scholarly journals INCOMPLETE BESSEL FUNCTIONS. II. ASYMPTOTIC EXPANSIONS FOR LARGE ARGUMENT

2007 ◽  
Vol 50 (3) ◽  
pp. 711-723 ◽  
Author(s):  
D. S. Jones
Keyword(s):  
Saddle Point ◽  
Bessel Function ◽  
Asymptotic Expansions ◽  
Bessel Functions

AbstractAsymptotic expansions for an incomplete Bessel function of large argument are derived when the parametric point (a) is well away from any saddle point, (b) coincides with a saddle point and (c) is in the neighbourhood of a saddle point.

Asymptotic behaviour of the inflection points of Bessel functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 509-518 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Behaviour ◽  
Recent Result ◽  
Bessel Function ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Inflection Points

Asymptotic expansions are derived for the inflection points j " vk of the Bessel function J v ( x ), as k → ∞ for fixed v and as v → ∞ for fixed k . Also derived is an asymptotic expansion of J v ( j" vk ) as v → ∞. Finally, we prove that j" vʎ ≽ v √2 if ʎ ≽ (0.07041) v + 0.25 and v ≽ 7, which implies by a recent result of Lorch & Szego that the sequence {| J v ( j" vk )|} is decreasing, for k ═ ʎ , ʎ + 1, ʎ + 2,....

Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 284-292 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Lommel Functions ◽  
Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

Uniform asymptotic expansions for oblate spheroidal functions I: positive separation parameter λ

1992 ◽  
Vol 121 (3-4) ◽  
pp. 303-320 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Wave Equation ◽  
Half Plane ◽  
Positive Constant ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Small Positive Constant ◽  
Separation Parameter ◽  
Oblate Spheroidal ◽  
Uniform Asymptotic Expansions

SynopsisUniform asymptotic expansions are derived for solutions of the spheroidal wave equation, in the oblate case where the parameter µ is real and nonnegative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u →∞, three types of expansions are derived for oblate spheroidal functions, which involve elementary, Airy and Bessel functions. Let δ be an arbitrary small positive constant. The expansions are uniformly valid for λ/u2 fixed and lying in the interval (0,2), and for λ / u2when 0<λ/u2 < 1, and when 1 = 1≦λ/u2 < 2. The union of the domains of validity of the various expansions cover the half- plane arg (z)≦ = π/2.

Higher monotonicity properties of normalized Bessel functions

2014 ◽  
Vol 12 (05) ◽  
pp. 1461007
Author(s):  
Yongping Liu
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Positive Zero ◽  
Local Maxima ◽  
Monotonicity Properties

Denote by Jν the Bessel function of the first kind of order ν and μν,k is its kth positive zero. For ν > ½, a theorem of Lorch, Muldoon and Szegö states that the sequence [Formula: see text] is decreasing, another theorem of theirs states that the sequence [Formula: see text] has higher monotonicity properties. In the present paper, we proved that when ν > ½ the sequence [Formula: see text] has higher monotonicity properties and the properties imply those of the sequence of the local maxima of the function x-ν+1|Jν-1(x)|, x ∈ (0, ∞), i.e. the sequence [Formula: see text] has higher monotonicity properties.

On the Points of Inflection of Bessel Functions of Positive Order, I

1990 ◽  
Vol 42 (5) ◽  
pp. 933-948 ◽  
Author(s):  
Lee Lorch ◽  
Peter Szego
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Primary Concern ◽  
Second Derivative ◽  
Positive Order ◽  
Fixed Rank

The primary concern addressed here is the variation with respect to the order v > 0 of the zeros jʺvk of fixed rank of the second derivative of the Bessel function Jv(x) of the first kind. It is shown that jʺv1 increases 0 < v < ∞ (Theorem 4.1) and that jʺvk increases in 0 < v ≤ 3838 for fixed k = 2, 3,… (Theorem 10.1).

scholarly journals The Asymptotic Expansions of the Spherical Harmonics

1922 ◽  
Vol 41 ◽  
pp. 82-93
Author(s):  
T. M. MacRobert
Keyword(s):  
Real Axis ◽  
Spherical Harmonics ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
The Real ◽  
Image Position

Associated Legendre Functions as Integrals involving Bessel Functions. Let,where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

scholarly journals The zeros of bessel functions.

1918 ◽  
Vol 94 (659) ◽  
pp. 190-206 ◽  
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Practical Importance ◽  
Zeros Of Bessel Functions ◽  
Order Of Magnitude ◽  
Low Order

1. Although many results are known concerning the zeros of Bessel functions,* the greater number of these results are of practical importance only in the case of functions of comparatively low order. For example, McMahon has given a formula† for calculating the zeros of the Bessel function J n ( x ), namely that, if k 1 , k 2 , k 3 , ..., are the positive zeros arranged in ascending order of magnitude, then ks = β- 4 n 2 -1/8β - 4(4 n 2 -1)(28 n 2 -31)/3.(8β) 3 -..., where β = 1/4 π (2 n +4 s —1).

scholarly journals K-Bessel functions associated to a 3-rank Jordan algebra

2005 ◽  
Vol 2005 (18) ◽  
pp. 2863-2870
Author(s):  
Hacen Dib
Keyword(s):  
Bessel Function ◽  
Linear Combination ◽  
Jordan Algebra ◽  
Bessel Functions

Using the Bessel-Muirhead system, we can express theK-Bessel function defined on a Jordan algebra as a linear combination of the J-solutions. We determine explicitly the coefficients when the rank of this Jordan algebra is three after a reduction to the rank two. The main tools are some algebraic identities developed for this occasion.

scholarly journals Unified integrals involving product of multivariable polynomials and generalized Bessel functions

2019 ◽  
Vol 38 (6) ◽  
pp. 73-83
Author(s):  
K. S. Nisar ◽  
D. L. Suthar ◽  
Sunil Dutt Purohit ◽  
Hafte Amsalu
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
General Character ◽  
Polynomial Functions ◽  
Finite Product ◽  
Generalized Bessel Function ◽  
Integral Formulas ◽  
Generalized Bessel Functions ◽  
Multivariable Polynomial ◽  
Integral Formulae

The aim of this paper is to evaluate two integral formulas involving a finite product of the generalized Bessel function of the first kind and multivariable polynomial functions which results are expressed in terms of the generalized Lauricella functions. The major results presented here are of general character and easily reducible to unique and well-known integral formulae.

A family of hyper-Bessel functions and convergent series in them

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Power Series ◽  
Bessel Function ◽  
Classical Theory ◽  
Bessel Functions ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Convergent Series ◽  
Multi Index ◽  
Fatou Theorem ◽  
Convergence Of Series

AbstractThe Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

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