RFC: DLMF Content Dictionaries Bruce R. Miller National Institute of Standards and Technology Gaithersburg, MD, USA bruce.miller@nist.gov 1 Overview The Digital Library of Mathematical Functions (DLMF1 ) covers the definitions and properties of a wide variety of special functions with applications in physics and engineering — several hundred, depending on how you count them. Although initially focused as a resource for human readers, the DLMF’s long-term goal is to support machine-readable content to enable interoperability between other digital libraries, computer algebra and theorem proving systems. But, the long history of special functions across and among different communities of interest has led to the potential for confusion and error. When different practitioners speak about apparently the same function, their different histories and conventions may include different assumptions of scalings, arguments (order or definition), branch cuts, assumed values in special cases, and so on. A simple example is the Jacobian elliptic function sn seen as a function either of the modulus k or the parameter m = k 2 ; each form is preferred for certain purposes. While neither party is necessarily wrong, failure to account for their differences in communications is guaranteed to lead to error. It is thus critical for interoperability between systems to establish each system’s conventions and assumptions. This is true enough for a human reader transcribing DLMF’s results into a computer algebra system, but all the more so when these processes are automated and hidden from view. Ideally these differences can be formalized to the extent that enables automatic conversion of formula across the different views. Indeed, the World Digital Mathematics Library2 has founded an effort to develop such a Special Function Concordance. Towards these ends, this note presents a proposed set of (virtual) OpenMath3 Content Dictionaries (CD) to characterize the choices made in the DLMF. 2 Organization and Naming Conventions An unexpected challenge was an organization and naming of the CDs and symbols in a fashion appropriate to OpenMath applications. The functions can be grouped according to mathematical or historical features, or applications. The proper names of functions can become quite verbose with strings of significant adjectives before they become sufficiently unique. Consider “Legendre’s incomplete elliptic integral of the first kind” and then add modifiers such as “zeros of the derivatives of”. Given that some functions are ubiquitous while others are truly esoteric, one would even hope for a Huffman-type encoding. Yet, the functions have already been grouped into chapters in a way appropriate for the DLMF’s purposes. Moreover, each function has a unique LATEX macro defined for it to simplify the markup and preserve the semantics during conversion to web formats 4 . For example the two functions mentioned above have, macros \Jacobiellsnk (encoding “the Jacobian elliptic function sn, of modulus k”) and \incellintFk (encoding “(Leg- endre’s) incomplete elliptic integral (of the first kind) of modulus k”; See Appendix A for details). While these Copyright c by the paper’s authors. Copying permitted for private and academic purposes. In: O. Hasan, J. Davenport, M. Kohlhase (eds.): Proceedings of the 29th OpenMath Workshop, Hagenberg, Austria, 13-Aug-2018, published at http://ceur-ws.org 1 https://dlmf.nist.gov/ 2 https://www.mathunion.org/ceic/library/world-digital-mathematics-library-wdml 3 https://openmath.org/ 4 a DLMF Macro set, to be released, is under development Table 1: Types used in the special function type signature, where T stands for any arbitrary set or type. Notation Meaning T →T0 function (mapping) from type T to T 0 T ×T0 product set of multiple types R the set of real numbers (excluding ∞) C the set of complex numbers (excluding ∞) Z the set of integers Z+ the set of integers > 0 Z∗ the set of integers ≥ 0 Q set of rational numbers D the complex numbers in the open unit disc, |z| < 1 {a0 , . . .} one of a finite set (of symbols) Tn n-tuples with elements of type T (e.g. R2 for pairs of reals) T• tuples with elements of type T , unspecified length Tn vectors of dimension n, with elements of type T T• vectors with elements of type T , unspecified dimension T n×m n × m matrices with elements of type T T •×• matrices with elements of type T (unspecified dimension) L lattices in the complex plane (in the sense of elliptic functions) may not roll off the tongue, they are unique, reasonably type-able and blend with TEX’s macro conventions. And while this organization and naming may not be optimized for OpenMath purposes, it seems better to reuse the one scheme than to introduce redundant ones. We have therefore followed DLMF’s organization for the primary grouping of functions. The most important functions in a chapter are covered in a base CD, such as DLMF BS (for Bessel functions). In most cases, progres- sively esoteric functions are grouped into subcategories according to: generalizations (DLMF BS gen), q-analogs (DLMF BS q), magnitudes, zeros, matrix argument and so on, as well as some special case such as DLMF GH Appell for Appell functions. 3 Characterizing the Functions The more fundamental challenge is to properly characterize the functions. This, of course, is exactly what any proper ‘definition’ ought to be. But here the point is that the definition be sufficiently complete, explicit and formalized, to enable easily determining the equivalancy of functions from different systems. Ultimately, the goal would be to enable automatic conversion between, for example, the two different ‘flavors’ of elliptic functions, sn. At this stage of development, we are providing URLs as the definitions of each function, being pointers into the DLMF where the definition is to be found. This is obviously an informal definition, and may require digging for some details. Definitions may be either explicit or implicit (such as a function defined by a differential equation along with boundary conditions). Additionally, we have provided a simple type signature for each function to characterize its domain and range (See Table 1). Note that in many cases functions are undefined for isolated values of some arguments, e.g. singularities; these cases are not always reflected in the current signatures. Other properties, such as branch cuts, multivaluedness, have not yet been made explicit. 4 Conclusions and Request for Comments We have provided here a catalog of the special functions covered by the DLMF — a set of informal, virtual Content Dictionaries. It should serve as a reasonable starting point for establishing a concordance between the sets of functions covered by the several interested parties. This is a continuing process; our CDs will continue to be refined and gradually extended and formalized as needed. The current status can be found at https://math.nist.gov/~BMiller/DLMF-CDS/, where also a JSON en- coding of the data may be downloaded for processing. We welcome suggestions about which features and characteristics function are important to the notion of a concordance, as well as how best to encode and formalize that information. Any other comments about the catalog are also welcome. Acknowledgements: The author would like to thank Patrick Ion, Howard Cohl and Florian Rabe for constructive comments. A DLMF Macro Naming conventions Briefly, the names of the various mathematical function macros are derived from the descriptive ‘Proper Name’ of the function according to: macro ≡ \ prefix ∗ name class ? symbol ? suffix ∗ The name is the ‘conventional’ name or based on the “inventor’s” name. The class indicates function (generally omitted), integrals, polynomials, and so on. The symbol is the latinized form of the notation, upper or lower case as appropriate. The prefix modifier includes all significant characteristics that may distinguish functions (e.g. ‘modified Bessel’ vs. simply ‘Bessel’). The suffix generally indicates limitations or special cases regarding arguments. The abbreviations used for prefix , class and suffix are given in Table 2. For predictability, we avoid abbreviating people’s names. Table 2: Abbreviations used for DLMF macros class Meaning prefix Meaning function (omitted by default) a arc, inverse (circular functions) char characteristic A arc, multi-valued-inverse eigval eigenvalues aff affine eigvec eigenvectors ass associated int integral aux auxiliary mod modulus big big number number canon canonical phase phase (or phase shift) comp complete poly polynomial ccomp complete complementary sum sum cont continuous sym symbol cusp cuspoid trans transform deriv derivative(s) of wave wavefunction diff differential diffr diffraction dil dilated disc discrete div dividing dual dual ell elliptic env envelope of exp exponential gen general | generalized suffix Meaning hyper hyperbolic | hypergeometric imag imaginary argument or order inc incomplete k elliptic functions of k, modulus inv inverse m elliptic functions of parameter m = k2 irreg irregular mat matrix argument little little real of real argument or order log logarithm(ic) invar on invariants (Weierstrass) mod modified (or modular?) latt on lattice (Weierstrass) multivar multivariate q functions of q, nome n number of tau functions of τ norm normalized or normalization para parabolic per periodic q q-variant of rad radial reg regular rest restricted sc scaled shift shifted sph spherical | spheroidal sym symmetric umb umbilic usph ultraspherical z zeros (of)