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This book brings together two streams of research in mathematics and computing that were begun in the nineteenth century and made possible through results brought to fruition in the twentieth century.
Methods for indefinite integration have been important ever since the invention of the calculus in the 1700s. In the 1800s Abel and Liouville began the earliest mathematical research on algorithmic methods on integration in finite terms leading to what might be considered today as an early mathematical vision of a complete algorithmic solution for integrating elementary functions. In an 1842 publication Lady Ada Augusta, Countess of Lovelace, describing the capabilities of Babbage's analytical engine put forth the vision that computational devices could do algebraic as well as numerical calculations when she said that "[Babbage's Analytical Engine] can arrange and combine its numerical quantities exactly as if they were letters or any other general symbols and in fact it might bring out its results in algebraical notation were provisions made accordingly." Thus these two visions set the stage for a century and a half of research that partially culminates in this book.
Progress in the mathematical realm continued throughout the nineteenth and twentieth centuries. The Russian mathematician Mordukhai-Boltovskoi wrote the first two books on this subject in 1910 and 1913. With the invention of electronic computers in the late 1930s and early 1940s, a new impetus was given to both the mathematical and computational streams of work. In the meantime in the mathematical world important progress had been made on algebraic methods of research. Ritt began to apply the new algebraic techniques to the problem of integration in finite terms, an approach that has proven crucially important. In 1948 he published the results of his research in a little book. Integration in Finite Terms, The use of these algebraic ideas were brought to further fruition by Kolchin, Rosenlicht, and, particularly for problems of symbolic integration, by three of Rosenlicht's Ph.D. students — Risch, Singer, and Bronstein.
On the computational side, matters rested until 1953 when two early programs were written, one by Kahrimanian at Temple University and another by Nolan at Massachusetts Institute of Technology, to do analytic differentiation — the inverse of indefinite integration. There was active research in the late 1950s and early 1960s on list processing packages and languages that laid the implementation foundations for today's computer algebra systems. Slagle's 1961 thesis was an early effort to write a program, in LiSP, to do symbolic integration. With the advent of general computer algebra systems, some kind of symbolic integration facility was implemented in most. These integration capabilities opened the eyes of many early users of symbolic mathematical computation to the amazing potential of this form of computation. But yet none of the systems had a complete implementation of the full algorithm that Risch had announced in barest outline in 1970. There were a number of reasons for this. First and foremost, no one had worked out the many aspects of the problem that Risch's announcement left incomplete.
Starting with his Ph.D. dissertation and continuing in a series of beautiful and important papers, Bronstein set out to fill in the missing components of Risch's 1970 announcement. Meanwhile working at the IBM T. J. Watson Research Center, he carried out an almost complete implementation of the integration algorithms for elementary functions. It is the most complete implementation of symbolic integration algorithms to date.
In this book, Bronstein brings these mathematical and computational streams of research together in a highly effective manner. He presents the algorithmic details in pseudo-code that is easy to implement in most of the general computer algebra systems. Indeed, my students and I have implemented and tested many of the algorithms in MAPLE and MACSYMA. Bronstein's style and appropriate level of detail makes this a straightforward task, and I expect this book to be the standard starting place for future implementers of symbolic integration algorithms. Along with the algorithms, he presents the mathematics necessary to show that the algorithms work correctly. This is a very interesting story in its own right and Bronstein tells it well. Nonetheless, for those primarily interested in the algorithms, much of the mathematics can be skipped at least in a first study. But the full beauty of the subject is to be most appreciated by studying both aspects.
Algebraic Preliminaries
Integration of Rational Functions
Differential Fields
The Order Function
Integration of Transcendental Functions
The Risch Differential Equation
Parametric Problems
The Coupled Differential System
Structure Theorems
Parallel Integration