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,....

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

1991 ◽  
Vol 43 (3) ◽  
pp. 628-651 ◽  
Author(s):  
R. Wong ◽  
T. Lang
Keyword(s):  
Bessel Function ◽  
Critical Points ◽  
Bessel Functions ◽  
Positive Order ◽  
Inflection Points ◽  
Increasing Functions

Let jν, 1, jν,2, … denote the positive zeros of the Bessel function Jν(x), and similarly, let j'v,1, j'v,2, … denote the positive zeros of J'v(x), which are the positive critical points of Jv(x). It is well-known that when v is positive, both jν ,k. it and j'ν k are increasing functions of ν; see, e.g., [12, pp. 246 and 248]. Recently, Lorch and Szego [6] have attempted to show that the same is true for the positive zeros j″v,1, j″v,2, … of j″v(x), which are the positive inflection points of Jv(x).

scholarly journals Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales

2014 ◽  
Vol 470 (2162) ◽  
pp. 20130529 ◽  
Author(s):  
Eric A. Galapon ◽  
Kay Marie L. Martinez
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Asymptotic Expansions ◽  
Integral Method ◽  
Asymptotic Series ◽  
Poincaré Series ◽  
Poincare Series ◽  
Half Integer ◽  
Distributional Method ◽  
Finite Sum

We obtain an exactification of the Poincaré asymptotic expansion (PAE) of the Hankel integral, as , using the distributional approach of McClure & Wong. We find that, for half-integer orders of the Bessel function, the exactified asymptotic series terminates, so that it gives an exact finite sum representation of the Hankel integral. For other orders, the asymptotic series does not terminate and is generally divergent, but is amenable to superasymptotic summation, i.e. by optimal truncation. For specific examples, we compare the accuracy of the optimally truncated asymptotic series owing to the McClure–Wong distributional method with owing to the Mellin–Barnes integral method. We find that the former is spectacularly more accurate than the latter, by, in some cases, more than 70 orders of magnitude for the same moderate value of b . Moreover, the exactification can lead to a resummation of the PAE when it is exact, with the resummed Poincaré series exhibiting again the same spectacular accuracy. More importantly, the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences.

Some new asymptotic expansions for Bessel functions of large orders

1952 ◽  
Vol 48 (3) ◽  
pp. 414-427 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Asymptotic Expansion ◽  
Recent Work ◽  
Asymptotic Expansions ◽  
Bessel Functions

During the course of recent work (6) on the zeros of the Bessel functions Jn(x) and Yn(x), it became evident that the theory of the asymptotic expansion of Bessel functions whose arguments and orders are of comparable magnitudes was incomplete. The existing expansions for large orders are those of Debye and Meissel, detailed derivations of both of which are given by Watson ((8), pp. 237–48).

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.

scholarly journals The Modified Bessel Function Kn(z)

1919 ◽  
Vol 38 ◽  
pp. 10-19
Author(s):  
T. M. MacRobert
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Bessel Functions ◽  
Modified Bessel Function ◽  
Definition Of ◽  
Image Position

Gray and Mathews, in their treatise on Bessel Functions, define the function Kn(z) to beWe shall denote this function by Vn(z). This definition only holds when z is real, and R(n)≧0. The asymptotic expansion of the function is also given; but the proof, which is said to be troublesome and not very satisfactory, is omitted. Basset (Proc. Camb. Phil. Soc., Vol. 6) gives a similar definition of the function.

Inflection Points of Bessel Functions of Negative Order

1991 ◽  
Vol 43 (6) ◽  
pp. 1309-1322 ◽  
Author(s):  
Lee Lorch ◽  
Martin E. Muldoon ◽  
Peter Szego
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Positive Zero ◽  
Second Derivative ◽  
Monotonicity Properties ◽  
Negative Order ◽  
Inflection Points

AbstractWe consider the positive zeros j″vk, k = 1, 2,…, of the second derivative of the Bessel function Jν(x). We are interested first in how many zeros there are on the interval (0,jν1), where jν1 is the smallest positive zero of Jν(x). We show that there exists a number ƛ = —0.19937078… such that and . Moreover, j″v1 decreases to 0 and j″ν2 increases to j″01 as ν increases from ƛ to 0. Further, j″vk increases in —1 < ν< ∞, for k = 3,4,… Monotonicity properties are established also for ordinates, and the slopes at the ordinates, of the points of inflection when — 1 < ν < 0.

scholarly journals Eisenstein series and an asymptotic for the K-Bessel function

2021 ◽  
Author(s):  
Jimmy Tseng
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Eisenstein Series ◽  
Uniform Estimate ◽  
Positive Real ◽  
Dominant Term ◽  
Complex Order ◽  
Fixed Parameter ◽  
Real Argument ◽  
Real Analytic

AbstractWe produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ y → ∞ . When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

scholarly journals Accurate analytic approximation for the Chapman grazing incidence function

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Dmytro Vasylyev
Keyword(s):  
Asymptotic Expansion ◽  
Radiative Transfer ◽  
Bessel Functions ◽  
Grazing Incidence ◽  
Analytical Approximation ◽  
Analytic Approximation ◽  
Atmospheric Physics ◽  
Uniform Asymptotic Expansion ◽  
Grazing Angles ◽  
Physical And Chemical

AbstractA new analytical approximation for the Chapman mapping integral, $${\text {Ch}}$$ Ch , for exponential atmospheres is proposed. This formulation is based on the derived relation of the Chapman function to several classes of the incomplete Bessel functions. Application of the uniform asymptotic expansion to the incomplete Bessel functions allowed us to establish the precise analytical approximation to $${\text {Ch}}$$ Ch , which outperforms established analytical results. In this way the resource consuming numerical integration can be replaced by the derived approximation with higher accuracy. The obtained results are useful for various branches of atmospheric physics such as the calculations of optical depths in exponential atmospheres at large grazing angles, physical and chemical aeronomy, atmospheric optics, ionospheric modeling, and radiative transfer theory.

Asymptotic Expansions in the Exponent: a Compound Poisson Approach

1997 ◽  
Vol 29 (02) ◽  
pp. 374-387 ◽  
Author(s):  
V. Čekanavičius
Keyword(s):  
Asymptotic Expansion ◽  
Large Deviations ◽  
Poisson Distribution ◽  
Asymptotic Expansions ◽  
Compound Poisson ◽  
Poisson Measures ◽  
Poisson Approximations

The accuracy of the Normal or Poisson approximations can be significantly improved by adding part of an asymptotic expansion in the exponent. The signed-compound-Poisson measures obtained in this manner can be of the same structure as the Poisson distribution. For large deviations we prove that signed-compound-Poisson measures enlarge the zone of equivalence for tails.

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