We show that an exact expression for the GreenÏs function in cylindrical coordinates is 1 o x [ x@ o \ 1 nJRR@ ; m/~= = eim(Õ~Õ {)Q m~1@2 (s), where and is the half-integer degree Legendre function of the s 4 [R2]R@ 2 ](z[z@)2]/(2RR@), Q m~1@2 second kind. This expression is signiÐcantly more compact and easier to evaluate numerically than the more familiar cylindrical GreenÏs function expression, which involves inÐnite integrals over products of Bessel functions and exponentials. It also contains far fewer terms in its series expansionÈand is therefore more amenable to accurate evaluationÈthan does the familiar expression for o x [ x@ o~1 that is given in terms of spherical harmonics. This compact GreenÏs function expression is well suited for the solution of potential problems in a wide variety of astrophysical contexts because it adapts readily to extremely Ñattened (or extremely elongated), isolated mass distributions.

Mathematical formulae are essential in science, but face challenges of ambiguity, due to the use of a small number of identifiers to represent an immense number of concepts. Corresponding to word sense disambiguation in Natural Language Processing, we disambiguate mathematical identifiers. By regarding formulae and natural text as one monolithic information source, we are able to extract the semantics of identifiers in a process we term Mathematical Language Processing (MLP). As scientific communities tend to establish standard (identifier) notations, we use the document domain to infer the actual meaning of an identifier. Therefore, we adapt the software development concept of namespaces to mathematical notation. Thus, we learn namespace definitions by clustering the MLP results and mapping those clusters to subject classification schemata. In addition, this gives fundamental insights into the usage of mathematical notations in science, technology, engineering and mathematics. Our gold standard based evaluation shows that MLP extracts relevant identifierdefinitions. Moreover, we discover that identifier namespaces improve the performance of automated identifier-definition extraction, and elevate it to a level that cannot be achieved within the document context alone.

Cohl & Tohline (1999) have shown how the integration/summation expression for the Green's function in cylindrical coordinates can be written as an azimuthal Fourier series expansion, with toroidal functions as expansion coefficients. In this paper, we show how this compact representation can be extended to other rotationally invariant coordinate systems which are known to admit separable solutions for Laplace's equation.

Mathematical formulae represent complex semantic information in a concise form. Especially in Science, Technology, Engineering, and Mathematics, mathematical formulae are crucial to communicate information, e.g., in scientific papers, and to perform computations using computer algebra systems. Enabling computers to access the information encoded in mathematical formulae requires machine-readable formats that can represent both the presentation and content, i.e., the semantics, of formulae. Exchanging such information between systems additionally requires conversion methods for mathematical representation formats. We analyze how the semantic enrichment of formulae improves the format conversion process and show that considering the textual context of formulae reduces the error rate of such conversions. Our main contributions are: (1) providing an openly available benchmark dataset for the mathematical format conversion task consisting of a newly created test collection, an extensive, manually curated gold standard and task-specific evaluation metrics; (2) performing a quantitative evaluation of state-of-the-art tools for mathematical format conversions; (3) presenting a new approach that considers the textual context of formulae to reduce the error rate for mathematical format conversions. Our benchmark dataset facilitates future research on mathematical format conversions as well as research on many problems in mathematical information retrieval. Because we annotated and linked all components of formulae, e.g., identifiers, operators and other entities, to Wikidata entries, the gold standard can, for instance, be used to train methods for formula concept discovery and recognition. Such methods can then be applied to improve mathematical information retrieval systems, e.g., for semantic formula search, recommendation of mathematical content, or detection of mathematical plagiarism.

Few-body problems involving Coulomb or gravitational interactions between pairs of particles, whether in classical or quantum physics, are generally handled through a standard multipole expansion of the two-body potentials. We develop an alternative based on an old, but hitherto forgotten, expression for the inverse distance between two points that builds on azimuthal symmetry. This alternative should have wide applicability throughout physics and astronomy, both for computation and for the insights it provides through its emphasis on different symmetries and structures than are familiar from the standard treatment. We compare and contrast the two methods, develop new addition theorems for Legendre functions of the second kind, and a number of useful analytical expressions for these functions. Two-electron "direct" and "exchange" integrals in many-electron quantum systems are evaluated to illustrate the procedure which is more compact than the standard one using Wigner coefficients and Slater integrals.