1. Mathematical Constants, D. S. Liepman.

2. Physical Constants and Conversion Factors, A. G.

McNish.

3. Elementary Analytical Methods, M. Abramowitz. 4. Elementary Transcendental Functions, R. Zucker. 5. Exponential Integral and Related Functions, W. Gautschi (American University) and William F. Cahill.

6. Gamma Function and Related Functions, P. J.

Davis.

7. Error Function and Fresnel Integrals, W. Gautschi (American University).

8. Legendre Functions, I. A. Stegun.

9. Bessel Functions of Integer Order, F. W. J. Olver. 10. Bessel Functions of Fractional Order, H. A. Antosiewicz.

11. Integrals of Bessel Functions, Y. L. Luke.

12. Struve Functions and Related Functions, M. Abramowitz.

13. Confluent Hypergeometric Functions, L. J. Slater (Cambridge University).

14. Coulomb Wave Functions, M. Abramowitz. 15. Hypergeometric Functions, F. Oberhettinger. 16. Jacobian Elliptic Functions and Theta Functions, L. M. Milne-Thomson (University of Arizona). 17. Elliptic Integrals, L. M. Milne-Thomson (University of Arizona).

18. Weierstrass Elliptic and Related Functions, T. H. Southard.

19. Parabolic Cylinder Functions, J. C. P. Miller (Cambridge University).

20. Mathieu Functions, G. Blanch (Wright-Patterson Air Force Base).

21. Spheroidal Wave Functions, A. N. Lowan (Yeshiva University).

22. Orthogonal Polynomials, U. W. Hochstrasser

(American University).

23. Bernoulli and Euler Polynomials-Riemann Zeta Function, E. V. Haynsworth and K. Goldberg. 24. Combinatorial Analysis, K. Goldberg, M. Newman, and E. Haynsworth.

25. Numerical Interpolation, Differentiation, and Integration, P. J. Davis and I. Polonsky.

26. Probability Functions, M. Zelen and N. C. Severo 27. Miscellaneous Functions, I. A. Stegun. 28. Scales of Notation, S. Peavy and A. Schopf (American University).

29. Laplace Transforms.

The public reaction to the publication of the Handbook was overwhelmingly positive. In a preface to the ninth printing in November 1970, NBS Director Lewis Branscomb wrote

Fig. 3. Photograph of Handbook.

"The enthusiastic reception accorded the 'Handbook of Mathematical Functions' is little short of unprecedented in the long history of mathematical tables that began when John Napier published his tables of logarithms in 1614. Only four and one-half years after the first copy came from the press in 1964, Myron Tribus, the Assistant Secretary for Commerce for Science and Technology, presented the 100,000th copy of the Handbook to Lee A. DuBridge, then Science Advisor to the President."

The Handbook has had enormous impact on science and engineering. Likely the most widely distributed NBS/NIST technical publication of all time, the government edition has never gone out of print, and it has appeared as a Dover reprint since 1965. It has been reprinted (in all or part) by other publishers, such as Moscow Nauka, Verlag Harri Deutsch, and Wiley Interscience. Government sales exceed 150,000 copies, with commercial sales estimated at three to six times this number. The Handbook's citation record is also remarkable. More than 23,000 citations have been logged by Science Citation Index (SCI) since 1973. Remarkably, the number of citations to the Handbook continues to grow, not only in absolute numbers, but also as a

fraction of the total number of citations made in the sciences and engineering. During the mid-1990s, for example, about once every 1.5 hours of each working day some author, somewhere, made sufficient use of the Handbook to list it as a reference. The success of the Handbook was due to several factors. It collected in one place, and in a well-organized way, the most important information needed to make use of mathematical functions in practical applications. It served to standardize notations and normalizations for the special functions of applied mathematics, thus easing the communication of scientific results. In 1965, Irene Stegun was awarded a Gold Medal from the Department of Commerce for her efforts in completing the project.

A number of difficult mathematical problems that emerged in the course of developing the Handbook engaged researchers in the NBS Applied Mathematics Division for a number of years after its publication. Two of these are especially noteworthy, the first having to do with stability of computations and the second with precision.

Mathematical functions often satisfy recurrence relations (difference equations) that have great potential for use in computations. However, if used improperly, recurrence relations can quickly lead to ruinous errors. This phenomenon, known as instability, has tripped up many a computation that appeared, superficially, to be straightforward. The errors are the result of subtle interactions in the set of all possible solutions of the difference equation. Frank Olver, who wrote the Handbook's chapter on Bessel functions of integer order, studied this problem in great detail. In a paper published in 1967 [5], Olver provided the first (and only) stable algorithm for computing all types of solutions of a difference equation with three different kinds of behavior: strongly growing, strongly decaying, and showing moderate growth or decay. Part of the impact of this work is reflected today in the existence of robust software for higher mathematical functions. Olver worked on such topics in the Mathematical Analysis Division of NBS, and this work provided the foundation for his very influential later book on asymptotic analysis

and special functions [6]. This book has been cited more than 800 times, according to SCI.

Another important problem in mathematical computation is the catastrophic loss of significance caused by the fixed length requirement for numbers stored in computer memory. Morris Newman, who co-authored the Handbook's chapter on combinatorial analysis, sought to remedy this situation. He proposed storing numbers in a computer as integers and performing operations on them exactly. This contrasts with the standard approach in which rounding errors accumulate with each arithmetic operation. Newman's approach had its roots in classical number theory: First perform the computations modulo a selected set of small prime numbers, where the number of primes required is determined by the problem. These computations furnish a number of local solutions, done using computer numbers represented in the normal way. At the end, only one multilength computation is required to construct the global solution (the exact answer) by means of the Chinese Remainder Theorem. This technique was first described in a paper by Newman in 1967 [7]; it was employed with great success in computing and checking the tables in Chapter 24 of the Handbook. Today, this technique remains a standard method by which exact computations are performed. Newman's research on this and other topics, performed at NBS, formed the basis for his 1972 book [8], which quickly became a standard reference in the applications of number theory to computation.

Research into the functions of applied mathematics has continued actively in the 36 years since the Handbook appeared. New functions have emerged in importance, and new properties of well-known functions have been discovered. In spite of the fact that sophisticated numerical methods have been embodied in well

designed commercial software for many functions, there continues to be a need for a compendium of information on the properties of mathematical functions. To address this need, NIST is currently developing a successor to the Handbook to be known as the Digital Library of Mathematical Functions (DLMF) [9]. Based upon a completely new survey of the literature, the DLMF will provide reference data in the style of the Handbook in a freely available online format, with sophisticated mathematical search facilities and interactive threedimensional graphics.

Prepared by Ronald F. Boisvert and Daniel W. Lozier.

Bibliography

[1] Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, NBS Applied Mathematics Series 55, National Bureau of Standards, Washington, DC (1964).

[2] Arnold N. Lowan, The Computation Laboratory of the National Bureau of Standards, Scripta Math. 15, 33-63 (1949).

[3] Tables of the Bessel Functions Yo(x), Y¡(x), Ko(x), K¡(x) 0≤x≤1, NBS Applied Mathematics Series 1, National Bureau of Standards, Washington, DC (1948).

[4] Philip J. Davis, Leonhard Euler's Integral: A Historical Profile of the Gamma Function, Am. Math. Monthly 66, 849-869 (1959). [5] F. W. J. Olver, Numerical Solution of Second-Order Linear Difference Equations, J. Res. Natl. Bur. Stand. 71B, 111-129 (1967). [6] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York (1974). Reprinted by A. K. Peters, Wellesley, MA (1997).

[7] Morris Newman, Solving Equations Exactly, J. Res. Natl. Bur. Stand. 71B, 171-179 (1967).

[8] Morris Newman, Integral Matrices, Academic Press, New York (1972).

[9] D. Lozier, F. W. J. Olver, C. Clark, and R. Boisvert (eds.), Digital Library of Mathematical Functions, (http://dlmf.nist.gov) National Institute of Standards and Technology, .

"Jack Edmonds has been one of the creators of the field of combinatorial optimization and polyhedral combinatorics. His 1965 paper 'Paths, Trees, and Flowers' [1] was one of the first papers to suggest the possibility of establishing a mathematical theory of efficient combinatorial algorithms ..." [from the award citation of the 1985 John von Neumann Theory Prize, awarded annually since 1975 by the Institute for Operations Research and the Management Sciences].

In 1962, the Chinese mathematician Mei-Ko Kwan proposed a method that could be used to minimize the lengths of routes walked by mail carriers. Much earlier, in 1736, the eminent Swiss mathematician Leonhard Euler, who had elevated calculus to unprecedented heights, had investigated whether there existed a walk across all seven bridges that connected two islands in the river Pregel with the rest of the city of Königsberg on the adjacent shores, a walk that would cross each bridge exactly once.

The commonality between the two graph-theoretical problem formulations reaches deeper. Given any graph, for one of its nodes, say io, find an arc a, that joins node io and another node, say i1. One may try to repeat this process and find a second arc a2 which joins node i, to a node iz and so on. This process would yield a sequence of nodes-some may be revisited-successive pairs of which are joined by arcs (Fig. 3). The first and the last node in that sequence are then connected by this "path."

Fig. 1. The bridges of Köningsberg.

Different as they may sound, these two inquiries have much in common. They both admit schematic representations based on the concept of graphs. Thus they are problems in graph theory, a twentieth century discipline which combines aspects of combinatorics and topology.

Given a set of items called nodes and a second set of items called arcs, a graph is defined as a relationship between such sets: For each arc, two of the nodes are specified to be joined by this arc. The actual, say physical, meaning of such nodes and arcs is not what distinguishes graphs. Only formal relationships count. In particular, the bridges of Königsberg may be consid

Fig. 3. Open and closed paths in the graph.

(Think of a route along street segments joining intersections.) If each pair of nodes in a graph can be connected by a path, then the whole graph is considered connected. A path in a graph is called closed if it returns to its starting point (Fig. 3).

C-C-C-C-C-C

C

C-C-C-C-C

c-c-&-c-c

over the bridges of Königsberg. Euler's examination typifies the classical combinatorial query: Do certain constructs exist and if so, how many?

The following question also illustrates the flavor of classical combinatorics: how many isomers are there of the hydrocarbon CnH2n+2? Each such isomer (Fig. 4) is characterized by the joining patterns of the carbon atoms as nodes and bonds as arcs. The corresponding arrangements, therefore, represent graphs. These graphs have a very special property: For any two of their nodes there exists a unique connecting path without repeat arcs. Such a graph is called a

"tree."

Every isomer defines a tree of carbon atoms whose nodes are of degree four or less. Conversely, every such tree uniquely characterizes an isomer. In graph-theoretical terms, the question is: How many trees with n nodes are there with node degrees not exceeding four? (For n = 40, the number is 62,481,801,147,341, as determined by ingenious use of "generating functions.")

с с 11 C-C-C-C

C

C-C-C-C C

Fig. 4. The carbon graphs of the isomers of C6 H14.

Mei-Ko Kwan's "Chinese Postman Problem," as it is now generally called (since first suggested by Alan Goldman, then of NBS), is to determine, in a given connected graph, a shortest closed path which traverses every arc at least once-delivers mail on every assigned street block. Euler also considers closed paths meeting all arcs, but aims at characterizing all graphs which accommodate an Euler tour, a closed path that crosses each arc exactly once. As Euler found, they are precisely those connected graphs in which each node attaches to an even number of arcs, or in other words, every node is of even degree. By this result, there can be no Euler tour

Fig. 5. A matching in a graph; two of the augmenting paths are highlighted by..

Here is one more topic in the same vein. A matchmaker has a list of k single men and I single women. Her experience of many years enables her to discern which pairs are compatible. She considers a number of compatible matches that might be arranged simultaneously. How can she be sure that she arrived at the largest possible number of such matches?

A theorem by one of the founders of graph theory, the Hungarian mathematician Dénes König in 1931, suggests an answer to that question. He considered the