Paper start single-particle part
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Marek Nečada
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@article{xu_efficient_1998,
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title = {Efficient {{Evaluation}} of {{Vector Translation Coefficients}} in {{Multiparticle Light}}-{{Scattering Theories}}},
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volume = {139},
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issn = {0021-9991},
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abstract = {Vector addition theorems are a necessary ingredient in the analytical solution of electromagnetic multiparticle-scattering problems. These theorems include a large number of vector addition coefficients. There exist three basic types of analytical expressions for vector translation coefficients: Stein's (Quart. Appl. Math.19, 15 (1961)), Cruzan's (Quart. Appl. Math.20, 33 (1962)), and Xu's (J. Comput. Phys.127, 285 (1996)). Stein's formulation relates vector translation coefficients with scalar translation coefficients. Cruzan's formulas use the Wigner 3jm symbol. Xu's expressions are based on the Gaunt coefficient. Since the scalar translation coefficient can also be expressed in terms of the Gaunt coefficient, the key to the expeditious and reliable calculation of vector translation coefficients is the fast and accurate evaluation of the Wigner 3jm symbol or the Gaunt coefficient. We present highly efficient recursive approaches to accurately evaluating Wigner 3jm symbols and Gaunt coefficients. Armed with these recursive approaches, we discuss several schemes for the calculation of the vector translation coefficients, using the three general types of formulation, respectively. Our systematic test calculations show that the three types of formulas produce generally the same numerical results except that the algorithm of Stein's type is less accurate in some particular cases. These extensive test calculations also show that the scheme using the formulation based on the Gaunt coefficient is the most efficient in practical computations.},
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number = {1},
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journal = {Journal of Computational Physics},
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doi = {10.1006/jcph.1997.5867},
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author = {Xu, Yu-lin},
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month = jan,
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year = {1998},
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pages = {137-165},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/STV5263F/Xu - 1998 - Efficient Evaluation of Vector Translation Coeffic.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/VMGZRSAA/S0021999197958678.html}
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}
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@book{jackson_classical_1998,
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address = {{New York}},
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edition = {3 edition},
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title = {Classical {{Electrodynamics Third Edition}}},
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isbn = {978-0-471-30932-1},
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abstract = {A revision of the defining book covering the physics and classical mathematics necessary to understand electromagnetic fields in materials and at surfaces and interfaces. The third edition has been revised to address the changes in emphasis and applications that have occurred in the past twenty years.},
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language = {English},
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publisher = {{Wiley}},
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author = {Jackson, John David},
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month = aug,
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year = {1998},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/3BWPD4BK/John David Jackson-Classical Electrodynamics-Wiley (1999).djvu}
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}
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@article{mie_beitrage_1908,
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title = {Beitr{\"a}ge Zur {{Optik}} Tr{\"u}ber {{Medien}}, Speziell Kolloidaler {{Metall{\"o}sungen}}},
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volume = {330},
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copyright = {Copyright \textcopyright{} 1908 WILEY-VCH Verlag GmbH \& Co. KGaA, Weinheim},
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issn = {1521-3889},
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language = {en},
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number = {3},
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journal = {Ann. Phys.},
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doi = {10.1002/andp.19083300302},
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author = {Mie, Gustav},
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month = jan,
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year = {1908},
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pages = {377-445},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/RM9J9RYH/Mie - 1908 - Beiträge zur Optik trüber Medien, speziell kolloid.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/F5A7EX6R/abstract.html}
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}
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@book{kristensson_scattering_2016,
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address = {{Edison, NJ}},
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title = {Scattering of {{Electromagnetic Waves}} by {{Obstacles}}},
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isbn = {978-1-61353-221-8},
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abstract = {This book is an introduction to some of the most important properties of electromagnetic waves and their interaction with passive materials and scatterers. The main purpose of the book is to give a theoretical treatment of these scattering phenomena, and to illustrate numerical computations of some canonical scattering problems for different geometries and materials. The scattering theory is also important in the theory of passive antennas, and this book gives several examples on this topic. Topics covered include an introduction to the basic equations used in scattering; the Green functions and dyadics; integral representation of fields; introductory scattering theory; scattering in the time domain; approximations and applications; spherical vector waves; scattering by spherical objects; the null-field approach; and propagation in stratified media. The book is organised along two tracks, which can be studied separately or together. Track 1 material is appropriate for a first reading of the textbook, while Track 2 contains more advanced material suited for the second reading and for reference. Exercises are included for each chapter.},
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language = {English},
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publisher = {{Scitech Publishing}},
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author = {Kristensson, Gerhard},
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month = jul,
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year = {2016},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/3R7VYZUK/Kristensson - 2016 - Scattering of Electromagnetic Waves by Obstacles.pdf}
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}
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@ -207,6 +207,31 @@
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\thespace}{\reals^{3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\particle}{\mathrm{\Omega}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\medium}{\thespace\backslash\particle}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\epsbg}{\mathrm{\epsilon_{b}}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\mubg}{\mathrm{\mu_{b}}}
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\end_inset
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\end_layout
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\begin_layout Title
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@ -445,6 +470,17 @@ filename "examples.lyx"
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset bibtex
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LatexCommand bibtex
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bibfiles "T-matrix paper"
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options "plain"
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\end_inset
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\end_layout
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\end_body
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@ -83,5 +83,189 @@
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Finite systems
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\end_layout
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\begin_layout Itemize
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\lang english
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motivation (classes of problems that this can solve: response to external
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radiation, resonances, ...)
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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\lang english
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theory
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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\lang english
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T-matrix definition, basics
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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\lang english
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How to get it?
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\end_layout
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\end_deeper
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\begin_layout Itemize
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\lang english
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translation operators (TODO think about how explicit this should be, but
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I guess it might be useful to write them to write them explicitly (but
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in the shortest possible form) in the normalisation used in my program)
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\end_layout
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\begin_layout Itemize
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\lang english
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employing point group symmetries and decomposing the problem to decrease
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the computational complexity (maybe separately)
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\end_layout
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\end_deeper
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\end_deeper
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\begin_layout Subsection
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\lang english
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Motivation
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\end_layout
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\begin_layout Subsection
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\lang english
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Single-particle scattering
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\end_layout
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\begin_layout Standard
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In order to define the basic concepts, let us first consider the case of
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EM radiation scattered by a single particle.
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We assume that the scatterer lies inside a closed sphere
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\begin_inset Formula $\particle$
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\end_inset
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, the space outside this volume
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\begin_inset Formula $\medium$
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\end_inset
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is filled with an homogeneous isotropic medium with relative electric permittiv
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ity
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\begin_inset Formula $\epsilon(\vect r,\omega)=\epsbg(\omega)$
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\end_inset
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and magnetic permeability
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\begin_inset Formula $\mu(\vect r,\omega)=\mubg(\omega)$
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\end_inset
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, and that the whole system is linear, i.e.
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the material properties of neither the medium nor the scatterer depend
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on field intensities.
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Under these assumptions, the EM fields in
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\begin_inset Formula $\medium$
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\end_inset
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must satisfy the homogeneous vector Helmholtz equation
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\begin_inset Formula $\left(\nabla^{2}+k^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0$
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\end_inset
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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todo define
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\begin_inset Formula $\Psi$
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\end_inset
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, mention transversality
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\end_layout
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\end_inset
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with
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\begin_inset Formula $k=TODO$
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\end_inset
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[TODO REF Jackson?].
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Its solutions (TODO under which conditions? What vector space do the SVWFs
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actually span? Check Comment 9.2 and Appendix f.9.1 in Kristensson)
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\end_layout
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\begin_layout Standard
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\lang english
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Throughout this text, we will use the same normalisation conventions as
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in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "kristensson_scattering_2016"
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\end_inset
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.
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\end_layout
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\begin_layout Subsubsection
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\lang english
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Spherical waves
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\end_layout
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\begin_layout Standard
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\lang english
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\lang english
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TODO small note about cartesian multipoles, anapoles etc.
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(There should be some comparing paper that the Russians at META 2018 mentioned.)
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Subsubsection
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\lang english
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T-matrix definition
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\end_layout
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\begin_layout Subsubsection
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Absorbed power
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\end_layout
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\begin_layout Subsubsection
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\lang english
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T-matrix compactness, cutoff validity
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\end_layout
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\begin_layout Subsection
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\lang english
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Multiple scattering
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\end_layout
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\begin_layout Subsubsection
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\lang english
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Translation operator
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\end_layout
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\begin_layout Subsubsection
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\lang english
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Numerical complexity, comparison to other methods
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\end_layout
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\end_body
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\end_document
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@ -83,5 +83,409 @@
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Infinite periodic systems
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\end_layout
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\begin_layout Subsection
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\lang english
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Formulation of the problem
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\end_layout
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\begin_layout Standard
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\lang english
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Assume a system of compact EM scatterers in otherwise homogeneous and isotropic
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medium, and assume that the system, i.e.
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both the medium and the scatterers, have linear response.
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A scattering problem in such system can be written as
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\begin_inset Formula
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\[
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A_{α}=T_{α}P_{α}=T_{α}(\sum_{β}S_{α\leftarrowβ}A_{β}+P_{0α})
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\]
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\end_inset
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where
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\begin_inset Formula $T_{α}$
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\end_inset
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is the
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\begin_inset Formula $T$
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\end_inset
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-matrix for scatterer α,
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\begin_inset Formula $A_{α}$
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\end_inset
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is its vector of the scattered wave expansion coefficient (the multipole
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indices are not explicitely indicated here) and
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\begin_inset Formula $P_{α}$
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\end_inset
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is the local expansion of the incoming sources.
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\begin_inset Formula $S_{α\leftarrowβ}$
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\end_inset
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is ...
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and ...
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is ...
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\end_layout
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\begin_layout Standard
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\lang english
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...
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\end_layout
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\begin_layout Standard
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\lang english
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\begin_inset Formula
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\[
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\sum_{β}(\delta_{αβ}-T_{α}S_{α\leftarrowβ})A_{β}=T_{α}P_{0α}.
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\lang english
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Now suppose that the scatterers constitute an infinite lattice
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\end_layout
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\begin_layout Standard
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\lang english
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\begin_inset Formula
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\[
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=T_{\vect aα}P_{0\vect aα}.
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\]
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\end_inset
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Due to the periodicity, we can write
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\begin_inset Formula $S_{\vect aα\leftarrow\vect bβ}=S_{α\leftarrowβ}(\vect b-\vect a)$
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\end_inset
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and
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\begin_inset Formula $T_{\vect aα}=T_{\alpha}$
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\end_inset
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.
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In order to find lattice modes, we search for solutions with zero RHS
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\begin_inset Formula
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\[
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
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\]
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\end_inset
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and we assume periodic solution
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\begin_inset Formula $A_{\vect b\beta}(\vect k)=A_{\vect a\beta}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$
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\end_inset
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, yielding
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\begin_inset Formula
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\begin{eqnarray*}
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
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\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect 0α\leftarrow\vect bβ})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
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\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
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A_{\vect 0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
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\end{eqnarray*}
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\end_inset
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Therefore, in order to solve the modes, we need to compute the
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\begin_inset Quotes eld
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\end_inset
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lattice Fourier transform
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\begin_inset Quotes erd
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\end_inset
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of the translation operator,
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\begin_inset Formula
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\begin{equation}
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W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Subsection
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\lang english
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Computing the Fourier sum of the translation operator
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\end_layout
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\begin_layout Standard
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\lang english
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The problem evaluating
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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is the asymptotic behaviour of the translation operator,
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\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
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\end_inset
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that makes the convergence of the sum quite problematic for any
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\begin_inset Formula $d>1$
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\end_inset
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-dimensional lattice.
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\begin_inset Foot
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status open
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\begin_layout Plain Layout
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\lang english
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Note that
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\begin_inset Formula $d$
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\end_inset
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here is dimensionality of the lattice, not the space it lies in, which
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I for certain reasons assume to be three.
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(TODO few notes on integration and reciprocal lattices in some appendix)
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\end_layout
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\end_inset
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In electrostatics, one can solve this problem with Ewald summation.
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Its basic idea is that if what asymptoticaly decays poorly in the direct
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space, will perhaps decay fast in the Fourier space.
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I use the same idea here, but everything will be somehow harder than in
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electrostatics.
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\end_layout
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\begin_layout Standard
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\lang english
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||||
Let us re-express the sum in
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\begin_inset CommandInset ref
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LatexCommand eqref
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||||
reference "eq:W definition"
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||||
\end_inset
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||||
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in terms of integral with a delta comb
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\end_layout
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|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
\begin_inset Formula
|
||||
\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}
|
||||
|
||||
\end_inset
|
||||
|
||||
The translation operator
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
is now a function defined in the whole 3d space;
|
||||
\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$
|
||||
\end_inset
|
||||
|
||||
are the displacements of scatterers
|
||||
\begin_inset Formula $\alpha$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $\beta$
|
||||
\end_inset
|
||||
|
||||
in a unit cell.
|
||||
The arrow notation
|
||||
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$
|
||||
\end_inset
|
||||
|
||||
means
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
translation operator for spherical waves originating in
|
||||
\begin_inset Formula $\vect r+\vect r_{\beta}$
|
||||
\end_inset
|
||||
|
||||
evaluated in
|
||||
\begin_inset Formula $\vect r_{\alpha}$
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
and obviously
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
is in fact a function of a single 3d argument,
|
||||
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Expression
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:W integral"
|
||||
|
||||
\end_inset
|
||||
|
||||
can be rewritten as
|
||||
\begin_inset Formula
|
||||
\[
|
||||
W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
where changed the sign of
|
||||
\begin_inset Formula $\vect r/\vect{\bullet}$
|
||||
\end_inset
|
||||
|
||||
has been swapped under integration, utilising evenness of
|
||||
\begin_inset Formula $\dc{\basis u}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Fourier transform of product is convolution of Fourier transforms, so (using
|
||||
formula
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:Dirac comb uaFt"
|
||||
|
||||
\end_inset
|
||||
|
||||
for the Fourier transform of Dirac comb)
|
||||
\begin_inset Formula
|
||||
\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}\\
|
||||
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\nonumber
|
||||
\end{eqnarray}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
|
||||
\lang english
|
||||
Factor
|
||||
\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
|
||||
\end_inset
|
||||
|
||||
cancels out with the
|
||||
\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$
|
||||
\end_inset
|
||||
|
||||
factor appearing in the convolution/product formula in the unitary angular
|
||||
momentum convention.
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
As such, this is not extremely helpful because the the
|
||||
\emph on
|
||||
whole
|
||||
\emph default
|
||||
translation operator
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
has singularities in origin, hence its Fourier transform
|
||||
\begin_inset Formula $\uaft S$
|
||||
\end_inset
|
||||
|
||||
will decay poorly.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
||||
\lang english
|
||||
However, Fourier transform is linear, so we can in principle separate
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
in two parts,
|
||||
\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
|
||||
\begin_inset Formula $S^{\textup{S}}$
|
||||
\end_inset
|
||||
|
||||
is a short-range part that decays sufficiently fast with distance so that
|
||||
its direct-space lattice sum converges well;
|
||||
\begin_inset Formula $S^{\textup{S}}$
|
||||
\end_inset
|
||||
|
||||
must as well contain all the singularities of
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
in the origin.
|
||||
The other part,
|
||||
\begin_inset Formula $S^{\textup{L}}$
|
||||
\end_inset
|
||||
|
||||
, will retain all the slowly decaying terms of
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
but it also has to be smooth enough in the origin, so that its Fourier
|
||||
transform
|
||||
\begin_inset Formula $\uaft{S^{\textup{L}}}$
|
||||
\end_inset
|
||||
|
||||
decays fast enough.
|
||||
(The same idea lies behind the Ewald summation in electrostatics.) Using
|
||||
the linearity of Fourier transform and formulae
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:W definition"
|
||||
|
||||
\end_inset
|
||||
|
||||
and legendre
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:W sum in reciprocal space"
|
||||
|
||||
\end_inset
|
||||
|
||||
, the operator
|
||||
\begin_inset Formula $W_{\alpha\beta}$
|
||||
\end_inset
|
||||
|
||||
can then be re-expressed as
|
||||
\begin_inset Formula
|
||||
\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}
|
||||
|
||||
\end_inset
|
||||
|
||||
where both sums should converge nicely.
|
||||
\end_layout
|
||||
|
||||
\end_body
|
||||
\end_document
|
||||
|
|
|
|||
Loading…
Reference in New Issue