Technical Aspects of theDigital Library of Mathematical Functions†Bruce R. MillerNational Institute of Standards and Technology; Gaithersburg, MD 20899, USA;[email protected] YoussefDept. of Computer Science; The George Washington University; Washington, DC20052; [email protected]. The NIST Digital Library of Mathematical Functions (DLMF) Project,begun in 1997, is preparing a handbook and Web site intended for wide communitiesof users. The contents are primarily mathematical formulas, graphs, methods of com-putation, references, and links to software. The task of developing a Web handbookof this nature presents several technical challenges. We describe the goals of theDigital Library of Mathematical Functions Project and the realities that constrainthose goals. We propose practical initial solutions, in order to ease the authoringof adaptable content: a LATEX class which encourages a modestly semantic markupstyle; and a mathematical search engine that adapts a text search engine to the task.Keywords: Digital Library, Mathematical markup, Mathematical notation, Math-ematical SearchAMS(MOS) Codes: 33-00, 68T301. IntroductionThe NIST Digital Library of Mathematical Functions (DLMF) Project1,begun in 1997, is preparing a handbook and Web site intended forwide communities of users. The contents are primarily mathematicalformulas, graphs, methods of computation, references, and links tosoftware. This project revises and extends Abramowitz and Stegun’sHandbook of Mathematical Functions [1]. An overview of the projectcan be found in [11]. This paper addresses some of the technical aspectsof the project. The route that we will take is to first describe the goalsof the DLMF. That some of these goals conflict with reality leads todiscussion of how we have adapted to those conflicts.There are two underlying themes to this paper. The first theme isfinding the most practical way of obtaining a variety of information†Official contribution of the National Institute of Standards and Technology; notsubject to copyright in the United States. Research was supported in part by NSFGrant 9980036 and in part by the SIMA, SRD and ATP Programs at NIST.1http://dlmf.nist.gov/c 2002 Kluwer Academic Publishers. Printed in the Netherlands.MKM-Miller-Youssef.pub.tex; 17/12/2002; 15:41; p.12 Bruce Miller and Abdou Yousseffrom the project’s authors, much of which may seem unusual to them.The second theme is how to make the information available to usersthrough different media and via appropriate search capabilities. Wemust extract from the author’s manuscripts a close approximation tothe semantic content, especially the mathematical content, in a formthat will allow the broadest, most long-term usage. It must be notedthat this project is a work-in-progress; many of the most interestingproblems have not yet been completely solved; and even the notion of‘best solution’ will evolve with the evolution of the web, itself.2. Goals and Challenges2.1. GoalsThat the material must be of the highest quality — authoritative andvalidated — should go without saying. We aim to produce a book, aweb site with extensive search capabilities and a CD-ROM version. Thematerial must therefore adapt to various media.Clearly, the electronic formats must provide search capabilities. Andsince the DLMF is primarily mathematics with little text, providinga traditional search for the textual components will not be enough.We must provide the capability to search for formulas that match auser’s criteria as well; searching for formulas according to keywordsor properties of the formula: e.g. ‘addition theorem for elliptic func-tions’. More intriguing would be searches using mathematical patterns.These should be flexible regarding syntax, not restricted to TEX[8],say. And the matching process must understand the basic properties ofcommutativity, associativity and so forth.Along with search capabilities, it would be extremely useful to beable to extract virtual documents. For example, one might wish to createa page of addition theorems, or a short booklet of differential equationsof certain classes. Thus one must not only find pages, but collections ofdocument fragments and formulas, and synthesize a document fromthem. These, along with existing sections or subsections should beviewable in a screen friendly format, or printable.The DLMF should provide layers of detail. Beyond the dry, tele-graphic, front facade of the identities, there should be associated withmost elements additional metadata. This data would point to refer-ences, original sources, or short tracts going into extra detail or moreesoterica.Often one finds a formula not quite in the terms one needs andone would like to transform the expressions. It would be exciting,MKM-Miller-Youssef.pub.tex; 17/12/2002; 15:41; p.2Technical Aspects of DLMF 3from within the online book, to substitute variables, or rearrange anexpression, to acquire exactly the identity required. One may need tore-expand, apply transformations, other identities, . . . . Transformed ornot, users will likely want to copy the formula into their own docu-ments, graphics programs or computer algebra system, without loss ofmeaning.For some people, a differential equation tells them all they need toknow about a special function. Other, more visually oriented people,will need interactive graphics to explore the function in different re-gions, on different scales. Graphics in 2D and 3D (or more) will beneeded. Above all, these graphics must be ‘Honest’ in the sense ofFateman [5]: they must not display artificial flat planes where clipped,nor smooth over narrow features, nor succumb to other sampling errors.2.2. Some SuccessesSome of these goals are well on their way to success. Dan Lozier de-scribes in [11] the project and the strategy for managing it. Given theproject’s Editorial Board and the authors they have contracted, thequality of the mathematical content is well in hand.Saunders and Wang [22, 18] have been making significant progresson 3D graphics, with some ‘honest’ features (see also [11]). We havebeen exploring 2D interactive honest graphics, as well.2.3. ChallengesOther elements remain difficult, however.What might be called ‘live mathematics’, direct manipulation of theformula within the browser, has been explored in teaching materials,where it is quite appropriate. But in our case, the capability to trans-form a given identity into an arbitrary form