1 Introduction

The body of knowledge in the sciences, engineering, and mathematics is vast and increasing at an exponential rate. By reading specialized journals and attending conferences regularly, an individual researcher can keep abreast of developments within a relatively narrow field. But when the need arises for results from quite different fields, a researcher may have to invest a tremendous amount of time immersed in the literature; even then, he or she may not succeed in arriving at, and assessing the reliability of, the pertinent information. Digital library technology, coupled with painstaking development and validation of comprehensive reference data, has the potential to minimize this general problem.

Pure and applied mathematics are the most pervasive disciplines in science and engineering. Mathematical definition is the key to uniform and accurate utilization of technical knowledge. Up-to-date refinements of fundamental mathematical techniques, such as approximation of functions, solution of ordinary and partial differential equations, and statistical analysis, provide the underpinning for all modern quantitative science. The increasing reliance of scientists and engineers on mathematical modeling and simulation, the growing use of symbolic and numerical software, and the rapidly developing capabilities of the Internet and World Wide Web, present challenges and opportunities for a comprehensive standardization of mathematical knowledge which supports new levels of multidisciplinary communication.

The authors are part of a NIST team effort to collect, organize, validate, develop, and disseminate a comprehensive and evolving digital library pertaining to mathematical functions. This library is being called the Digital Library of Mathematical Functions, or DLMF. The reason for beginning with this particular branch of applied mathematics, instead of, say, numerical analysis, is NIST's direct experience with the 1964 Handbook of Mathematical Functions [1], known also as AMS 55 (for Applied Mathematics Series No. 55). This work has had unique influence among individuals who apply mathematics to the solution of real-world problems, e.g. engineers, physical scientists, and statisticians. Such users have come to regard AMS 55 as the definitive source of reference information on the ``special functions'' of mathematics.

Figure 1: Selected yearly citations to AMS 55 (black bars) compared with total citations scaled by 0.00147 (white bars). Data from Science Citation Index.

AMS 55 is one of the most frequently cited works in the scientific literature. Even though it is 40 years out of date (never having been revised), the number of citations to it continues to rise annually, not only in absolute numbers but also as a fraction of the total number of citations made in the sciences and engineering each year; see Figure 1. Currently, about once every $1\tfrac{1}{2}$ hours of each working day some author, somewhere, makes sufficient use of this handbook to list it as a reference. Moreover, the journals in which these references appear range widely over the sciences and engineering; see Table 1.

The target date for completion of the public version of the DLMF, which will be freely accessible from a Web site at NIST, is late in 2002. A distinctive characteristic, in comparison with other initiatives in digital library research and development, is the emphasis on original development of detailed and authoritative content; see § 2 of this paper. An equally important thrust is the dissemination of mathematical reference data as a digital library on the Web with provisions for state-of-the-art indexing, searching, navigation, cross-referencing, linking, downloading, and so on. A prototype Web site with a newly written chapter on Airy functions [7] is described, and some of the issues involved in its construction are discussed in § 3. Advanced interactive graphics in two and three dimensions are a valuable aid to qualitative understanding of the properties of mathematical functions; § 4 discusses issues associated with this topic. § 5 discusses the issue of numerical and symbolic computation, including the location and downloading of software; many users will want easy-to-use support in these matters. Application and learning modules are the subject of § 6. These are auxiliary units tailored to the needs of fields outside mathematics itself, with links to the DLMF Web site. This paper concludes with a few final remarks in § 7.

Table 1: Journals with highest number of citations to AMS 55 in the 10-year period 1988-1997. Data from Science Citation Index.

Cit.	Journal
498	Phys. Rev. B: Condensed Matter Physics
462	Phys. Rev. A: Atomic, Molecular, Optical Phys.
381	Journal of Chemical Physics
262	J. Phys. A: Mathematical and General Physics
240	Phys. Rev. E: Statist. Phys., Plasmas, & Fluids
231	Journal of the Acoustical Society of America
205	Journal of Fluid Mechanics
183	Astrophysical Journal
182	Phys. Rev. D: Elementary Particles
153	J. Phys. B: Atomic, Molecular, & Optical Phys.