Tables of Normalized Associated Legendre Polynomials

1963 ◽  
Vol 17 (84) ◽  
pp. 470
Author(s):  
J. W. W. ◽  
S. L. Belousov
Keyword(s):  
Legendre Polynomials

scholarly journals Optimal solution of the fractional order breast cancer competition model

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
H. Hassani ◽  
J. A. Tenreiro Machado ◽  
Z. Avazzadeh ◽  
E. Safari ◽  
S. Mehrabi
Keyword(s):  
Breast Cancer ◽  
Fractional Order ◽  
Legendre Polynomials ◽  
Numerical Experiments ◽  
Optimal Solution ◽  
Optimization Method ◽  
Competition Model ◽  
Basis Functions ◽  
Cell Populations ◽  
Operational Matrices

AbstractIn this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the stem, tumor and healthy cells, as well as the effects of excess estrogen and the body’s natural immune response on the cell populations. After combining the operational matrices with the Lagrange multipliers technique we obtain an optimization method for solving the F-BCCM whose convergence is investigated. Several examples show that a few number of basis functions lead to the satisfactory results. In fact, numerical experiments not only confirm the accuracy but also the practicability and computational efficiency of the devised technique.

scholarly journals Experimental study of the $ \gamma p \rightarrow K^{0}\Sigma^{+}$, $ \gamma n \rightarrow K^{0} \Lambda$, and $ \gamma n \rightarrow K^{0} \Sigma^{0}$ reactions at the Mainz Microtron

2019 ◽  
Vol 55 (11) ◽  
Author(s):  
C. S. Akondi ◽  
K. Bantawa ◽  
D. M. Manley ◽  
S. Abt ◽  
P. Achenbach ◽  
...  
Keyword(s):  
Experimental Study ◽  
Cross Sections ◽  
Differential Cross ◽  
Legendre Polynomials ◽  
Neutral Kaon ◽  
Experimental Results ◽  
Differential Cross Sections ◽  
Theoretical Predictions ◽  
Photoproduction Reactions ◽  
Insight Into

Abstract.This work measured $ \mathrm{d}\sigma/\mathrm{d}\Omega$dσ/dΩ for neutral kaon photoproduction reactions from threshold up to a c.m. energy of 1855MeV, focussing specifically on the $ \gamma p\rightarrow K^0\Sigma^+$γp→K0Σ+, $ \gamma n\rightarrow K^0\Lambda$γn→K0Λ, and $ \gamma n\rightarrow K^0 \Sigma^0$γn→K0Σ0 reactions. Our results for $ \gamma n\rightarrow K^0 \Sigma^0$γn→K0Σ0 are the first-ever measurements for that reaction. These data will provide insight into the properties of $ N^{\ast}$N* resonances and, in particular, will lead to an improved knowledge about those states that couple only weakly to the $ \pi N$πN channel. Integrated cross sections were extracted by fitting the differential cross sections for each reaction as a series of Legendre polynomials and our results are compared with prior experimental results and theoretical predictions.

On the circle polynomials of Zernike and related orthogonal sets

1954 ◽  
Vol 50 (1) ◽  
pp. 40-48 ◽  
Author(s):  
A. B. Bhatia ◽  
E. Wolf
Keyword(s):  
Unit Circle ◽  
Generating Functions ◽  
Legendre Polynomials ◽  
Phase Contrast ◽  
Diffraction Theory ◽  
Zernike Polynomials ◽  
Optical Aberrations ◽  
Orthogonal Set ◽  
Complete Orthogonal Set ◽  
Explicit Expressions

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

Angular terms in the electron density for the phosphine molecule

1956 ◽  
Vol 52 (4) ◽  
pp. 703-711 ◽  
Author(s):  
R. A. Ballinger ◽  
N. H. March
Keyword(s):  
Variational Method ◽  
Charge Density ◽  
Electron Density ◽  
Legendre Polynomials ◽  
Molecular Axis ◽  
Thomas Fermi ◽  
Line Charge ◽  
Density Map ◽  
Circular Line

ABSTRACTAn attempt is made to calculate the first few angular terms in an expansion of the electron density for the phosphine molecule in Legendre polynomials. Such an expansion is appropriate for a model in which the three hydrogen nuclei are smeared to form a circular line charge. The Thomas–Fermi approximation has been used in conjunction with the variational method. The variational density employed includes p and f angular terms. An approximate charge density map is constructed for a plane containing the molecular axis in order to demonstrate the effect of the angular terms.

scholarly journals Szegö Kernels and Asymptotic Expansions for Legendre Polynomials

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Roberto Paoletti
Keyword(s):  
Asymptotic Expansions ◽  
Legendre Polynomials ◽  
Geometric Approach ◽  
Szegö Kernel ◽  
Quadric Hypersurface ◽  
Szegő Kernel ◽  
Szego Kernel

We present a geometric approach to the asymptotics of the Legendre polynomials Pk,n+1, based on the Szegö kernel of the Fermat quadric hypersurface, leading to complete asymptotic expansions holding on expanding subintervals of [-1,1].

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