scholarly journals Non-classical integrals of Bessel functions

1981 ◽  
Vol 22 (3) ◽  
pp. 368-378 ◽  
Author(s):  
S. N. Stuart
Keyword(s):  
Harmonic Function ◽  
Potential Theory ◽  
Spherical Harmonic ◽  
Bessel Functions ◽  
Delta Function ◽  
Generalized Function ◽  
Definite Integrals ◽  
Spherical Harmonic Function ◽  
Image Position ◽  
Fourier Integrals

AbstractCertain definite integrals involving spherical Bessel functions are treated by relating them to Fourier integrals of the point multipoles of potential theory. The main result (apparently new) concernswhere l1, l2 and N are non-negative integers, and r1 and r2 are real; it is interpreted as a generalized function derived by differential operations from the delta function δ(r1 − r2). An ancillary theorem is presented which expresses the gradient ∇2nYlm(∇) of a spherical harmonic function g(r)YLM(Ω) in a form that separates angular and radial variables. A simple means of translating such a function is also derived.

Some series and integrals involving associated Legendre functions

1931 ◽  
Vol 27 (2) ◽  
pp. 184-189 ◽  
Author(s):  
W. N. Bailey
Keyword(s):  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Definite Integrals ◽  
Image Position ◽  
Limiting Case

1. In a recent paper I have given some definite integrals involving Legendre functions which, as a limiting case, give known results involving Bessel functions. In another paper I have shown how some integrals involving Bessel functions can be obtained from Bateman's integraland the well-known expansion

Analysis of Light Scattering Properties about Dent Nanoparticles upon Wafer

2013 ◽  
Vol 464 ◽  
pp. 94-97
Author(s):  
Lei Gong ◽  
Hong Lu Hou ◽  
Jin Long Zou
Keyword(s):  
Light Scattering ◽  
Harmonic Function ◽  
Cross Section ◽  
Spherical Harmonic ◽  
Theoretical Foundation ◽  
Scattering Process ◽  
Differential Scattering Cross Section ◽  
Scattering Coefficients ◽  
Vector Spherical Harmonic ◽  
Spherical Harmonic Function

The light scattering properties of the dent nanoparticles upon wafers is discussed in this paper. Taking the advantage of the Bobbert-Vlieger (BV) theorem, the scattering model between wafer and dent nanoparticles is established. The scattering process is analyzed and the scattering coefficients are derived by using of the vector spherical harmonic function. The differential scattering cross section (DSCS) of the dent nanoparticles upon the wafer is calculated which is compared with the extended Mie method proved the validity of the method and the influences of the dent position, dent scale and scattering angle on the DSCS are analyzed numerically in details. The result is shown that the effect of the dielectric is smaller than the metal. Therefore, the material of the defect and the shape can be extracted by calculate the DSCS, which provide strong theoretical foundation to the nondestructive detector engineer.

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

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