Generalized Heine’s identity for complex Fourier series of binomials

2010 ◽  
Vol 467 (2126) ◽  
pp. 333-345 ◽  
Author(s):  
Howard S. Cohl ◽  
Diego E. Dominici
Keyword(s):  
Fourier Series ◽  
Closed Form ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Complex Fourier Series

In his treatise, Heine ( Heine 1881 In Theorie und Anwendungen ) gave an identity for the Fourier series of the function , with , and z >1, in terms of associated Legendre functions of the second kind . In this paper, we generalize Heine’s identity for the function , with , and , in terms of . We also compute closed-form expressions for some  .

scholarly journals Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 982
Author(s):  
Yujuan Huang ◽  
Jing Li ◽  
Hengyu Liu ◽  
Wenguang Yu
Keyword(s):  
Fourier Series ◽  
Ruin Probability ◽  
Risk Model ◽  
Expansion Method ◽  
Nonparametric Estimator ◽  
Insurance Risk ◽  
Numerical Examples ◽  
Complex Fourier Series ◽  
Premium Income ◽  
Insurance Risk Model

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

scholarly journals Spatially Restricted Integrals in Gradiometric Boundary Value Problems

2009 ◽  
Vol 44 (4) ◽  
pp. 131-148 ◽  
Author(s):  
M. Eshagh
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Legendre Functions ◽  
Gravity Gradiometry ◽  
Satellite Gravity Gradiometry ◽  
Satellite Gravity ◽  
Earth’S Gravity Field ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Slepian Functions

Spatially Restricted Integrals in Gradiometric Boundary Value ProblemsThe spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

scholarly journals Optimal multistage relaxation with automatic parameter selection

2021 ◽  
Author(s):  
Alvin Wong
Keyword(s):  
Fourier Series ◽  
Relaxation Method ◽  
Accurate Solution ◽  
Convergence Time ◽  
Fourier Series Expansion ◽  
End User ◽  
Local Flow ◽  
Flow Problems ◽  
User Expertise ◽  
Complex Fourier Series

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

scholarly journals Optimal multistage relaxation with automatic parameter selection

2021 ◽  
Author(s):  
Alvin Wong
Keyword(s):  
Fourier Series ◽  
Relaxation Method ◽  
Accurate Solution ◽  
Convergence Time ◽  
Fourier Series Expansion ◽  
End User ◽  
Local Flow ◽  
Flow Problems ◽  
User Expertise ◽  
Complex Fourier Series

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

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

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