notes/Ewald summation.bib

   1
   2 @article{leeuwElectrostaticLatticeSums1979,
   3   title = {Electrostatic Lattice Sums for Semi-Infinite Lattices},
   4   volume = {37},
   5   issn = {0026-8976},
   6   doi = {10.1080/00268977900100951},
   7   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.},
   8   number = {4},
   9   journal = {Molecular Physics},
  10   author = {Leeuw, Simon W. De and Perram, John W.},
  11   month = apr,
  12   year = {1979},
  13   pages = {1313-1322},
  14   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}
  15 }
  16
  17 @article{mazarsLeknerSummationsEwald2005,
  18   archivePrefix = {arXiv},
  19   eprinttype = {arxiv},
  20   eprint = {cond-mat/0301161},
  21   title = {Lekner Summations and {{Ewald}} Summations for Quasi-Two Dimensional Systems},
  22   volume = {103},
  23   issn = {0026-8976, 1362-3028},
  24   doi = {10.1080/00268970412331332934},
  25   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.},
  26   number = {9},
  27   journal = {Molecular Physics},
  28   author = {Mazars, M.},
  29   month = may,
  30   year = {2005},
  31   keywords = {Condensed Matter - Statistical Mechanics},
  32   pages = {1241-1260},
  33   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}
  34 }
  35
  36 @incollection{weisSimpleDipolarFluids,
  37   series = {Advances in Polymer Science},
  38   title = {Simple {{Dipolar Fluids}} as {{Generic Models}} for {{Soft Matter}}},
  39   isbn = {978-3-540-26091-2 978-3-540-31581-0},
  40   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.},
  41   language = {en},
  42   booktitle = {Advanced {{Computer Simulation Approaches}} for {{Soft Matter Sciences II}}},
  43   publisher = {{Springer, Berlin, Heidelberg}},
  44   author = {Weis, J.-J. and Levesque, D.},
  45   pages = {163-225},
  46   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},
  47   doi = {10.1007/b136796}
  48 }
  49
  50 @incollection{arnoldEfficientMethodsCompute,
  51   series = {Advances in Polymer Science},
  52   title = {Efficient {{Methods}} to {{Compute Long}}-{{Range Interactions}} for {{Soft Matter Systems}}},
  53   isbn = {978-3-540-26091-2 978-3-540-31581-0},
  54   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},
  55   language = {en},
  56   booktitle = {Advanced {{Computer Simulation Approaches}} for {{Soft Matter Sciences II}}},
  57   publisher = {{Springer, Berlin, Heidelberg}},
  58   author = {Arnold, Axel and Holm, Christian},
  59   pages = {59-109},
  60   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},
  61   doi = {10.1007/b136793}
  62 }
  63
  64 @article{harrisEwaldSummationsSystems1998,
  65   title = {Ewald Summations in Systems with Two-Dimensional Periodicity},
  66   volume = {68},
  67   issn = {1097-461X},
  68   doi = {10.1002/(SICI)1097-461X(1998)68:6<385::AID-QUA2>3.0.CO;2-R},
  69   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},
  70   language = {en},
  71   number = {6},
  72   journal = {International Journal of Quantum Chemistry},
  73   author = {Harris, Frank E.},
  74   month = jan,
  75   year = {1998},
  76   keywords = {Ewald method,electrostatic energy,lattice sums,two-dimensional},
  77   pages = {385-404},
  78   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}
  79 }
  80
  81 @article{gaoVaporLiquidCoexistence1997,
  82   title = {Vapor\textendash{}Liquid Coexistence of Quasi-Two-Dimensional {{Stockmayer}} Fluids},
  83   volume = {106},
  84   issn = {0021-9606},
  85   doi = {10.1063/1.473079},
  86   number = {8},
  87   journal = {The Journal of Chemical Physics},
  88   author = {Gao, G. T. and Zeng, X. C. and Wang, Wenchuan},
  89   month = feb,
  90   year = {1997},
  91   pages = {3311-3317},
  92   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}
  93 }
  94
  95 @article{hummerMolecularTheoriesSimulation1998,
  96   title = {Molecular {{Theories}} and {{Simulation}} of {{Ions}} and {{Polar Molecules}} in {{Water}}},
  97   volume = {102},
  98   issn = {1089-5639},
  99   doi = {10.1021/jp982195r},
 100   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.},
 101   number = {41},
 102   journal = {The Journal of Physical Chemistry A},
 103   author = {Hummer, Gerhard and Pratt, Lawrence R. and Garc\'ia, Angel E.},
 104   month = oct,
 105   year = {1998},
 106   pages = {7885-7895},
 107   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}
 108 }
 109
 110 @article{rokhlinRapidSolutionIntegral1985,
 111   title = {Rapid Solution of Integral Equations of Classical Potential Theory},
 112   volume = {60},
 113   issn = {0021-9991},
 114   doi = {10.1016/0021-9991(85)90002-6},
 115   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.},
 116   number = {2},
 117   journal = {Journal of Computational Physics},
 118   author = {Rokhlin, V},
 119   month = sep,
 120   year = {1985},
 121   pages = {187-207},
 122   file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/IG8HX6QD/rokhlin1985.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/276H2VVD/0021999185900026.html}
 123 }
 124
 125 @article{baddourApplicationGeneralizedShift2014,
 126   title = {Application of the Generalized Shift Operator to the {{Hankel}} Transform},
 127   volume = {3},
 128   issn = {2193-1801},
 129   doi = {10.1186/2193-1801-3-246},
 130   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.},
 131   journal = {SpringerPlus},
 132   author = {Baddour, Natalie},
 133   month = may,
 134   year = {2014},
 135   pages = {246},
 136   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}
 137 }
 138
 139 @article{wangFourierAnalysisPolar2008,
 140   title = {Fourier {{Analysis}} in {{Polar}} and {{Spherical Coordinates}}},
 141   author = {Wang, Qing and Ronneberger, Olaf and Burkhardt, Hans},
 142   year = {2008},
 143   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}
 144 }
 145
 146 @article{adkinsThreedimensionalFourierTransforms2013,
 147   archivePrefix = {arXiv},
 148   eprinttype = {arxiv},
 149   eprint = {1302.1830},
 150   primaryClass = {math-ph},
 151   title = {Three-Dimensional {{Fourier}} Transforms, Integrals of Spherical {{Bessel}} Functions, and Novel Delta Function Identities},
 152   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).},
 153   journal = {arXiv:1302.1830 [math-ph]},
 154   author = {Adkins, Gregory S.},
 155   month = feb,
 156   year = {2013},
 157   keywords = {Mathematical Physics,42B10},
 158   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}
 159 }
 160
 161 @article{baddourOperationalConvolutionProperties2010,
 162   title = {Operational and Convolution Properties of Three-Dimensional {{Fourier}} Transforms in Spherical Polar Coordinates},
 163   volume = {27},
 164   copyright = {\textcopyright{} 2010 Optical Society of America},
 165   issn = {1520-8532},
 166   doi = {10.1364/JOSAA.27.002144},
 167   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.},
 168   language = {EN},
 169   number = {10},
 170   journal = {JOSA A},
 171   author = {Baddour, Natalie},
 172   month = oct,
 173   year = {2010},
 174   keywords = {Spectrum Analysis,Continuous optical signal processing,Tomographic image processing,Transforms},
 175   pages = {2144-2155},
 176   file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/RDN4K7GW/baddour2010.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/9R32Q9ND/abstract.html}
 177 }
 178
 179 @article{sheppardHankelTransformNdimensions2015,
 180   title = {The {{Hankel Transform}} in N-Dimensions and {{Its Applications}} in {{Optical Propagation}} and {{Imaging}}},
 181   volume = {188},
 182   issn = {1076-5670},
 183   doi = {10.1016/bs.aiep.2015.02.003},
 184   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.},
 185   journal = {Advances in Imaging and Electron Physics},
 186   author = {Sheppard, Colin J. R. and Kou, Shan S. and Lin, Jiao},
 187   month = jan,
 188   year = {2015},
 189   keywords = {Green function,diffraction,Hankel transform,Fourier transform,propagation,planar waveguides,fringe analysis,Plasmonics},
 190   pages = {135-184},
 191   file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/MDUDQB8Q/sheppard2015.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/CJEIER7D/S107656701500021X.html}
 192 }
 193
 194 @article{bloomfieldIndefiniteIntegralsSpherical2017,
 195   archivePrefix = {arXiv},
 196   eprinttype = {arxiv},
 197   eprint = {1703.06428},
 198   primaryClass = {math},
 199   title = {Indefinite {{Integrals}} of {{Spherical Bessel Functions}}},
 200   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.},
 201   journal = {arXiv:1703.06428 [math]},
 202   author = {Bloomfield, Jolyon K. and Face, Stephen H. P. and Moss, Zander},
 203   month = mar,
 204   year = {2017},
 205   keywords = {Mathematics - Classical Analysis and ODEs,Mathematics - Numerical Analysis},
 206   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}
 207 }
 208
 209 @book{v.aIntegralnyePreobrazovaniyaOperacionnoe1962,
 210   series = {\cyrchar\CYRS\cyrchar\CYRM\cyrchar\CYRB},
 211   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},
 212   publisher = {{\cyrchar\CYRF\cyrchar\cyri\cyrchar\cyrz\cyrchar\cyrm\cyrchar\cyra\cyrchar\cyrt\cyrchar\cyrg\cyrchar\cyri\cyrchar\cyrz}},
 213   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},
 214   year = {1962}
 215 }
 216
 217 @incollection{WorpitzkyNumbers2015,
 218   title = {Worpitzky {{Numbers}}},
 219   isbn = {978-981-4725-26-2},
 220   booktitle = {Combinatorial {{Identities}} for {{Stirling Numbers}}},
 221   publisher = {{WORLD SCIENTIFIC}},
 222   month = aug,
 223   year = {2015},
 224   pages = {147-163},
 225   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},
 226   doi = {10.1142/9789814725286_0011}
 227 }
 228
 229 @book{a.pIntegralyRyadySpecialnye2003,
 230   edition = {2. \cyrchar\cyri\cyrchar\cyrz\cyrchar\cyrd., \cyrchar\cyri\cyrchar\cyrs\cyrchar\cyrp\cyrchar\cyrr},
 231   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},
 232   volume = {\cyrchar\CYRT\cyrchar\cyro\cyrchar\cyrm{} 3},
 233   isbn = {978-5-9221-0322-0},
 234   publisher = {{\cyrchar\CYRF\cyrchar\CYRI\cyrchar\CYRZ\cyrchar\CYRM\cyrchar\CYRA\cyrchar\CYRT\cyrchar\CYRL\cyrchar\CYRI\cyrchar\CYRT}},
 235   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},
 236   year = {2003},
 237   file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/DKE6VC44/[Prudnikov_A.P.,_Bruechkov_YU.A.,_Marichev_O.I.]_I(BookFi)(1).djvu}
 238 }
 239
 240 @misc{CombinatorialIdentitiesStirling,
 241   title = {Combinatorial {{Identities}} for {{Stirling Numbers}}},
 242   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.},
 243   howpublished = {http://www.worldscientific.com/worldscibooks/10.1142/9821},
 244   journal = {World Scientific Publishing Company},
 245   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}
 246 }
 247
 248 @article{zivFastEvaluationElementary1991,
 249   title = {Fast {{Evaluation}} of {{Elementary Mathematical Functions}} with {{Correctly Rounded Last Bit}}},
 250   volume = {17},
 251   issn = {0098-3500},
 252   doi = {10.1145/114697.116813},
 253   number = {3},
 254   journal = {ACM Trans. Math. Softw.},
 255   author = {Ziv, Abraham},
 256   month = sep,
 257   year = {1991},
 258   keywords = {compatibility,correct rounding,mathematical library},
 259   pages = {410--423},
 260   file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/VRVEUFSA/Ziv - 1991 - Fast Evaluation of Elementary Mathematical Functio.pdf}
 261 }
 262
 263 @book{a.pIntegralyRyadyElementarnye2002,
 264   edition = {2. \cyrchar\cyri\cyrchar\cyrz\cyrchar\cyrd., \cyrchar\cyri\cyrchar\cyrs\cyrchar\cyrp\cyrchar\cyrr},
 265   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},
 266   volume = {\cyrchar\CYRT\cyrchar\cyro\cyrchar\cyrm{} 1},
 267   isbn = {978-5-9221-0322-0},
 268   publisher = {{\cyrchar\CYRF\cyrchar\CYRI\cyrchar\CYRZ\cyrchar\CYRM\cyrchar\CYRA\cyrchar\CYRT\cyrchar\CYRL\cyrchar\CYRI\cyrchar\CYRT}},
 269   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},
 270   year = {2002},
 271   file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/R3QJRT5W/[Prudnikov_A.P.,_Bruechkov_YU.A.,_Marichev_O.I.]_I(BookFi).djvu}
 272 }
 273
 274 @book{a.pIntegralyRyadySpecialnye2003a,
 275   edition = {2. \cyrchar\cyri\cyrchar\cyrz\cyrchar\cyrd., \cyrchar\cyri\cyrchar\cyrs\cyrchar\cyrp\cyrchar\cyrr},
 276   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},
 277   volume = {\cyrchar\CYRT\cyrchar\cyro\cyrchar\cyrm{} 2},
 278   isbn = {978-5-9221-0322-0},
 279   publisher = {{\cyrchar\CYRF\cyrchar\CYRI\cyrchar\CYRZ\cyrchar\CYRM\cyrchar\CYRA\cyrchar\CYRT\cyrchar\CYRL\cyrchar\CYRI\cyrchar\CYRT}},
 280   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},
 281   year = {2003},
 282   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}
 283 }
 284
 285 @article{wuAccurateComputationHypergeometric1993,
 286   title = {An {{Accurate Computation}} of the {{Hypergeometric Distribution Function}}},
 287   volume = {19},
 288   issn = {0098-3500},
 289   doi = {10.1145/151271.151274},
 290   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.},
 291   number = {1},
 292   journal = {ACM Trans. Math. Softw.},
 293   author = {Wu, Trong},
 294   month = mar,
 295   year = {1993},
 296   keywords = {Ada programming language,Peizer approximations,exception handling,hypergeometric distribution function,prime number factorization,tasking},
 297   pages = {33--43},
 298   file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/CC3GQ79I/Wu - 1993 - An Accurate Computation of the Hypergeometric Dist.pdf}
 299 }
 300
 301 @article{nicoroviciPhotonicBandGaps1995,
 302   title = {Photonic Band Gaps for Arrays of Perfectly Conducting Cylinders},
 303   volume = {52},
 304   doi = {10.1103/PhysRevE.52.1135},
 305   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.},
 306   number = {1},
 307   journal = {Physical Review E},
 308   author = {Nicorovici, N. A. and McPhedran, R. C. and Botten, L. C.},
 309   month = jul,
 310   year = {1995},
 311   pages = {1135-1145},
 312   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}
 313 }
 314
 315 @article{linton_lattice_2010,
 316   title = {Lattice {{Sums}} for the {{Helmholtz Equation}}},
 317   volume = {52},
 318   issn = {0036-1445},
 319   doi = {10.1137/09075130X},
 320   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.},
 321   number = {4},
 322   journal = {SIAM Review},
 323   author = {Linton, C.},
 324   month = jan,
 325   year = {2010},
 326   pages = {630-674},
 327   file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/T86ATKYB/09075130x.pdf;/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/ETB8X4S9/09075130X.html}
 328 }
 329
 330 @article{linton_one-_2009,
 331   title = {One- and Two-Dimensional Lattice Sums for the Three-Dimensional {{Helmholtz}} Equation},
 332   volume = {228},
 333   issn = {0021-9991},
 334   doi = {10.1016/j.jcp.2008.11.013},
 335   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.},
 336   number = {6},
 337   journal = {Journal of Computational Physics},
 338   author = {Linton, C. M. and Thompson, I.},
 339   month = apr,
 340   year = {2009},
 341   keywords = {Helmholtz equation,Clausen function,Ewald summation,Lattice reduction,Lattice sum,Schlömilch series},
 342   pages = {1815-1829},
 343   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}
 344 }
 345
 346 @article{wei_broadband_2017,
 347   title = {A {{Broadband ML}}-{{FMA}} for 3-{{D Periodic Green}}'s {{Function}} in 2-{{D Lattice Using Ewald Summation}}},
 348   volume = {65},
 349   issn = {0018-926X},
 350   doi = {10.1109/TAP.2017.2690533},
 351   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.},
 352   number = {6},
 353   journal = {IEEE Transactions on Antennas and Propagation},
 354   author = {Wei, M. and Chew, W. C.},
 355   month = jun,
 356   year = {2017},
 357   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},
 358   pages = {3134-3145},
 359   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}
 360 }
 361
 362 @article{moroz_quasi-periodic_2006,
 363   title = {Quasi-Periodic {{Green}}'s Functions of the {{Helmholtz}} and {{Laplace}} Equations},
 364   volume = {39},
 365   issn = {0305-4470},
 366   doi = {10.1088/0305-4470/39/36/009},
 367   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.},
 368   language = {en},
 369   number = {36},
 370   journal = {Journal of Physics A: Mathematical and General},
 371   author = {Moroz, Alexander},
 372   year = {2006},
 373   pages = {11247},
 374   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}
 375 }
 376
 377 @article{mcrae_multiplescattering_1966,
 378   title = {Multiple-{{Scattering Treatment}} of {{Low}}-{{Energy Electron}}-{{Diffraction Intensities}}},
 379   volume = {45},
 380   issn = {0021-9606},
 381   doi = {10.1063/1.1728101},
 382   number = {9},
 383   journal = {The Journal of Chemical Physics},
 384   author = {McRae, E. G.},
 385   month = nov,
 386   year = {1966},
 387   pages = {3258-3276},
 388   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}
 389 }
 390
 391 @article{kambe_theory_2014,
 392   title = {Theory of {{Electron Diffraction}} by {{Crystals}}},
 393   volume = {22},
 394   issn = {1865-7109},
 395   doi = {10.1515/zna-1967-0402},
 396   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.},
 397   number = {4},
 398   journal = {Zeitschrift f\"ur Naturforschung A},
 399   author = {Kambe, Kyozaburo},
 400   year = {2014},
 401   pages = {422--431},
 402   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}
 403 }
 404
 405 @article{enoch_sums_2001,
 406   title = {Sums of Spherical Waves for Lattices, Layers, and Lines},
 407   volume = {42},
 408   issn = {0022-2488},
 409   doi = {10.1063/1.1409348},
 410   number = {12},
 411   journal = {Journal of Mathematical Physics},
 412   author = {Enoch, S. and McPhedran, R. C. and Nicorovici, N. A. and Botten, L. C. and Nixon, J. N.},
 413   month = nov,
 414   year = {2001},
 415   pages = {5859-5870},
 416   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}
 417 }
 418
 419