ewald notes references
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@article{leeuwElectrostaticLatticeSums1979,
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title = {Electrostatic Lattice Sums for Semi-Infinite Lattices},
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volume = {37},
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issn = {0026-8976},
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doi = {10.1080/00268977900100951},
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abstract = {The techniques for the rapid computation of energies of three-dimensional neutral periodic assemblies of charged particles are extended to semi-infinite arrays and assemblies of ions in infinite filsm. The results will be useful for simulation of ionic movements in fast-ion conductors and dense colloidal dispersions.},
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number = {4},
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journal = {Molecular Physics},
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author = {Leeuw, Simon W. De and Perram, John W.},
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month = apr,
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year = {1979},
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pages = {1313-1322},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/KDUKWQIW/10.1080@00268977900100951.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/BSCI3PVD/00268977900100951.html}
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}
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@article{mazarsLeknerSummationsEwald2005,
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archivePrefix = {arXiv},
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eprinttype = {arxiv},
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eprint = {cond-mat/0301161},
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title = {Lekner Summations and {{Ewald}} Summations for Quasi-Two Dimensional Systems},
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volume = {103},
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issn = {0026-8976, 1362-3028},
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doi = {10.1080/00268970412331332934},
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abstract = {Using the specific model of a bilayer of classical charged particles (bilayer Wigner crystal), we compare the predictions for energies and pair distribution functions obtained by Monte Carlo simulations using three different methods available to treat the long range Coulomb interactions in systems periodic in two directions but bound in the third one. The three methods compared are: the Ewald method for quasi-two dimensional systems [D.E. Parry, Surf. Sci. \$$\backslash$bm\{49\}\$, 433 (1975); $\backslash$it\{ibid.\}, \$$\backslash$bm\{54\}\$, 195 (1976)], the Hautman-Klein method [J. Hautman and M.L. Klein, Mol. Phys. \$$\backslash$bm\{75\}\$, 379 (1992)] and the Lekner summations method [J. Lekner, Physica A\$$\backslash$bm\{176\}\$, 485 (1991)]. All of the three method studied in this paper may be applied to any quasi-two dimensional systems, including those having not the specific symmetry of slab systems. For the particular system used in this work, the Ewald method for quasi-two dimensional systems is exact and may be implemented with efficiency; results obtained with the other two methods are systematically compared to results found with the Ewald method. General recommendations to implement with accuracy, but not always with efficiency, the Lekner summations technique in Monte Carlo algorithms are given.},
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number = {9},
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journal = {Molecular Physics},
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author = {Mazars, M.},
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month = may,
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year = {2005},
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keywords = {Condensed Matter - Statistical Mechanics},
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pages = {1241-1260},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/26XXWVMR/Mazars - 2005 - Lekner summations and Ewald summations for quasi-t.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/7TE92NAR/0301161.html}
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}
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@incollection{weisSimpleDipolarFluids,
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series = {Advances in Polymer Science},
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title = {Simple {{Dipolar Fluids}} as {{Generic Models}} for {{Soft Matter}}},
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isbn = {978-3-540-26091-2 978-3-540-31581-0},
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abstract = {The physical properties, based on simulation results, of model fluids and solids bearing an electric or magnetic point dipole moment are described. Comparison is made with experimental data on ferrofluids and electro- or magneto-rheological fluids. The qualitative agreement between experiment and simulation shows the interest of these simple models for the comprehension of physical systems where the dipolar interaction dominates.},
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language = {en},
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booktitle = {Advanced {{Computer Simulation Approaches}} for {{Soft Matter Sciences II}}},
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publisher = {{Springer, Berlin, Heidelberg}},
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author = {Weis, J.-J. and Levesque, D.},
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pages = {163-225},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/ESRTJRFN/10.1007@b136796(1).pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/TWV6D7D3/b136796.html},
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doi = {10.1007/b136796}
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}
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@incollection{arnoldEfficientMethodsCompute,
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series = {Advances in Polymer Science},
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title = {Efficient {{Methods}} to {{Compute Long}}-{{Range Interactions}} for {{Soft Matter Systems}}},
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isbn = {978-3-540-26091-2 978-3-540-31581-0},
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abstract = {An extensive introduction to the topic of how to compute long-range interactions efficiently is presented. First, the traditional Ewald sum for 3D Coulomb systems is reviewed, then the P3M method of Hockney and Eastwood is discussed in some detail, and alternative ways of dealing with the Coulomb sum are briefly mentioned. The best strategies to perform the sum under partially periodic boundary conditions are discussed, and two recently developed methods are presented, namely the MMM2D and ELC methods for two-dimensionally periodic boundary conditions, and the MMM1D method for systems with only one periodic coordinate. The dipolar Ewald sum is also reviewed. For some of the methods, error formulas are provided which enable the algorithm to be tuned at a predefined accuracy. Open image in new window},
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language = {en},
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booktitle = {Advanced {{Computer Simulation Approaches}} for {{Soft Matter Sciences II}}},
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publisher = {{Springer, Berlin, Heidelberg}},
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author = {Arnold, Axel and Holm, Christian},
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pages = {59-109},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/6BFTZWI7/10.1007@b136793(1).pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/99GCVXNR/b136793.html},
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doi = {10.1007/b136793}
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}
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@article{harrisEwaldSummationsSystems1998,
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title = {Ewald Summations in Systems with Two-Dimensional Periodicity},
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volume = {68},
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issn = {1097-461X},
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doi = {10.1002/(SICI)1097-461X(1998)68:6<385::AID-QUA2>3.0.CO;2-R},
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abstract = {This study presents formulas for the electrostatic energy of lattices with two-dimensional periodicity, based on Fourier representations and alternatively on the Ewald procedure for convergence acceleration. The work extends the contributions of previous investigators by taking full advantage of plane-group symmetry and by providing analytical formulas for all derivatives of the energy through second order. The derivatives considered include those with respect to the positions of all charges within the unit cell, those with respect to the lattice vectors (cell deformations), and those involving both types of variables.~\textcopyright{} 1998 John Wiley \& Sons, Inc. Int J Quant Chem 68: 385\textendash{}404, 1998},
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language = {en},
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number = {6},
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journal = {International Journal of Quantum Chemistry},
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author = {Harris, Frank E.},
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month = jan,
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year = {1998},
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keywords = {Ewald method,electrostatic energy,lattice sums,two-dimensional},
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pages = {385-404},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/JAZ7DZXT/Harris - 1998 - Ewald summations in systems with two-dimensional p.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/KZ577FAF/abstract.html}
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}
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@article{gaoVaporLiquidCoexistence1997,
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title = {Vapor\textendash{}Liquid Coexistence of Quasi-Two-Dimensional {{Stockmayer}} Fluids},
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volume = {106},
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issn = {0021-9606},
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doi = {10.1063/1.473079},
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number = {8},
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journal = {The Journal of Chemical Physics},
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author = {Gao, G. T. and Zeng, X. C. and Wang, Wenchuan},
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month = feb,
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year = {1997},
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pages = {3311-3317},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/P9P9D9CJ/Gao ym. - 1997 - Vapor–liquid coexistence of quasi-two-dimensional .pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/M835BVF8/1.html}
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}
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@article{hummerMolecularTheoriesSimulation1998,
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title = {Molecular {{Theories}} and {{Simulation}} of {{Ions}} and {{Polar Molecules}} in {{Water}}},
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volume = {102},
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issn = {1089-5639},
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doi = {10.1021/jp982195r},
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abstract = {Recent developments in molecular theories and simulation of ions and polar molecules in water are reviewed. The hydration of imidazole and imidazolium is used to exemplify the theoretical issues. The treatment of long-ranged electrostatic interactions in simulations is discussed extensively. It is argued that the Ewald approach is an easy way to get correct hydration free energies corresponding to thermodynamic limit from molecular calculations. Molecular simulations with Ewald interactions and periodic boundary conditions can also be more efficient than many common alternatives. The Ewald treatment permits a conclusive extrapolation to infinite system size. Accurate results for well-defined models have permitted careful testing of simple theories of electrostatic hydration free energies, such as dielectric continuum models. The picture that emerges from such testing is that the most prominent failings of the simplest theories are associated with solvent proton conformations that lead to non-Gaussian fluctuations of electrostatic potentials. Thus, the most favorable cases for second-order perturbation theories are monoatomic positive ions. For polar and anionic solutes, continuum or Gaussian theories are less accurate. The appreciation of the specific deficiencies of those simple models have led to new concepts, multistate Gaussian and quasi-chemical theories, which address the cases for which the simpler theories fail. It is argued that, relative to direct dielectric continuum treatments, the quasi-chemical theories provide a better theoretical organization for the computational study of the electronic structure of solution species.},
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number = {41},
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journal = {The Journal of Physical Chemistry A},
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author = {Hummer, Gerhard and Pratt, Lawrence R. and Garc\'ia, Angel E.},
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month = oct,
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year = {1998},
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pages = {7885-7895},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/WU5N5TMT/Hummer ym. - 1998 - Molecular Theories and Simulation of Ions and Pola.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/I65M7FSZ/jp982195r.html}
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}
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@article{rokhlinRapidSolutionIntegral1985,
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title = {Rapid Solution of Integral Equations of Classical Potential Theory},
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volume = {60},
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issn = {0021-9991},
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doi = {10.1016/0021-9991(85)90002-6},
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abstract = {An algorithm is described for rapid solution of classical boundary value problems (Dirichlet an Neumann) for the Laplace equation based on iteratively solving integral equations of potential theory. CPU time requirements for previously published algorithms of this type are proportional to n2, where n is the number of nodes in the discretization of the boundary of the region. The CPU time requirements for the algorithm of the present paper are proportional to n, making it considerably more practical for large scale problems.},
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number = {2},
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journal = {Journal of Computational Physics},
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author = {Rokhlin, V},
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month = sep,
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year = {1985},
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pages = {187-207},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/IG8HX6QD/rokhlin1985.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/276H2VVD/0021999185900026.html}
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}
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@article{baddourApplicationGeneralizedShift2014,
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title = {Application of the Generalized Shift Operator to the {{Hankel}} Transform},
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volume = {3},
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issn = {2193-1801},
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doi = {10.1186/2193-1801-3-246},
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abstract = {It is well known that the Hankel transform possesses neither a shift-modulation nor a convolution-multiplication rule, both of which have found many uses when used with other integral transforms. In this paper, the generalized shift operator, as defined by Levitan, is applied to the Hankel transform. It is shown that under this generalized definition of shift, both convolution and shift theorems now apply to the Hankel transform. The operation of a generalized shift is compared to that of a simple shift via example.},
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journal = {SpringerPlus},
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author = {Baddour, Natalie},
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month = may,
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year = {2014},
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pages = {246},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/IXG9H85Q/Baddour - 2014 - Application of the generalized shift operator to t.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/IJ3X5XXC/2193-1801-3-246.html}
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}
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@article{wangFourierAnalysisPolar2008,
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title = {Fourier {{Analysis}} in {{Polar}} and {{Spherical Coordinates}}},
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author = {Wang, Qing and Ronneberger, Olaf and Burkhardt, Hans},
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year = {2008},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/I36FI8XD/Wang et al. - 2008 - Fourier Analysis in Polar and Spherical Coordinate.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/62QT93PD/WRB08.html}
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}
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@article{adkinsThreedimensionalFourierTransforms2013,
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archivePrefix = {arXiv},
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eprinttype = {arxiv},
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eprint = {1302.1830},
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primaryClass = {math-ph},
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title = {Three-Dimensional {{Fourier}} Transforms, Integrals of Spherical {{Bessel}} Functions, and Novel Delta Function Identities},
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abstract = {We present a general approach for evaluating a large variety of three-dimensional Fourier transforms. The transforms considered include the useful cases of the Coulomb and dipole potentials, and include situations where the transforms are singular and involve terms proportional to the Dirac delta function. Our approach makes use of the Rayleigh expansion of exp(i p.r) in terms of spherical Bessel functions, and we study a number of integrals, including singular integrals, involving a power of the independent variable times a spherical Bessel function. We work through several examples of three-dimensional Fourier transforms using our approach and show how to derive a number of identities involving multiple derivatives of 1/r, 1/r\^2, and delta($\backslash$vec r).},
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journal = {arXiv:1302.1830 [math-ph]},
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author = {Adkins, Gregory S.},
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month = feb,
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year = {2013},
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keywords = {Mathematical Physics,42B10},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/X7KQ8EMV/Adkins - 2013 - Three-dimensional Fourier transforms, integrals of.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/FVQ2QGIJ/1302.html}
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}
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@article{baddourOperationalConvolutionProperties2010,
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title = {Operational and Convolution Properties of Three-Dimensional {{Fourier}} Transforms in Spherical Polar Coordinates},
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volume = {27},
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copyright = {\textcopyright{} 2010 Optical Society of America},
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issn = {1520-8532},
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doi = {10.1364/JOSAA.27.002144},
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abstract = {For functions that are best described with spherical coordinates, the three-dimensional Fourier transform can be written in spherical coordinates as a combination of spherical Hankel transforms and spherical harmonic series. However, to be as useful as its Cartesian counterpart, a spherical version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the spherical version of the standard Fourier operation toolset. In particular, convolution in various forms is discussed in detail as this has important consequences for filtering. It is shown that standard multiplication and convolution rules do apply as long as the correct definition of convolution is applied.},
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language = {EN},
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number = {10},
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journal = {JOSA A},
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author = {Baddour, Natalie},
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month = oct,
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year = {2010},
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keywords = {Spectrum Analysis,Continuous optical signal processing,Tomographic image processing,Transforms},
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pages = {2144-2155},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/RDN4K7GW/baddour2010.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/9R32Q9ND/abstract.html}
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}
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@article{sheppardHankelTransformNdimensions2015,
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title = {The {{Hankel Transform}} in N-Dimensions and {{Its Applications}} in {{Optical Propagation}} and {{Imaging}}},
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volume = {188},
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issn = {1076-5670},
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doi = {10.1016/bs.aiep.2015.02.003},
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abstract = {Wave propagation is considered in multidimensional reciprocal space. For the first Rayleigh-Sommerfeld diffraction integral, the propagating field can be represented by homogeneous and inhomogeneous components. These add up to give a propagating component on a hemispherical surface in reciprocal space, and an evanescent component that lies totally outside the corresponding sphere. If evanescent waves can be neglected, the 3D angular spectrum method, entailing inverse Fourier transformation of the weighted hemisphere, can be used to calculate efficiently the propagated field. This basic concept is applied in spaces of different dimensionality. For functions displaying hyperspherical symmetry in nD space, the corresponding Hankel transformation leads to Hankel-transform pairs. Tables of functions relevant in wave propagation, diffraction, and information optics are presented. The two-dimensional (2D) case is particularly important as it can be applied to propagation in planar wave guides, surface plasmonics, and cross sections of propagationally invariant fields, as well as to fringe analysis and image processing in two dimensions.},
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journal = {Advances in Imaging and Electron Physics},
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author = {Sheppard, Colin J. R. and Kou, Shan S. and Lin, Jiao},
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month = jan,
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year = {2015},
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keywords = {Green function,diffraction,Hankel transform,Fourier transform,propagation,planar waveguides,fringe analysis,Plasmonics},
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pages = {135-184},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/MDUDQB8Q/sheppard2015.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/CJEIER7D/S107656701500021X.html}
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}
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@article{bloomfieldIndefiniteIntegralsSpherical2017,
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archivePrefix = {arXiv},
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eprinttype = {arxiv},
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eprint = {1703.06428},
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primaryClass = {math},
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title = {Indefinite {{Integrals}} of {{Spherical Bessel Functions}}},
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abstract = {Highly oscillatory integrals, such as those involving Bessel functions, are best evaluated analytically as much as possible, as numerical errors can be difficult to control. We investigate indefinite integrals involving monomials in \$x\$ multiplying one or two spherical Bessel functions of the first kind \$j\_l(x)\$ with integer order \$l\$. Closed-form solutions are presented where possible, and recursion relations are developed that are guaranteed to reduce all integrals in this class to closed-form solutions. These results allow for definite integrals over spherical Bessel functions to be computed quickly and accurately. For completeness, we also present our results in terms of ordinary Bessel functions, but in general, the recursion relations do not terminate.},
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journal = {arXiv:1703.06428 [math]},
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author = {Bloomfield, Jolyon K. and Face, Stephen H. P. and Moss, Zander},
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month = mar,
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year = {2017},
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keywords = {Mathematics - Classical Analysis and ODEs,Mathematics - Numerical Analysis},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/253VUGDB/Bloomfield et al. - 2017 - Indefinite Integrals of Spherical Bessel Functions.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/UQ4RZDZ2/1703.html}
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}
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@book{v.aIntegralnyePreobrazovaniyaOperacionnoe1962,
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series = {\cyrchar\CYRS\cyrchar\CYRM\cyrchar\CYRB},
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||||||
|
title = {\cyrchar\CYRI\cyrchar\cyrn\cyrchar\cyrt\cyrchar\cyre\cyrchar\cyrg\cyrchar\cyrr\cyrchar\cyra\cyrchar\cyrl\cyrchar\cyrsftsn\cyrchar\cyrn\cyrchar\cyrery\cyrchar\cyre{} \cyrchar\CYRP\cyrchar\cyrr\cyrchar\cyre\cyrchar\cyro\cyrchar\cyrb\cyrchar\cyrr\cyrchar\cyra\cyrchar\cyrz\cyrchar\cyro\cyrchar\cyrv\cyrchar\cyra\cyrchar\cyrn\cyrchar\cyri\cyrchar\cyrya{} \cyrchar\cyri{} \cyrchar\CYRO\cyrchar\cyrp\cyrchar\cyre\cyrchar\cyrr\cyrchar\cyra\cyrchar\cyrc\cyrchar\cyri\cyrchar\cyro\cyrchar\cyrn\cyrchar\cyrn\cyrchar\cyro\cyrchar\cyre{} \cyrchar\CYRI\cyrchar\cyrs\cyrchar\cyrch\cyrchar\cyri\cyrchar\cyrs\cyrchar\cyrl\cyrchar\cyre\cyrchar\cyrn\cyrchar\cyri\cyrchar\cyre},
|
||||||
|
publisher = {{\cyrchar\CYRF\cyrchar\cyri\cyrchar\cyrz\cyrchar\cyrm\cyrchar\cyra\cyrchar\cyrt\cyrchar\cyrg\cyrchar\cyri\cyrchar\cyrz}},
|
||||||
|
author = {\cyrchar\CYRV.\cyrchar\CYRA, \cyrchar\CYRD\cyrchar\cyri\cyrchar\cyrt\cyrchar\cyrk\cyrchar\cyri\cyrchar\cyrn{} and \cyrchar\CYRA.\cyrchar\CYRP, \cyrchar\CYRP\cyrchar\cyrr\cyrchar\cyru\cyrchar\cyrd\cyrchar\cyrn\cyrchar\cyri\cyrchar\cyrk\cyrchar\cyro\cyrchar\cyrv},
|
||||||
|
year = {1962}
|
||||||
|
}
|
||||||
|
|
||||||
|
@incollection{WorpitzkyNumbers2015,
|
||||||
|
title = {Worpitzky {{Numbers}}},
|
||||||
|
isbn = {978-981-4725-26-2},
|
||||||
|
booktitle = {Combinatorial {{Identities}} for {{Stirling Numbers}}},
|
||||||
|
publisher = {{WORLD SCIENTIFIC}},
|
||||||
|
month = aug,
|
||||||
|
year = {2015},
|
||||||
|
pages = {147-163},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/PPWCVB9C/10.1142@97898147252860011.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/PETENVWN/9789814725286_0011.html},
|
||||||
|
doi = {10.1142/9789814725286_0011}
|
||||||
|
}
|
||||||
|
|
||||||
|
@book{a.pIntegralyRyadySpecialnye2003,
|
||||||
|
edition = {2. \cyrchar\cyri\cyrchar\cyrz\cyrchar\cyrd., \cyrchar\cyri\cyrchar\cyrs\cyrchar\cyrp\cyrchar\cyrr},
|
||||||
|
title = {\cyrchar\CYRI\cyrchar\cyrn\cyrchar\cyrt\cyrchar\cyre\cyrchar\cyrg\cyrchar\cyrr\cyrchar\cyra\cyrchar\cyrl\cyrchar\cyrery{} \cyrchar\cyri{} \cyrchar\CYRR\cyrchar\cyrya\cyrchar\cyrd\cyrchar\cyrery. {{\cyrchar\CYRS\cyrchar\cyrp\cyrchar\cyre\cyrchar\cyrc\cyrchar\cyri\cyrchar\cyra\cyrchar\cyrl\cyrchar\cyrsftsn\cyrchar\cyrn\cyrchar\cyrery\cyrchar\cyre}} \cyrchar\CYRF\cyrchar\cyru\cyrchar\cyrn\cyrchar\cyrk\cyrchar\cyrc\cyrchar\cyri\cyrchar\cyri. {{\cyrchar\CYRD\cyrchar\cyro\cyrchar\cyrp\cyrchar\cyro\cyrchar\cyrl\cyrchar\cyrn\cyrchar\cyri\cyrchar\cyrt\cyrchar\cyre\cyrchar\cyrl\cyrchar\cyrsftsn\cyrchar\cyrn\cyrchar\cyrery\cyrchar\cyre}} \cyrchar\CYRG\cyrchar\cyrl\cyrchar\cyra\cyrchar\cyrv\cyrchar\cyrery},
|
||||||
|
volume = {\cyrchar\CYRT\cyrchar\cyro\cyrchar\cyrm{} 3},
|
||||||
|
isbn = {978-5-9221-0322-0},
|
||||||
|
publisher = {{\cyrchar\CYRF\cyrchar\CYRI\cyrchar\CYRZ\cyrchar\CYRM\cyrchar\CYRA\cyrchar\CYRT\cyrchar\CYRL\cyrchar\CYRI\cyrchar\CYRT}},
|
||||||
|
author = {\cyrchar\CYRA.\cyrchar\CYRP, \cyrchar\CYRP\cyrchar\cyrr\cyrchar\cyru\cyrchar\cyrd\cyrchar\cyrn\cyrchar\cyri\cyrchar\cyrk\cyrchar\cyro\cyrchar\cyrv{} and \cyrchar\CYRYU.\cyrchar\CYRA, \cyrchar\CYRB\cyrchar\cyrr\cyrchar\cyrery\cyrchar\cyrch\cyrchar\cyrk\cyrchar\cyro\cyrchar\cyrv{} and \cyrchar\CYRO.\cyrchar\CYRI, \cyrchar\CYRM\cyrchar\cyra\cyrchar\cyrr\cyrchar\cyri\cyrchar\cyrch\cyrchar\cyre\cyrchar\cyrv},
|
||||||
|
year = {2003},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/DKE6VC44/[Prudnikov_A.P.,_Bruechkov_YU.A.,_Marichev_O.I.]_I(BookFi)(1).djvu}
|
||||||
|
}
|
||||||
|
|
||||||
|
@misc{CombinatorialIdentitiesStirling,
|
||||||
|
title = {Combinatorial {{Identities}} for {{Stirling Numbers}}},
|
||||||
|
abstract = {This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical knowledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving binomial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Gould's techniques also provide connections between both types of Stirling numbers and Bernoulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities. This book will appeal to a wide audience and may be used either as lecture notes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics. Sample Chapter(s)Foreword (94 KB)Chapter 1: Basic Properties of Series (183 KB) Contents: Basic Properties of Series The Binomial Theorem Iterative Series Two of Professor Gould's Favorite Algebraic Techniques Vandermonde Convolution The nth Difference Operator and Euler's Finite Difference Theorem Melzak's Formula Generalized Derivative Formulas Stirling Numbers of the Second Kind S(n; k) Eulerian Numbers Worpitzky Numbers Stirling Numbers of the First Kind s(n; k) Explicit Formulas for s(n; n \textemdash{} k) Number Theoretic Definitions of Stirling Numbers Bernoulli Numbers Appendix A: Newton-Gregory Expansions Appendix B: Generalized Bernoulli and Euler Polynomials Readership: Undergraduates, graduates and researchers interested in combinatorial and algebraic techniques.},
|
||||||
|
howpublished = {http://www.worldscientific.com/worldscibooks/10.1142/9821},
|
||||||
|
journal = {World Scientific Publishing Company},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/K555Q8NT/Jocelyn_Quaintance,_Henry_W._Gould_Combinatorial_Identities_for_Stirling_Numbers_The_Unpublished_Notes_of_H_W_Gould.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/HHV7BB6G/9821.html}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{zivFastEvaluationElementary1991,
|
||||||
|
title = {Fast {{Evaluation}} of {{Elementary Mathematical Functions}} with {{Correctly Rounded Last Bit}}},
|
||||||
|
volume = {17},
|
||||||
|
issn = {0098-3500},
|
||||||
|
doi = {10.1145/114697.116813},
|
||||||
|
number = {3},
|
||||||
|
journal = {ACM Trans. Math. Softw.},
|
||||||
|
author = {Ziv, Abraham},
|
||||||
|
month = sep,
|
||||||
|
year = {1991},
|
||||||
|
keywords = {compatibility,correct rounding,mathematical library},
|
||||||
|
pages = {410--423},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/VRVEUFSA/Ziv - 1991 - Fast Evaluation of Elementary Mathematical Functio.pdf}
|
||||||
|
}
|
||||||
|
|
||||||
|
@book{a.pIntegralyRyadyElementarnye2002,
|
||||||
|
edition = {2. \cyrchar\cyri\cyrchar\cyrz\cyrchar\cyrd., \cyrchar\cyri\cyrchar\cyrs\cyrchar\cyrp\cyrchar\cyrr},
|
||||||
|
title = {\cyrchar\CYRI\cyrchar\cyrn\cyrchar\cyrt\cyrchar\cyre\cyrchar\cyrg\cyrchar\cyrr\cyrchar\cyra\cyrchar\cyrl\cyrchar\cyrery{} \cyrchar\cyri{} \cyrchar\CYRR\cyrchar\cyrya\cyrchar\cyrd\cyrchar\cyrery. {{\cyrchar\CYREREV\cyrchar\cyrl\cyrchar\cyre\cyrchar\cyrm\cyrchar\cyre\cyrchar\cyrn\cyrchar\cyrt\cyrchar\cyra\cyrchar\cyrr\cyrchar\cyrn\cyrchar\cyrery\cyrchar\cyre}} \cyrchar\CYRF\cyrchar\cyru\cyrchar\cyrn\cyrchar\cyrk\cyrchar\cyrc\cyrchar\cyri\cyrchar\cyri},
|
||||||
|
volume = {\cyrchar\CYRT\cyrchar\cyro\cyrchar\cyrm{} 1},
|
||||||
|
isbn = {978-5-9221-0322-0},
|
||||||
|
publisher = {{\cyrchar\CYRF\cyrchar\CYRI\cyrchar\CYRZ\cyrchar\CYRM\cyrchar\CYRA\cyrchar\CYRT\cyrchar\CYRL\cyrchar\CYRI\cyrchar\CYRT}},
|
||||||
|
author = {\cyrchar\CYRA.\cyrchar\CYRP, \cyrchar\CYRP\cyrchar\cyrr\cyrchar\cyru\cyrchar\cyrd\cyrchar\cyrn\cyrchar\cyri\cyrchar\cyrk\cyrchar\cyro\cyrchar\cyrv{} and \cyrchar\CYRYU.\cyrchar\CYRA, \cyrchar\CYRB\cyrchar\cyrr\cyrchar\cyrery\cyrchar\cyrch\cyrchar\cyrk\cyrchar\cyro\cyrchar\cyrv{} and \cyrchar\CYRO.\cyrchar\CYRI, \cyrchar\CYRM\cyrchar\cyra\cyrchar\cyrr\cyrchar\cyri\cyrchar\cyrch\cyrchar\cyre\cyrchar\cyrv},
|
||||||
|
year = {2002},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/R3QJRT5W/[Prudnikov_A.P.,_Bruechkov_YU.A.,_Marichev_O.I.]_I(BookFi).djvu}
|
||||||
|
}
|
||||||
|
|
||||||
|
@book{a.pIntegralyRyadySpecialnye2003a,
|
||||||
|
edition = {2. \cyrchar\cyri\cyrchar\cyrz\cyrchar\cyrd., \cyrchar\cyri\cyrchar\cyrs\cyrchar\cyrp\cyrchar\cyrr},
|
||||||
|
title = {\cyrchar\CYRI\cyrchar\cyrn\cyrchar\cyrt\cyrchar\cyre\cyrchar\cyrg\cyrchar\cyrr\cyrchar\cyra\cyrchar\cyrl\cyrchar\cyrery{} \cyrchar\cyri{} \cyrchar\CYRR\cyrchar\cyrya\cyrchar\cyrd\cyrchar\cyrery. {{\cyrchar\CYRS\cyrchar\cyrp\cyrchar\cyre\cyrchar\cyrc\cyrchar\cyri\cyrchar\cyra\cyrchar\cyrl\cyrchar\cyrsftsn\cyrchar\cyrn\cyrchar\cyrery\cyrchar\cyre}} \cyrchar\CYRF\cyrchar\cyru\cyrchar\cyrn\cyrchar\cyrk\cyrchar\cyrc\cyrchar\cyri\cyrchar\cyri},
|
||||||
|
volume = {\cyrchar\CYRT\cyrchar\cyro\cyrchar\cyrm{} 2},
|
||||||
|
isbn = {978-5-9221-0322-0},
|
||||||
|
publisher = {{\cyrchar\CYRF\cyrchar\CYRI\cyrchar\CYRZ\cyrchar\CYRM\cyrchar\CYRA\cyrchar\CYRT\cyrchar\CYRL\cyrchar\CYRI\cyrchar\CYRT}},
|
||||||
|
author = {\cyrchar\CYRA.\cyrchar\CYRP, \cyrchar\CYRP\cyrchar\cyrr\cyrchar\cyru\cyrchar\cyrd\cyrchar\cyrn\cyrchar\cyri\cyrchar\cyrk\cyrchar\cyro\cyrchar\cyrv{} and \cyrchar\CYRYU.\cyrchar\CYRA, \cyrchar\CYRB\cyrchar\cyrr\cyrchar\cyrery\cyrchar\cyrch\cyrchar\cyrk\cyrchar\cyro\cyrchar\cyrv{} and \cyrchar\CYRO.\cyrchar\CYRI, \cyrchar\CYRM\cyrchar\cyra\cyrchar\cyrr\cyrchar\cyri\cyrchar\cyrch\cyrchar\cyre\cyrchar\cyrv},
|
||||||
|
year = {2003},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/CJ7AKG2J/Прудников_А.П.,_Брычков_Ю.А.,_Маричев_О.И._Интегралы_и_ряды._Специальные_функции.djvu}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{wuAccurateComputationHypergeometric1993,
|
||||||
|
title = {An {{Accurate Computation}} of the {{Hypergeometric Distribution Function}}},
|
||||||
|
volume = {19},
|
||||||
|
issn = {0098-3500},
|
||||||
|
doi = {10.1145/151271.151274},
|
||||||
|
abstract = {The computation of the cumulative hypergeometric distribution function is of interest to many researchers who are working in the computational sciences and related areas. Presented here is a new method for computing this function that applies prime number factorization to the factorials. We also apply cancellation to the numerator and denominator to reduce the computational complexity of the initial, the tail end, or weighted probabilities to achieve maximum accuracy. The new method includes two algorithms, one using recursion and the other using iteration. These two algorithms are machine independent; precision is arbitrary, subject to storage limitation. The development of the algorithms is discussed, and some test results and the comparison of these two algorithms are given. To implement both algorithms, we use the Ada programming language that is an American National Standard Institute standardized language. The language has special features such as exception handling and tasks. Exception handling is used to make programming easier and to prevent overflow or underflow conditions during the execution of the program. Tasks are used to compute the numerator and denominator concurrently, and to maximize the possible number of integer multiplications in the numerator and denominator. All of the computations can be done on currently available machines, and the time consumed by these computations remains reasonably small.},
|
||||||
|
number = {1},
|
||||||
|
journal = {ACM Trans. Math. Softw.},
|
||||||
|
author = {Wu, Trong},
|
||||||
|
month = mar,
|
||||||
|
year = {1993},
|
||||||
|
keywords = {Ada programming language,Peizer approximations,exception handling,hypergeometric distribution function,prime number factorization,tasking},
|
||||||
|
pages = {33--43},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/CC3GQ79I/Wu - 1993 - An Accurate Computation of the Hypergeometric Dist.pdf}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{nicoroviciPhotonicBandGaps1995,
|
||||||
|
title = {Photonic Band Gaps for Arrays of Perfectly Conducting Cylinders},
|
||||||
|
volume = {52},
|
||||||
|
doi = {10.1103/PhysRevE.52.1135},
|
||||||
|
abstract = {We study the propagation of electromagnetic waves through arrays of perfectly conducting cylinders for both fundamental polarization cases s and p. We use a generalized Rayleigh identity method and show that for p polarization the fundamental band defines an effective refractive index not in keeping with electrostatics. We exhibit the photonic band structures for very dilute arrays, where they tend towards the expected free-propagation form. We also study them for arrays approaching touching, where very interesting differences between s and p polarization behavior are manifest.},
|
||||||
|
number = {1},
|
||||||
|
journal = {Physical Review E},
|
||||||
|
author = {Nicorovici, N. A. and McPhedran, R. C. and Botten, L. C.},
|
||||||
|
month = jul,
|
||||||
|
year = {1995},
|
||||||
|
pages = {1135-1145},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/44QN6K4W/PhysRevE.52.1135.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/USYDNPNY/Nicorovici ym. - 1995 - Photonic band gaps for arrays of perfectly conduct.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/RTPI6HEA/PhysRevE.52.html}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{linton_lattice_2010,
|
||||||
|
title = {Lattice {{Sums}} for the {{Helmholtz Equation}}},
|
||||||
|
volume = {52},
|
||||||
|
issn = {0036-1445},
|
||||||
|
doi = {10.1137/09075130X},
|
||||||
|
abstract = {A survey of different representations for lattice sums for the Helmholtz equation is made. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension \$d\_$\backslash$Lambda\$. Lattice sums are related to, and can be calculated from, the quasi-periodic Green's function and this object serves as the starting point of the analysis.},
|
||||||
|
number = {4},
|
||||||
|
journal = {SIAM Review},
|
||||||
|
author = {Linton, C.},
|
||||||
|
month = jan,
|
||||||
|
year = {2010},
|
||||||
|
pages = {630-674},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/T86ATKYB/09075130x.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/ETB8X4S9/09075130X.html}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{linton_one-_2009,
|
||||||
|
title = {One- and Two-Dimensional Lattice Sums for the Three-Dimensional {{Helmholtz}} Equation},
|
||||||
|
volume = {228},
|
||||||
|
issn = {0021-9991},
|
||||||
|
doi = {10.1016/j.jcp.2008.11.013},
|
||||||
|
abstract = {The accurate and efficient computation of lattice sums for the three-dimensional Helmholtz equation is considered for the cases where the underlying lattice is one- or two-dimensional. We demonstrate, using careful numerical computations, that the reduction method, in which the sums for a two-dimensional lattice are expressed as a sum of one-dimensional lattice sums leads to an order-of-magnitude improvement in performance over the well-known Ewald method. In the process we clarify and improve on a number of results originally formulated by Twersky in the 1970s.},
|
||||||
|
number = {6},
|
||||||
|
journal = {Journal of Computational Physics},
|
||||||
|
author = {Linton, C. M. and Thompson, I.},
|
||||||
|
month = apr,
|
||||||
|
year = {2009},
|
||||||
|
keywords = {Helmholtz equation,Clausen function,Ewald summation,Lattice reduction,Lattice sum,Schlömilch series},
|
||||||
|
pages = {1815-1829},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/YMRZHBY4/Linton ja Thompson - 2009 - One- and two-dimensional lattice sums for the thre.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/Z8CFQ6S9/S0021999108005962.html}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{wei_broadband_2017,
|
||||||
|
title = {A {{Broadband ML}}-{{FMA}} for 3-{{D Periodic Green}}'s {{Function}} in 2-{{D Lattice Using Ewald Summation}}},
|
||||||
|
volume = {65},
|
||||||
|
issn = {0018-926X},
|
||||||
|
doi = {10.1109/TAP.2017.2690533},
|
||||||
|
abstract = {A periodic fast multipole algorithm (P-FMA) is devised for evaluating 3-D periodic Green's function (PGF) for a 2-D lattice which can be used to solve scattering by a structure with 2-D periodicity. The introduction of periodicity in the Green's function formulation produces image sources at each lattice site. Like multilevel FMA (ML-FMA), P-FMA takes advantage of the distance between image sources and observation points to factorize the field using multipoles. By substituting known factorizations of the free-space Green's function into the expression for PGF, one can isolate the summation over the lattice into the translation phase of the FMA. For both plane wave and multipole factorizations, a common term known as lattice constant appears. The lattice constant is an infinite sum over the lattice which does not converge absolutely when expressed as a spatial sum. Using the Ewald summation technique, the lattice constants can be evaluated with exponential convergence and high accuracy. The resulting P-FMA is between O(N) and O(N log N) in memory use and computational complexity, depending on the object size relative to the wavelength.},
|
||||||
|
number = {6},
|
||||||
|
journal = {IEEE Transactions on Antennas and Propagation},
|
||||||
|
author = {Wei, M. and Chew, W. C.},
|
||||||
|
month = jun,
|
||||||
|
year = {2017},
|
||||||
|
keywords = {computational complexity,Green's function methods,Geometry,Convergence,Scattering,periodic structures,Ewald summation,2D lattice,2D periodicity,3D periodic Green function,3D PGF evaluation,Broadband antennas,Broadband communication,broadband ML-FMA,Ewald summation technique,fast multipole method (ML-FMA),FMA translation phase,free-space Green function,lattice constant,lattice sum,Lattices,method of moments (MoM),multilevel,multilevel FMA,multipole factorization,P-FMA,periodic fast multipole algorithm,periodic Green’s function (PGF),periodic scattering,plane wave factorization},
|
||||||
|
pages = {3134-3145},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/33MFL85V/Wei ja Chew - 2017 - A Broadband ML-FMA for 3-D Periodic Green’s Functi.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/RN9AUGNQ/7891536.html}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{moroz_quasi-periodic_2006,
|
||||||
|
title = {Quasi-Periodic {{Green}}'s Functions of the {{Helmholtz}} and {{Laplace}} Equations},
|
||||||
|
volume = {39},
|
||||||
|
issn = {0305-4470},
|
||||||
|
doi = {10.1088/0305-4470/39/36/009},
|
||||||
|
abstract = {A classical problem of free-space Green's function G 0$\Lambda$ representations of the Helmholtz equation is studied in various quasi-periodic cases, i.e., when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space. Exponentially convergent series for the free-space quasi-periodic G 0$\Lambda$ and for the expansion coefficients D L of G 0$\Lambda$ in the basis of regular (cylindrical in two dimensions and spherical in three dimension (3D)) waves, or lattice sums, are reviewed and new results for the case of a one-dimensional (1D) periodicity in 3D are derived. From a mathematical point of view, a derivation of exponentially convergent representations for Schl\"omilch series of cylindrical and spherical Hankel functions of any integer order is accomplished. Exponentially convergent series for G 0$\Lambda$ and lattice sums D L hold for any value of the Bloch momentum and allow G 0$\Lambda$ to be efficiently evaluated also in the periodicity plane. The quasi-periodic Green's functions of the Laplace equation are obtained from the corresponding representations of G 0$\Lambda$ of the Helmholtz equation by taking the limit of the wave vector magnitude going to zero. The derivation of relevant results in the case of a 1D periodicity in 3D highlights the common part which is universally applicable to any of remaining quasi-periodic cases. The results obtained can be useful for the numerical solution of boundary integral equations for potential flows in fluid mechanics, remote sensing of periodic surfaces, periodic gratings, and infinite arrays of resonators coupled to a waveguide, in many contexts of simulating systems of charged particles, in molecular dynamics, for the description of quasi-periodic arrays of point interactions in quantum mechanics, and in various ab initio first-principle multiple-scattering theories for the analysis of diffraction of classical and quantum waves.},
|
||||||
|
language = {en},
|
||||||
|
number = {36},
|
||||||
|
journal = {Journal of Physics A: Mathematical and General},
|
||||||
|
author = {Moroz, Alexander},
|
||||||
|
year = {2006},
|
||||||
|
pages = {11247},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/268RXLJ4/Moroz - 2006 - Quasi-periodic Green's functions of the Helmholtz .pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/MGA5XR44/dlserr.pdf}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{mcrae_multiplescattering_1966,
|
||||||
|
title = {Multiple-{{Scattering Treatment}} of {{Low}}-{{Energy Electron}}-{{Diffraction Intensities}}},
|
||||||
|
volume = {45},
|
||||||
|
issn = {0021-9606},
|
||||||
|
doi = {10.1063/1.1728101},
|
||||||
|
number = {9},
|
||||||
|
journal = {The Journal of Chemical Physics},
|
||||||
|
author = {McRae, E. G.},
|
||||||
|
month = nov,
|
||||||
|
year = {1966},
|
||||||
|
pages = {3258-3276},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/SK7KQSKF/McRae - 1966 - Multiple‐Scattering Treatment of Low‐Energy Electr.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/Q2S3495C/1.html}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{kambe_theory_2014,
|
||||||
|
title = {Theory of {{Electron Diffraction}} by {{Crystals}}},
|
||||||
|
volume = {22},
|
||||||
|
issn = {1865-7109},
|
||||||
|
doi = {10.1515/zna-1967-0402},
|
||||||
|
abstract = {A general theory of electron diffraction by crystals is developed. The crystals are assumed to be infinitely extended in two dimensions and finite in the third dimension. For the scattering problem by this structure two-dimensionally expanded forms of GREEN'S function and integral equation are at first derived, and combined in single three-dimensional forms. EWALD'S method is applied to sum up the series for GREEN'S function.},
|
||||||
|
number = {4},
|
||||||
|
journal = {Zeitschrift f\"ur Naturforschung A},
|
||||||
|
author = {Kambe, Kyozaburo},
|
||||||
|
year = {2014},
|
||||||
|
pages = {422--431},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/VEIUHCCD/Kambe - 2014 - Theory of Electron Diffraction by Crystals.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/WPKCSVZG/Kambe - 2014 - Theory of Electron Diffraction by Crystals.pdf}
|
||||||
|
}
|
||||||
|
|
||||||
|
@article{enoch_sums_2001,
|
||||||
|
title = {Sums of Spherical Waves for Lattices, Layers, and Lines},
|
||||||
|
volume = {42},
|
||||||
|
issn = {0022-2488},
|
||||||
|
doi = {10.1063/1.1409348},
|
||||||
|
number = {12},
|
||||||
|
journal = {Journal of Mathematical Physics},
|
||||||
|
author = {Enoch, S. and McPhedran, R. C. and Nicorovici, N. A. and Botten, L. C. and Nixon, J. N.},
|
||||||
|
month = nov,
|
||||||
|
year = {2001},
|
||||||
|
pages = {5859-5870},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/GZW5G2AY/Enoch ym. - 2001 - Sums of spherical waves for lattices, layers, and .pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/2ZQXY82F/1.html}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
|
@ -0,0 +1,19 @@
|
||||||
|
|
||||||
|
@article{NIST:DLMF,
|
||||||
|
title = {{{NIST Digital Library}} of {{Mathematical Functions}}},
|
||||||
|
url = {http://dlmf.nist.gov/},
|
||||||
|
key = {DLMF},
|
||||||
|
note = {F.~W.~J. Olver, A.~B. Olde Daalhuis, D.~W. Lozier, B.~I. Schneider, R.~F. Boisvert, C.~W. Clark, B.~R. Miller and B.~V. Saunders, eds.}
|
||||||
|
}
|
||||||
|
|
||||||
|
@book{olver_nist_2010,
|
||||||
|
edition = {1 Pap/Cdr},
|
||||||
|
title = {{{NIST}} Handbook of Mathematical Functions},
|
||||||
|
isbn = {978-0-521-14063-8},
|
||||||
|
publisher = {{Cambridge University Press}},
|
||||||
|
author = {Olver, Frank W. J. and Lozier, Daniel W. and Boisvert, Ronald F. and Clark, Charles W.},
|
||||||
|
year = {2010},
|
||||||
|
file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/ZJ5LBQ8W/Olver ym. - 2010 - NIST handbook of mathematical functions.pdf}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
|
|
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|
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|
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|
||||||
|
{hyperref}
|
||||||
|
\hypersetup{pdftitle={Accelerating lattice mode calculations with T-matrix method},
|
||||||
|
pdfauthor={Marek Nečada}}
|
||||||
|
|
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\makeatletter
|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
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|
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|
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|
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|
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|
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% Toto je trik, jimž se z fontspec získá familyname pro následující
|
||||||
|
\ExplSyntaxOn
|
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|
\DeclareExpandableDocumentCommand{\getfamilyname}{m}
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|
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\use:c { g__fontspec_ \cs_to_str:N #1 _family }
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|
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|
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|
\ExplSyntaxOff
|
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|
% definujeme novou rodinu, jež se volá pomocí \MyCyr pro běžné použití, avšak pro účely \DeclareSymbolFont je nutno získat název pomocí getfamilyname definovaného výše
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||||||
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\newfontfamily\MyCyr{CMU Serif}
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\DeclareSymbolFont{cyritletters}{EU1}{\getfamilyname\MyCyr}{m}{it}
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\newcommand{\makecyrmathletter}[1]{%
|
||||||
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\begingroup\lccode`a=#1\lowercase{\endgroup
|
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|
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\count255="409
|
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|
||||||
|
\setotherlanguage{russian}
|
||||||
|
\newfontfamily\russianfont[Script=Cyrillic]{URW Palladio L}
|
||||||
|
%\newfontfamily\russianfont[Script=Cyrillic]{DejaVu Sans}
|
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|
||||||
|
\makeatother
|
||||||
|
|
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|
\usepackage{xunicode}
|
||||||
|
\usepackage{polyglossia}
|
||||||
|
\setdefaultlanguage{english}
|
||||||
|
\begin{document}
|
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|
\global\long\def\uoft#1{\mathfrak{F}#1}
|
||||||
|
\global\long\def\uaft#1{\mathfrak{\mathbb{F}}#1}
|
||||||
|
\global\long\def\usht#1#2{\mathbb{S}_{#1}#2}
|
||||||
|
\global\long\def\bsht#1#2{\mathrm{S}_{#1}#2}
|
||||||
|
\global\long\def\pht#1#2{\mathfrak{\mathbb{H}}_{#1}#2}
|
||||||
|
\global\long\def\vect#1{\mathbf{#1}}
|
||||||
|
\global\long\def\ud{\mathrm{d}}
|
||||||
|
\global\long\def\basis#1{\mathfrak{#1}}
|
||||||
|
\global\long\def\dc#1{\lyxmathsym{Ш}_{#1}}
|
||||||
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|
||||||
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|
||||||
|
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|
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|
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||||||
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|
||||||
|
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|
||||||
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||||||
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|
||||||
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|
||||||
|
\global\long\def\kor#1{\underline{#1}}
|
||||||
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|
||||||
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|
||||||
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|
||||||
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|
||||||
|
\title{Accelerating lattice mode calculations with $T$-matrix method}
|
||||||
|
|
||||||
|
|
||||||
|
\author{Marek Nečada}
|
||||||
|
\maketitle
|
||||||
|
\begin{abstract}
|
||||||
|
The $T$-matrix approach is the method of choice for simulating optical
|
||||||
|
response of a reasonably small system of compact linear scatterers
|
||||||
|
on isotropic background. However, its direct utilisation for problems
|
||||||
|
with infinite lattices is problematic due to slowly converging sums
|
||||||
|
over the lattice. Here I develop a way to compute the problematic
|
||||||
|
sums in the reciprocal space, making the $T$-matrix method very suitable
|
||||||
|
for infinite periodic systems as well.
|
||||||
|
\end{abstract}
|
||||||
|
|
||||||
|
\section{Formulation of the problem}
|
||||||
|
|
||||||
|
Assume a system of compact EM scatterers in otherwise homogeneous
|
||||||
|
and isotropic medium, and assume that the system, i.e. both the medium
|
||||||
|
and the scatterers, have linear response. A scattering problem in
|
||||||
|
such system can be written as
|
||||||
|
\[
|
||||||
|
A_{\alpha}=T_{\alpha}P_{\alpha}=T_{\alpha}(\sum_{\beta}S_{\alpha\leftarrow\beta}A_{\beta}+P_{0\alpha})
|
||||||
|
\]
|
||||||
|
where $T_{\alpha}$ is the $T$-matrix for scatterer α, $A_{\alpha}$
|
||||||
|
is its vector of the scattered wave expansion coefficient (the multipole
|
||||||
|
indices are not explicitely indicated here) and $P_{\alpha}$ is the
|
||||||
|
local expansion of the incoming sources. $S_{\alpha\leftarrow\beta}$
|
||||||
|
is ... and ... is ...
|
||||||
|
|
||||||
|
...
|
||||||
|
|
||||||
|
\[
|
||||||
|
\sum_{\beta}(\delta_{\alpha\beta}-T_{\alpha}S_{\alpha\leftarrow\beta})A_{\beta}=T_{\alpha}P_{0\alpha}.
|
||||||
|
\]
|
||||||
|
|
||||||
|
|
||||||
|
Now suppose that the scatterers constitute an infinite lattice
|
||||||
|
|
||||||
|
\[
|
||||||
|
\sum_{\vect b\beta}(\delta_{\vect{ab}}\delta_{\alpha\beta}-T_{\vect a\alpha}S_{\vect a\alpha\leftarrow\vect b\beta})A_{\vect b\beta}=T_{\vect a\alpha}P_{0\vect a\alpha}.
|
||||||
|
\]
|
||||||
|
Due to the periodicity, we can write $S_{\vect a\alpha\leftarrow\vect b\beta}=S_{\alpha\leftarrow\beta}(\vect b-\vect a)$
|
||||||
|
and $T_{\vect a\alpha}=T_{\alpha}$. In order to find lattice modes,
|
||||||
|
we search for solutions with zero RHS
|
||||||
|
\[
|
||||||
|
\sum_{\vect b\beta}(\delta_{\vect{ab}}\delta_{\alpha\beta}-T_{\alpha}S_{\vect a\alpha\leftarrow\vect b\beta})A_{\vect b\beta}=0
|
||||||
|
\]
|
||||||
|
and we assume periodic solution $A_{\vect b\beta}(\vect k)=A_{\vect a\beta}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$,
|
||||||
|
yielding
|
||||||
|
\begin{eqnarray*}
|
||||||
|
\sum_{\vect b\beta}(\delta_{\vect{ab}}\delta_{\alpha\beta}-T_{\alpha}S_{\vect a\alpha\leftarrow\vect b\beta})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
|
||||||
|
\sum_{\vect b\beta}(\delta_{\vect{0b}}\delta_{\alpha\beta}-T_{\alpha}S_{\vect 0\alpha\leftarrow\vect b\beta})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
|
||||||
|
\sum_{\beta}(\delta_{\alpha\beta}-T_{\alpha}\underbrace{\sum_{\vect b}S_{\vect 0\alpha\leftarrow\vect b\beta}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
|
||||||
|
A_{\vect 0\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
|
||||||
|
\end{eqnarray*}
|
||||||
|
Therefore, in order to solve the modes, we need to compute the ``lattice
|
||||||
|
Fourier transform'' of the translation operator,
|
||||||
|
\begin{equation}
|
||||||
|
W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0\alpha\leftarrow\vect b\beta}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\section{Computing the Fourier sum of the translation operator}
|
||||||
|
|
||||||
|
The problem evaluating (\ref{eq:W definition}) is the asymptotic
|
||||||
|
behaviour of the translation operator, $S_{\vect 0\alpha\leftarrow\vect b\beta}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
|
||||||
|
that makes the convergence of the sum quite problematic for any $d>1$-dimensional
|
||||||
|
lattice.%
|
||||||
|
\footnote{Note that $d$ here is dimensionality of the lattice, not the space
|
||||||
|
it lies in, which I for certain reasons assume to be three. (TODO
|
||||||
|
few notes on integration and reciprocal lattices in some appendix)%
|
||||||
|
} In electrostatics, one can solve this problem with Ewald summation.
|
||||||
|
Its basic idea is that if what asymptoticaly decays poorly in the
|
||||||
|
direct space, will perhaps decay fast in the Fourier space. I use
|
||||||
|
the same idea here, but everything will be somehow harder than in
|
||||||
|
electrostatics.
|
||||||
|
|
||||||
|
Let us re-express the sum in (\ref{eq:W definition}) in terms of
|
||||||
|
integral with a delta comb
|
||||||
|
|
||||||
|
\begin{equation}
|
||||||
|
W_{\alpha\beta}(\vect k)=\int\ud^{d}\vect r\dc{\basis u}(\vect r)S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})e^{i\vect k\cdot\vect r}.\label{eq:W integral}
|
||||||
|
\end{equation}
|
||||||
|
The translation operator $S$ is now a function defined in the whole
|
||||||
|
3d space; $\vect r_{\alpha},\vect r_{\beta}$ are the displacements
|
||||||
|
of scatterers $\alpha$ and $\beta$ in a unit cell. The arrow notation
|
||||||
|
$S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$ means ``translation
|
||||||
|
operator for spherical waves originating in $\vect r+\vect r_{\beta}$
|
||||||
|
evaluated in $\vect r_{\alpha}$'' and obviously $S$ is in fact
|
||||||
|
a function of a single 3d argument, $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$.
|
||||||
|
Expression (\ref{eq:W integral}) can be rewritten as
|
||||||
|
\[
|
||||||
|
W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
|
||||||
|
\]
|
||||||
|
where changed the sign of $\vect r/\vect{\bullet}$ has been swapped
|
||||||
|
under integration, utilising evenness of $\dc{\basis u}$. Fourier
|
||||||
|
transform of product is convolution of Fourier transforms, so (using
|
||||||
|
formula (\ref{eq:Dirac comb uaFt}) for the Fourier transform of Dirac
|
||||||
|
comb)
|
||||||
|
\begin{eqnarray}
|
||||||
|
W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
|
||||||
|
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
|
||||||
|
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right).\label{eq:W sum in reciprocal space}
|
||||||
|
\end{eqnarray}
|
||||||
|
As such, this is not extremely helpful because the the \emph{whole}
|
||||||
|
translation operator $S$ has singularities in origin, hence its Fourier
|
||||||
|
transform $\uaft S$ will decay poorly.
|
||||||
|
|
||||||
|
However, Fourier transform is linear, so we can in principle separate
|
||||||
|
$S$ in two parts, $S=S^{\textup{L}}+S^{\textup{S}}$. $S^{\textup{S}}$
|
||||||
|
is a short-range part that decays sufficiently fast with distance
|
||||||
|
so that its direct-space lattice sum converges well; $S^{\textup{S}}$
|
||||||
|
must as well contain all the singularities of $S$ in the origin.
|
||||||
|
The other part, $S^{\textup{L}}$, will retain all the slowly decaying
|
||||||
|
terms of $S$ but it also has to be smooth enough in the origin, so
|
||||||
|
that its Fourier transform $\uaft{S^{\textup{L}}}$ decays fast enough.
|
||||||
|
(The same idea lies behind the Ewald summation in electrostatics.)
|
||||||
|
Using the linearity of Fourier transform and formulae (\ref{eq:W definition})
|
||||||
|
and (\ref{eq:W sum in reciprocal space}), the operator $W_{\alpha\beta}$
|
||||||
|
can then be re-expressed as
|
||||||
|
\begin{eqnarray}
|
||||||
|
W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
|
||||||
|
W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
|
||||||
|
W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
|
||||||
|
\end{eqnarray}
|
||||||
|
where both sums should converge nicely.
|
||||||
|
|
||||||
|
|
||||||
|
\section{Finding a good decomposition}
|
||||||
|
|
||||||
|
The remaining challenge is therefore finding a suitable decomposition
|
||||||
|
$S^{\textup{L}}+S^{\textup{S}}$ such that both $S^{\textup{S}}$
|
||||||
|
and $\uaft{S^{\textup{L}}}$ decay fast enough with distance and are
|
||||||
|
expressable analytically. With these requirements, I do not expect
|
||||||
|
to find gaussian asymptotics as in the electrostatic Ewald formula—having
|
||||||
|
$\sim x^{-t}$, $t>d$ asymptotics would be nice, making the sums
|
||||||
|
in (\ref{eq:W Short definition}), (\ref{eq:W Long definition}) absolutely
|
||||||
|
convergent.
|
||||||
|
|
||||||
|
The translation operator $S$ for compact scatterers in 3d can be
|
||||||
|
expressed as
|
||||||
|
\[
|
||||||
|
S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(k_{0}\left|\vect r\right|\right)
|
||||||
|
\]
|
||||||
|
where $Y_{l,m}\left(\theta,\phi\right)$ are the spherical harmonics,
|
||||||
|
$z_{p}^{(J)}\left(r\right)$ some of the Bessel or Hankel functions
|
||||||
|
(probably $h_{p}^{(1)}$ in all meaningful cases; TODO) and $c_{p}^{l,m,t\leftarrow l',m',t'}$
|
||||||
|
are some ugly but known coefficients (REF Xu 1996, eqs. 76,77).
|
||||||
|
|
||||||
|
The spherical Hankel functions can be expressed analytically as (REF
|
||||||
|
DLMF 10.49.6, 10.49.1)
|
||||||
|
\begin{equation}
|
||||||
|
h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!},\label{eq:spherical Hankel function series}
|
||||||
|
\end{equation}
|
||||||
|
so if we find a way to deal with the radial functions $s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$,
|
||||||
|
$q=1,2$ in 2d case or $q=1,2,3$ in 3d case, we get absolutely convergent
|
||||||
|
summations in the direct space.
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{2d}
|
||||||
|
|
||||||
|
Assume that all scatterers are placed in the plane $\vect z=0$, so
|
||||||
|
that the 2d Fourier transform of the long-range part of the translation
|
||||||
|
operator in terms of Hankel transforms, according to (\ref{eq:Fourier v. Hankel tf 2d}),
|
||||||
|
reads
|
||||||
|
|
||||||
|
\begin{multline*}
|
||||||
|
\uaft{S_{l',m',t'\leftarrow l,m,t}^{\textup{L}}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
|
||||||
|
\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{h_{p}^{(1)\textup{L}}\left(k_{0}\vect{\bullet}\right)}\left(\left|\vect k\right|\right)
|
||||||
|
\end{multline*}
|
||||||
|
Here $h_{p}^{(1)\textup{L}}=h_{p}^{(1)}-h_{p}^{(1)\textup{S}}$ is
|
||||||
|
a long range part of a given spherical Hankel function which has to
|
||||||
|
be found and which contains all the terms with far-field ($r\to\infty$)
|
||||||
|
asymptotics proportional to$\sim e^{ik_{0}r}\left(k_{0}r\right)^{-q}$,
|
||||||
|
$q\le Q$ where $Q$ is at least two in order to achieve absolute
|
||||||
|
convergence of the direct-space sum, but might be higher in order
|
||||||
|
to speed the convergence up.
|
||||||
|
|
||||||
|
Obviously, all the terms $\propto s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$,
|
||||||
|
$q>Q$ of the spherical Hankel function (\ref{eq:spherical Hankel function series})
|
||||||
|
can be kept untouched as part of $h_{p}^{(1)\textup{S}}$, as they
|
||||||
|
decay fast enough.
|
||||||
|
|
||||||
|
The remaining task is therefore to find a suitable decomposition of
|
||||||
|
$s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$, $q\le Q$ into
|
||||||
|
short-range and long-range parts, $s_{k_{0},q}(r)=s_{k_{0},q}^{\textup{S}}(r)+s_{k_{0},q}^{\textup{L}}(r)$,
|
||||||
|
such that $s_{k_{0},q}^{\textup{L}}(r)$ contains all the slowly decaying
|
||||||
|
asymptotics and its Hankel transforms decay desirably fast as well,
|
||||||
|
$\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$, $z\to\infty$.
|
||||||
|
The latter requirement calls for suitable regularisation functions—$s_{q}^{\textup{L}}$
|
||||||
|
must be sufficiently smooth in the origin, so that
|
||||||
|
\begin{equation}
|
||||||
|
\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{k_{0},q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{k_{0},q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement}
|
||||||
|
\end{equation}
|
||||||
|
exists and decays fast enough. $J_{\nu}(r)\sim\left(r/2\right)^{\nu}/\Gamma\left(\nu+1\right)$
|
||||||
|
(REF DLMF 10.7.3) near the origin, so the regularisation function
|
||||||
|
should be $\rho(r)=o(r^{q-n-1})$ only to make $\pht n{s_{q}^{\textup{L}}}$
|
||||||
|
converge. The additional decay speed requirement calls for at least
|
||||||
|
$\rho(r)=o(r^{q-n+Q-1})$, I guess. At the same time, $\rho(r)$ must
|
||||||
|
converge fast enough to one for $r\to\infty$.
|
||||||
|
|
||||||
|
The electrostatic Ewald summation uses regularisation with $1-e^{-cr^{2}}$.
|
||||||
|
However, such choice does not seem to lead to an analytical solution
|
||||||
|
(really? could not something be dug out of DLMF 10.22.54?) for the
|
||||||
|
current problem (\ref{eq:2d long range regularisation problem statement}).
|
||||||
|
But it turns out that the family of functions
|
||||||
|
\begin{equation}
|
||||||
|
\rho_{\kappa,c}(r)\equiv\left(1-e^{-cr}\right)^{\text{\ensuremath{\kappa}}},\quad c>0,\kappa\in\nats\label{eq:binom regularisation function}
|
||||||
|
\end{equation}
|
||||||
|
might lead to satisfactory results; see below.
|
||||||
|
|
||||||
|
|
||||||
|
\subsubsection{Hankel transforms of the long-range parts, „binomial“ regularisation\label{sub:Hankel-transforms-binom-reg}}
|
||||||
|
|
||||||
|
Let
|
||||||
|
|
||||||
|
\begin{eqnarray}
|
||||||
|
\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & \equiv & \int_{0}^{\infty}\frac{e^{ik_{0}r}}{\left(k_{0}r\right)^{q}}J_{n}\left(kr\right)\left(1-e^{-cr}\right)^{\kappa}r\,\ud r\nonumber \\
|
||||||
|
& = & k_{0}^{-q}\int_{0}^{\infty}r^{1-q}J_{n}\left(kr\right)\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}e^{-(\sigma c-ik_{0})r}\ud r\nonumber \\
|
||||||
|
& \underset{\equiv}{\textup{form.}} & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\pht n{s_{q,k_{0}}^{\textup{L}1,\sigma c}}\left(k\right).\label{eq:2D Hankel transform of regularized outgoing wave, decomposition}
|
||||||
|
\end{eqnarray}
|
||||||
|
From {[}REF DLMF 10.22.49{]} one digs
|
||||||
|
\begin{multline}
|
||||||
|
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}\Gamma\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right),\\
|
||||||
|
\Re\left(2-q+n\right)>0,\Re(c-ik_{0}\pm k)\ge0,\label{eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1}
|
||||||
|
\end{multline}
|
||||||
|
and using {[}REF DLMF 15.9.17{]} and {[}REF DLMF 14.9.5{]}
|
||||||
|
|
||||||
|
{\footnotesize{}
|
||||||
|
\begin{multline}
|
||||||
|
\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}\Gamma\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right),\\
|
||||||
|
k>0\wedge k_{0}>0\wedge c\ge0\wedge\lnot\left(c=0\wedge k_{0}=k\right)\label{eq:2D Hankel transform of exponentially suppressed outgoing wave expanded}
|
||||||
|
\end{multline}
|
||||||
|
}with principal branches of the hypergeometric functions, associated
|
||||||
|
Legendre functions, and fractional powers. The conditions from (\ref{eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1})
|
||||||
|
should hold, but we will use (\ref{eq:2D Hankel transform of exponentially suppressed outgoing wave expanded})
|
||||||
|
formally even if they are violated, with the hope that the divergences
|
||||||
|
eventually cancel in (\ref{eq:2D Hankel transform of regularized outgoing wave, decomposition}).
|
||||||
|
|
||||||
|
One problematic element here is the gamma function $\text{Γ}\left(2-q+n\right)$
|
||||||
|
which is singular if the arguments are negative integers, i.e. if
|
||||||
|
$q-n\ge3$; but at least the necessary minimum of $q=1,2$ would be
|
||||||
|
covered this way. The associated Legendre function can be expressed
|
||||||
|
as a finite ``polynomial'' if $q\ge n$. In other cases, different
|
||||||
|
expressions can be obtained from \ref{eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1}
|
||||||
|
using various transformation formulae from either DLMF or \begin{russian}Прудников\end{russian}.
|
||||||
|
|
||||||
|
In fact, Mathematica is usually able to calculate the transforms for
|
||||||
|
specific values of $\kappa,q,n$, but it did not find any general
|
||||||
|
formula for me. The resulting expressions are finite sums of algebraic
|
||||||
|
functions, Table \ref{tab:Asymptotical-behaviour-Mathematica} shows
|
||||||
|
how fast they decay with growing $k$ for some parameters. The only
|
||||||
|
case where Mathematica did not help at all is $q=2,n=0$, which is
|
||||||
|
unfortunately important. But if I have not made some mistake, the
|
||||||
|
expression (\ref{eq:2D Hankel transform of exponentially suppressed outgoing wave expanded})
|
||||||
|
is applicable for this case.
|
||||||
|
|
||||||
|
\begin{table}
|
||||||
|
\begin{centering}
|
||||||
|
{\footnotesize{}}%
|
||||||
|
\begin{tabular}{cc|ccc}
|
||||||
|
\multicolumn{2}{c|}{{\footnotesize{}$\kappa=0$}} & & {\footnotesize{}$n$} & \tabularnewline
|
||||||
|
\multicolumn{1}{c}{} & & {\footnotesize{}0} & {\footnotesize{}1} & {\footnotesize{}2}\tabularnewline
|
||||||
|
\hline
|
||||||
|
\multirow{2}{*}{{\footnotesize{}$q$}} & {\footnotesize{}1} & {\footnotesize{}2} & {\footnotesize{}1} & {\footnotesize{}1}\tabularnewline
|
||||||
|
& {\footnotesize{}2} & {\footnotesize{}x} & {\footnotesize{}w} & {\footnotesize{}0}\tabularnewline
|
||||||
|
\end{tabular}{\footnotesize{} \hspace*{\fill}}%
|
||||||
|
\begin{tabular}{cc|ccccc}
|
||||||
|
\multicolumn{2}{c|}{{\footnotesize{}$\kappa=1$}} & \multicolumn{5}{c}{{\footnotesize{}$n$}}\tabularnewline
|
||||||
|
& & {\footnotesize{}0} & {\footnotesize{}1} & {\footnotesize{}2} & {\footnotesize{}3} & {\footnotesize{}4}\tabularnewline
|
||||||
|
\hline
|
||||||
|
\multirow{2}{*}{{\footnotesize{}$q$}} & {\footnotesize{}1} & {\footnotesize{}w} & {\footnotesize{}3} & {\footnotesize{}2} & {\footnotesize{}2} & {\footnotesize{}2}\tabularnewline
|
||||||
|
& {\footnotesize{}2} & {\footnotesize{}x} & {\footnotesize{}1} & {\footnotesize{}w} & {\footnotesize{}1} & {\footnotesize{}1}\tabularnewline
|
||||||
|
\end{tabular}{\footnotesize{} \hspace*{\fill}}%
|
||||||
|
\begin{tabular}{cc|ccccc}
|
||||||
|
\multicolumn{2}{c|}{{\footnotesize{}$\kappa=2$}} & \multicolumn{5}{c}{{\footnotesize{}$n$}}\tabularnewline
|
||||||
|
& & {\footnotesize{}0} & {\footnotesize{}1} & {\footnotesize{}2} & {\footnotesize{}3} & {\footnotesize{}4}\tabularnewline
|
||||||
|
\hline
|
||||||
|
\multirow{2}{*}{{\footnotesize{}$q$}} & {\footnotesize{}1} & {\footnotesize{}0/w} & {\footnotesize{}3} & {\footnotesize{}4} & {\footnotesize{}3} & {\footnotesize{}3}\tabularnewline
|
||||||
|
& {\footnotesize{}2} & {\footnotesize{}x} & {\footnotesize{}3} & {\footnotesize{}2} & {\footnotesize{}2} & {\footnotesize{}1}\tabularnewline
|
||||||
|
\end{tabular}
|
||||||
|
\par\end{centering}{\footnotesize \par}
|
||||||
|
|
||||||
|
\protect\caption{Asymptotical behaviour of some (\ref{eq:2D Hankel transform of regularized outgoing wave, decomposition})
|
||||||
|
obtained by Mathematica for $k\to\infty$. The table entries are the
|
||||||
|
$N$ of $\protect\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right)=o\left(1/k^{N}\right)$.
|
||||||
|
The special entry ``x'' means that Mathematica was not able to calculate
|
||||||
|
the integral, and ``w'' denotes that the first returned term was
|
||||||
|
not simply of the kind $(\ldots)k^{-N-1}$.\label{tab:Asymptotical-behaviour-Mathematica}}
|
||||||
|
\end{table}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{3d (TODO)}
|
||||||
|
|
||||||
|
\begin{multline*}
|
||||||
|
\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
|
||||||
|
\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right)
|
||||||
|
\end{multline*}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\section{Major TODOs and open questions}
|
||||||
|
\begin{itemize}
|
||||||
|
\item Check if (\ref{eq:2D Hankel transform of exponentially suppressed outgoing wave expanded})
|
||||||
|
gives a satisfactory result for the case $q=2,n=0$.
|
||||||
|
\item Analyse the behaviour $k\to k_{0}$.
|
||||||
|
\item Find a general algorithm for generating the expressions of the Hankel
|
||||||
|
transforms.
|
||||||
|
\item Three-dimensional case.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\section{(Appendix) Fourier vs. Hankel transform}
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Three dimensions}
|
||||||
|
|
||||||
|
Given a nice enough function $f$ of a real 3d variable, assume its
|
||||||
|
factorisation into radial and angular parts
|
||||||
|
\[
|
||||||
|
f(\vect r)=\sum_{l,m}f_{l,m}(\left|\vect r\right|)\ush lm\left(\theta_{\vect r},\phi_{\vect r}\right).
|
||||||
|
\]
|
||||||
|
Acording to (REF Baddour 2010, eqs. 13, 16), its Fourier transform
|
||||||
|
can then be expressed in terms of Hankel transforms (CHECK normalisation
|
||||||
|
of $j_{n}$, REF Baddour (1))
|
||||||
|
\[
|
||||||
|
\uaft f(\vect k)=\frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sum_{l,m}\left(-i\right)^{l}\left(\bsht{f_{l,m}}{}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)
|
||||||
|
\]
|
||||||
|
where the spherical Hankel transform $\bsht l{}$ of degree $l$ is
|
||||||
|
defined as (REF Baddour eq. 2)
|
||||||
|
\[
|
||||||
|
\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).
|
||||||
|
\]
|
||||||
|
Using this convention, the inverse spherical Hankel transform is given
|
||||||
|
by (REF Baddour eq. 3)
|
||||||
|
\[
|
||||||
|
g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k),
|
||||||
|
\]
|
||||||
|
so it is not unitary.
|
||||||
|
|
||||||
|
An unitary convention would look like this:
|
||||||
|
\begin{equation}
|
||||||
|
\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition}
|
||||||
|
\end{equation}
|
||||||
|
Then $\usht l{}^{-1}=\usht l{}$ and the unitary, angular-momentum
|
||||||
|
Fourier transform reads
|
||||||
|
\begin{eqnarray}
|
||||||
|
\uaft f(\vect k) & = & \frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sqrt{\frac{\pi}{2}}\sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)\nonumber \\
|
||||||
|
& = & \sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right).\label{eq:Fourier v. Hankel tf 3d}
|
||||||
|
\end{eqnarray}
|
||||||
|
Cool.
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Two dimensions}
|
||||||
|
|
||||||
|
Similarly in 2d, let the expansion of $f$ be
|
||||||
|
\[
|
||||||
|
f\left(\vect r\right)=\sum_{m}f_{m}\left(\left|\vect r\right|\right)e^{im\phi_{\vect r}},
|
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|
\]
|
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|
its Fourier transform is then (CHECK this, it is taken from the Wikipedia
|
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|
article on Hankel transform)
|
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|
\begin{equation}
|
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|
\uaft f\left(\vect k\right)=\sum_{m}i^{m}e^{im\phi_{\vect k}}\pht mf_{m}\left(\left|\vect k\right|\right)\label{eq:Fourier v. Hankel tf 2d}
|
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|
\end{equation}
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|
where the Hankel transform of order $m$ is defined as
|
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|
\begin{equation}
|
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|
\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}
|
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|
\end{equation}
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|
which is already self-inverse, $\pht m{}^{-1}=\pht m{}$ (hence also
|
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|
unitary).
|
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|
|
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|
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|
\section{(Appendix) Multidimensional Dirac comb}
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|
|
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|
|
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|
\subsection{1D}
|
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|
|
||||||
|
This is all from Wikipedia
|
||||||
|
|
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|
|
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|
\subsubsection{Definitions}
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|
|
||||||
|
\begin{eqnarray*}
|
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|
\lyxmathsym{Ш}(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-k)\\
|
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|
\lyxmathsym{Ш}_{T}(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-kT)=\frac{1}{T}\lyxmathsym{Ш}\left(\frac{t}{T}\right)
|
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|
\end{eqnarray*}
|
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|
|
||||||
|
|
||||||
|
|
||||||
|
\subsubsection{Fourier series representation}
|
||||||
|
|
||||||
|
\begin{equation}
|
||||||
|
\lyxmathsym{Ш}_{T}(t)=\sum_{n=-\infty}^{\infty}e^{2\pi int/T}\label{eq:1D Dirac comb Fourier series}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsubsection{Fourier transform}
|
||||||
|
|
||||||
|
With unitary ordinary frequency Ft., i.e.
|
||||||
|
|
||||||
|
\[
|
||||||
|
\uoft f(\vect{\xi})\equiv\int_{\mathbb{R}^{n}}f(\vect x)e^{-2\pi i\vect x\cdot\vect{\xi}}\ud^{n}\vect x
|
||||||
|
\]
|
||||||
|
we have
|
||||||
|
\begin{equation}
|
||||||
|
\uoft{\lyxmathsym{Ш}_{T}}(f)=\frac{1}{T}\lyxmathsym{Ш}_{\frac{1}{T}}(f)=\sum_{n=-\infty}^{\infty}e^{-i2\pi fnT}\label{eq:1D Dirac comb Ft ordinary freq}
|
||||||
|
\end{equation}
|
||||||
|
and with unitary angular frequency Ft., i.e.
|
||||||
|
\begin{equation}
|
||||||
|
\uaft f(\vect k)\equiv\frac{1}{\left(2\pi\right)^{n/2}}\int_{\mathbb{R}^{n}}f(\vect x)e^{-i\vect x\cdot\vect k}\ud^{n}\vect x\label{eq:Ft unitary angular frequency}
|
||||||
|
\end{equation}
|
||||||
|
we have (CHECK)
|
||||||
|
\[
|
||||||
|
\uaft{\lyxmathsym{Ш}_{T}}(\omega)=\frac{\sqrt{2\pi}}{T}\lyxmathsym{Ш}_{\frac{2\pi}{T}}(\omega)=\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-i\omega nT}
|
||||||
|
\]
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Dirac comb for multidimensional lattices}
|
||||||
|
|
||||||
|
|
||||||
|
\subsubsection{Definitions}
|
||||||
|
|
||||||
|
Let $d$ be the dimensionality of the real vector space in question,
|
||||||
|
and let $\basis u\equiv\left\{ \vect u_{i}\right\} _{i=1}^{d}$ denote
|
||||||
|
a basis for some lattice in that space. Let the corresponding lattice
|
||||||
|
delta comb be
|
||||||
|
\[
|
||||||
|
\dc{\basis u}\left(\vect x\right)\equiv\sum_{n_{1}=-\infty}^{\infty}\ldots\sum_{n_{d}=-\infty}^{\infty}\delta\left(\vect x-\sum_{i=1}^{d}n_{i}\vect u_{i}\right).
|
||||||
|
\]
|
||||||
|
|
||||||
|
|
||||||
|
Furthemore, let $\rec{\basis u}\equiv\left\{ \rec{\vect u}_{i}\right\} _{i=1}^{d}$
|
||||||
|
be the reciprocal lattice basis, that is the basis satisfying $\vect u_{i}\cdot\rec{\vect u_{j}}=\delta_{ij}$.
|
||||||
|
This slightly differs from the usual definition of a reciprocal basis,
|
||||||
|
here denoted $\recb{\basis u}\equiv\left\{ \recb{\vect u_{i}}\right\} _{i=1}^{d}$,
|
||||||
|
which satisfies $\vect u_{i}\cdot\recb{\vect u_{j}}=2\pi\delta_{ij}$
|
||||||
|
instead.
|
||||||
|
|
||||||
|
|
||||||
|
\subsubsection{Factorisation of a multidimensional lattice delta comb}
|
||||||
|
|
||||||
|
By simple drawing, it can be seen that
|
||||||
|
\[
|
||||||
|
\dc{\basis u}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right)
|
||||||
|
\]
|
||||||
|
where $c_{\basis u}$ is some numerical volume factor. In order to
|
||||||
|
determine $c_{\basis u}$, let us consider only the ``zero tooth''
|
||||||
|
of the comb, leading to
|
||||||
|
\[
|
||||||
|
\delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\delta\left(\vect x\cdot\rec{\vect u_{i}}\right).
|
||||||
|
\]
|
||||||
|
From the scaling property of delta function, $\delta(ax)=\left|a\right|^{-1}\delta(x)$,
|
||||||
|
we get
|
||||||
|
\[
|
||||||
|
\delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert ^{-1}\delta\left(\vect x\cdot\frac{\rec{\vect u_{i}}}{\left\Vert \rec{\vect u_{i}}\right\Vert }\right).
|
||||||
|
\]
|
||||||
|
|
||||||
|
|
||||||
|
From the Osgood's book (p. 375):
|
||||||
|
|
||||||
|
\[
|
||||||
|
\dc A(\vect x)=\frac{1}{\left|\det A\right|}\dc{}^{(d)}\left(A^{-1}\vect x\right)
|
||||||
|
\]
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsubsection{Fourier series representation}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsubsection{Fourier transform (OK)}
|
||||||
|
|
||||||
|
From the Osgood's book https://see.stanford.edu/materials/lsoftaee261/chap8.pdf,
|
||||||
|
p. 379
|
||||||
|
|
||||||
|
\[
|
||||||
|
\uoft{\dc{\basis u}}\left(\vect{\xi}\right)=\left|\det\rec{\basis u}\right|\dc{\rec{\basis u}}^{(d)}\left(\vect{\xi}\right).
|
||||||
|
\]
|
||||||
|
And consequently, for unitary/angular frequency it is
|
||||||
|
|
||||||
|
\begin{eqnarray}
|
||||||
|
\uaft{\dc{\basis u}}\left(\vect k\right) & = & \frac{1}{\left(2\pi\right)^{\frac{d}{2}}}\uoft{\dc{\basis u}}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
|
||||||
|
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\rec{\basis u}}^{(d)}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
|
||||||
|
& = & \left(2\pi\right)^{\frac{d}{2}}\left|\det\rec{\basis u}\right|\dc{\recb{\basis u}}\left(\vect k\right)\nonumber \\
|
||||||
|
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\recb{\basis u}}\left(\vect k\right).\label{eq:Dirac comb uaFt}
|
||||||
|
\end{eqnarray}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\subsubsection{Convolution}
|
||||||
|
|
||||||
|
\[
|
||||||
|
\left(f\ast\dc{\basis u}\right)(\vect x)=\sum_{\vect t\in\basis u\ints^{d}}f(\vect x-\vect t)
|
||||||
|
\]
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\end{document}
|
||||||
Loading…
Reference in New Issue