scholarly journals Hearing the shape of an annular drum

1983 ◽  
Vol 24 (4) ◽  
pp. 435-438 ◽  
Author(s):  
H. P. W. Gottlieb
Keyword(s):  
Asymptotic Expansion ◽  
Spectral Function ◽  
Bessel Functions ◽  
Direct Calculation ◽  
Eigenvalue Equation ◽  
Connected Region ◽  
Laplacian Operator ◽  
Geometrical Properties ◽  
Annular Membrane

AbstractThe asymptotic expansion for a spectral function of the Laplacian operator, involving geometrical properties of the domain, is demonstrated by direct calculation for the case of a doubly-connected region in the form of a narrow annular membrane. By utilizing a known formula for the zeros of the eigenvalue equation containing Bessel functions, the area, total perimeter and connectivity are all extracted explicitly.

scholarly journals On hearing the shape of an arbitrary doubly-connected region in R2

1990 ◽  
Vol 31 (4) ◽  
pp. 472-483 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Asymptotic Expansion ◽  
Spectral Function ◽  
Basic Problem ◽  
Outer Boundary ◽  
Connected Region ◽  
Two Dimensional ◽  
Complete Knowledge ◽  
Impedance Condition

AbstractThe basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region in R2 together with an impedance condition on its inner boundary and another impedance condition on its outer boundary, from the complete knowledge of the eigenvalues for the two-dimensional Laplacian using the asymptotic expansion of the spectral function for small positive t.

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 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.

An asymptotic expansion for Bessel functions derived with respect to their order

1961 ◽  
Vol 57 (2) ◽  
pp. 284-287
Author(s):  
A. C. Sim
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Bessel Functions ◽  
Integral Forms ◽  
Previous Discussion

Formulae for Bessel derivatives with respect to order have generally been obtained by differentiating integral forms of the Bessel functions. In this note an asymptotic expansion will be derived from the differential equation satisfied by the functions, and it will be obtained in a form which terminates when the order is half an odd integer. Previous discussion of these functions has been restricted to the order one half. (See, for example, Ansell and Fisher (1) and Oberhettinger(2).)

scholarly journals Asymptotic Expressions for the Bessel Functions and the Fourier-Bessel Expansions

1920 ◽  
Vol 39 ◽  
pp. 13-20
Author(s):  
T. M. MacRobert
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Functions ◽  
Asymptotic Expressions ◽  
Image Position

Asymptotic Expressions for the Bessel Functions.From the asymptotic expansion for Ku(z) it follows that, if − π < amp z < π,This theorem is also true if amp z = ± π; to prove this consider the formula

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