and the constants are such that the integral converges.

scholarly journals The Distribution of Larvae of Randomly Moving Insects

1961 ◽  
Vol 14 (4) ◽  
pp. 598 ◽  
Author(s):  
EJ Williams
Keyword(s):  
Normal Distribution ◽  
Bessel Function ◽  
Theoretical Distribution ◽  
Frequency Function ◽  
Published Data ◽  
Modified Bessel Function ◽  
Theoretical Conclusion ◽  
Constant Rate ◽  
Spatially Distributed ◽  
Good Agreement

Though randomly moving insects released from a central point in a uniform environment are often found to be distributed according to a circular normal distribution, their larvae will not conform to this distribution. When such insects lay at a constant rate and are subject to constant mortality, their larvae are found to be spatially distributed according to a highly peaked frequency function, depending on the modified Bessel function of the second kind. This theoretical conclusion is in good agreement with published data. Some of the properties of the theoretical distribution are discussed.

A modified bessel function model in life testing

Metrika ◽  
1967 ◽  
Vol 11 (1) ◽  
pp. 133-144 ◽  
Author(s):  
Samir Kumar Bhattacharya
Keyword(s):  
Bessel Function ◽  
Life Testing ◽  
Modified Bessel Function ◽  
Function Model
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