scholarly journals Integrals involving products of modified Bessel functions of the second kind

1963 ◽  
Vol 6 (2) ◽  
pp. 70-74 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Image Position

It is proposed to establish the two following integrals.where n is a positive integer, x is real and positive, μi and ν are complex, and Δ (n; a) represents the set of parameterswhere n is a positive integer and x is real and positive.

A Note on Some Integrals Involving Bessel Functions

1929 ◽  
Vol 25 (2) ◽  
pp. 130-131
Author(s):  
C. Fox
Keyword(s):  
Positive Integer ◽  
Bessel Functions ◽  
Image Position

The object of this note is to prove the following results, all of which hold when |a| < 1.(2) If r is any positive integer other than zero, the

scholarly journals An Infinite Integral involving a Product of Two Modified Bessel Functions of the Second Kind

1953 ◽  
Vol 1 (4) ◽  
pp. 187-189 ◽  
Author(s):  
T. M. Macrobert
Keyword(s):  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Infinite Integral ◽  
Image Position

The formula to be established iswhere l, m, n. are any numbers real or complex and R(b)>0. A similar result, involving Bessel Functions of the First Kind, was obtained by Hanumanta Rao [Mess, of Maths., XLVII. (1918), pp. 134–137].

scholarly journals On some infinite series involving the zeros of Bessel functions of the first kind

1960 ◽  
Vol 4 (3) ◽  
pp. 144-156 ◽  
Author(s):  
Ian N. Sneddon
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Rational Function ◽  
Recurrence Relation ◽  
Infinite Series ◽  
Bessel Functions ◽  
Zeros Of Bessel Functions ◽  
Image Position

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sumsandwhere m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.

Error bounds for the Liouville–Green (or WKB) approximation

1961 ◽  
Vol 57 (4) ◽  
pp. 790-810 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Error Bounds ◽  
Bessel Functions ◽  
Approximate Solutions ◽  
Parabolic Cylinder ◽  
Real Interval ◽  
Positive Parameter ◽  
Modified Bessel Functions ◽  
Elementary Functions ◽  
Parabolic Cylinder Functions ◽  
Image Position

ABSTRACTError bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equationsin which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u. The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded.Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.

scholarly journals On certain new connections between Legendre and Bessel Functions

1935 ◽  
Vol 4 (3) ◽  
pp. 111-111
Author(s):  
S. C. Mitra
Keyword(s):  
Positive Integer ◽  
Bessel Functions ◽  
Image Position

Let n be a positive integer. Then we know that, if m>– 1,Consider the integralwhich is equal to

scholarly journals Integrals Involving E-Functions and Modified Bessel Functions of the Second Kind

1954 ◽  
Vol 2 (1) ◽  
pp. 52-56 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Image Position

The two following formulae are to be established.If R (m ± n) > 0, | amp z | < π,If p ≧ q + 1, R(k ± n + 2αr) > 0, r = 1, 2, …, p, | amp z | < π,For other values of p and q the formula holds if the integral is convergent.

On the function

1963 ◽  
Vol 59 (4) ◽  
pp. 735-737
Author(s):  
A. S. Meligy ◽  
E. M. EL Gazzy
Keyword(s):  
Positive Integer ◽  
General Formula ◽  
Bessel Functions ◽  
Exponential Integral ◽  
Euler’S Constant ◽  
Image Position ◽  
Euler's Constant

In a previous paper (3) one of us reported an expansion for the exponential integralin terms of Bessel functions. In this note, we shall obtain the more general formulawhere n is any positive integer, γ is Euler's constant andIt reduces to that in (3) when n = 1.

scholarly journals An Asymptotic Expansion for the Error Term in the Brent-McMillan Algorithm for Euler’s Constant

2019 ◽  
Vol 11 (3) ◽  
pp. 60
Author(s):  
R. B. Paris
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Error Estimate ◽  
High Precision ◽  
Error Term ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Optimal Error ◽  
Precise Information ◽  
Large Positive Integer

The Brent-McMillan algorithm is the fastest known procedure for the high-precision computation of Euler&rsquo;s constant &gamma; and is based on the modified Bessel functions I_0(2x) and K_0(2x). An error estimate for this algorithm relies on the optimally truncated asymptotic expansion for the product I_0(2x)K_0(2x) when x assumes large positive integer values. An asymptotic expansion for this optimal error term is derived by exploiting the techniques developed in hyperasymptotics, thereby enabling more precise information on the error term than recently obtained bounds and estimates.

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