The asymptotic expansion of legendre function of large degree and order

1957 ◽  
Vol 249 (971) ◽  
pp. 597-620 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Complex Variable ◽  
Bessel Functions ◽  
Legendre Function ◽  
Large Degree ◽  
Legendre Functions ◽  
Airy Functions ◽  
Exponential Functions ◽  
The Third ◽  
Shaped Domain

New expansions for the Legendre functions and are obtained; m and n are large positive numbers, is kept fixed as is an unrestricted complex variable. Three groups of expansions are obtained. The first is in terms of exponential functions. These expansions are uniformly valid as with respect to z for all z lying in except for the strips given by. The second set of expansions is in terms of Airy functions. These expansions are uniformly valid with respect to z throughout the whole z plane cut from except for a pear-shaped domain surrounding the point z = — 1 and a strip lying immediately below the real z axis for which . The third group of expansions is in terms of Bessel functions of order m . These expansions are valid uniformly with respect to z over the whole cut z plane except for the pear-shaped domain surrounding z = — 1. No expansions have been given before for the Legendre functions of large degree and order.

The asymptotic expansion of bessel functions of large order

1954 ◽  
Vol 247 (930) ◽  
pp. 328-368 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Functions ◽  
Airy Functions ◽  
Elementary Functions ◽  
Large Order ◽  
Hankel Functions ◽  
Rapid Calculation ◽  
Zeros Of Functions ◽  
Significant Figures ◽  
Complex Zeros

New expansions are obtained for the functions Iv{yz), ) and their derivatives in terms of elementary functions, and for the functions J v(vz), Yv{vz), H fvz) and their derivatives in terms of Airy functions, which are uniformly valid with respect to z when | | is large. New series for the zeros and associated values are derived by reversion and used to determine the distribution of the zeros of functions of large order in the z-plane. Particular attention is paid to the complex zeros of 7„(z) and the Hankel functions when the order n is an integer or half an odd integer, and for this purpose some new asymptotic expansions of the Airy functions are derived. Tables are given of complex zeros of Airy functions and other quantities which facilitate the rapid calculation of the smaller complex zeros of 7„(z), 7'(z), and the Hankel functions and their derivatives, when 2 n is an integer, to an accuracy of three or four significant figures.

Integrals with nearly coincident branch points: Gegenbauer polynomials of large degree

2006 ◽  
Vol 463 (2079) ◽  
pp. 697-710 ◽  
Author(s):  
F Ursell
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Functions ◽  
Complex Function ◽  
Maximum Modulus ◽  
Large Degree ◽  
Branch Points ◽  
Complex Function Theory ◽  
Complete Asymptotic Expansion ◽  
Uniform Asymptotic Expansion ◽  
Standard Techniques

The integral considered here is a loop integral of the form where the integrand has branch points at t =± θ , F ( t 2 ,  θ 2 ) is an analytic function of its arguments and N is a large positive parameter. When θ is not small, its complete asymptotic expansion can be found by standard techniques. When θ is small, the branch points are nearly coincident, and it will be shown that there is a uniform asymptotic expansion involving Bessel functions of argument Nθ . An inequality will be established and will be used to show that the expansion is valid in a region including a disc | θ |≤ m of the complex θ -plane, where m does not tend to 0, when N tends to ∞. The proof of this inequality uses the maximum-modulus principle of complex function theory.

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.

Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

2014 ◽  
Vol 11 (01) ◽  
pp. 39-49 ◽  
Author(s):  
Bin Wei
Keyword(s):  
Asymptotic Expansion ◽  
Cusp Form ◽  
Rational Number ◽  
Fourier Coefficients ◽  
Bessel Functions ◽  
Exponential Sums ◽  
Summation Formula ◽  
Cusp Forms ◽  
Real Numbers ◽  
Voronoi Summation

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

The Exponential Nature of Technology

2017 ◽  
Author(s):  
Arlindo Oliveira
Keyword(s):  
Industrial Revolution ◽  
Technological Development ◽  
Exponential Functions ◽  
The Third ◽  
Agricultural Revolution ◽  
Nature Of Technology ◽  
History Of ◽  
Third Industrial Revolution ◽  
Industrial Revolutions

This chapter provides a brief review of the history of technology, covering pre-historical technologies, the agricultural revolution, the first two industrial revolutions, and the third industrial revolution, based on information technology. Evidence is provided that technological development tends to follow an exponential curve, leading to technologies that typically were unpredictable just a few years before. An analysis of a number of exponential trends and behaviors is provided, in order to acquaint the reader with the sometimes surprising properties of exponential growth. In general, exponential functions tend to grow slower than expected in the short term, and faster than expected in the long term. It is this property that make technology evolution very hard to predict.

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

Landau constants and their q-analogues

2015 ◽  
Vol 13 (02) ◽  
pp. 217-231 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Xin Li ◽  
M. Rahman
Keyword(s):  
Asymptotic Expansion ◽  
Large Degree ◽  
Complete Asymptotic Expansion

We derive inequalities and a complete asymptotic expansion for the Landau constants Gn, as n → ∞ using the asymptotic sequence n!/(n + k)!. We also introduce a q-analogue of the Landau constants and calculate their large degree asymptotics. In the process, we also establish q-analogues of identities due to Ramanujan and Bailey.

scholarly journals Integrals involving Bessel and Legendre functions

1965 ◽  
Vol 14 (4) ◽  
pp. 269-272 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Plane Wave ◽  
Bessel Function ◽  
Scattered Wave ◽  
Approximate Expression ◽  
Apex Angle ◽  
Legendre Functions ◽  
The Third ◽  
Image Position

Felsen (1) has shown that when a plane wave is incident along the axis of a rigid cone of narrow apex angle an approximate expression for the scattered wave involves an integral of the formwhere is a Bessel function of the third kind, k a constant and , is the Legendre (conical) function of the first kind.

scholarly journals Some Integrals involving Legendre Functions

1959 ◽  
Vol 11 (3) ◽  
pp. 161-165 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Bessel Functions ◽  
Legendre Functions ◽  
Dirichlet Integral ◽  
Associated Legendre Functions ◽  
Different Order

In this note generalisations of certain integrals involving Legendre functions including the Mehler-Dirichlet integral for Legendre functions of the first kind are given, these new results expressing associated Legendre functions of the first or second kinds as integrals involving corresponding functions of the same degree but different order. These integrals appear to be analogous to Sonine's integral in the theory of Bessel functions.

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