» Bronstein M. - Integration of transcendental functions
Bronstein M. - Integration of transcendental functions
||Integration of transcendental functions
||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 through out the nineteenth and twentieth centuries. The Russian mathematician Mordukhai-Boltovskoi wrote the first two books on this subject in 1910 and 19131.
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 Bronstein2.
1 On the Integration in Finite Terms of Linear Differential Equations. Warsaw, 1910 (in Russian) and On the Integration of Transcendental Functions. Warsaw, 1913 (in Russian).
2 Let me hasten to add that there have been important contributions by many others and it is not my intention to give a complete history of the field in this short paragraph, but to indicate some of main streams of work that have led to the current book.