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Handbook of Continued Fractions for Special Functions

A. Cuyt, V. Brevik Petersen, B. Verdonk, H. Waadeland, and W. B. Jones
Publication Date: 
Number of Pages: 
[Reviewed by
Mehdi Hassani
, on

Special functions are the key to explicit solutions of many problems in computational mathematics with no elementary answers. They usually appear in integrating and solving differential equations, and branches of sciences like mathematics, physics, engineering, chemistry, all of which use differential equations, make use of special functions. Many books have been written about these functions, investigating them from various views of point, and there are a number of handbooks and web pages that list their main properties and discuss how to compute them.

The representation of numbers and functions as continued fractions is a classic and fruitful tool in the studying both numbers and functions. Among various formulas for special functions in the main sources, one can find some continued fraction representations for them. A few more are diffused in various papers. The aim of the book under review is to collect continued fractions for special functions in a friendly volume.

The book has three main parts: The first part investigates fundamental concepts and tools in the theory of continued fractions, described by way of various theorems, mostly given without proof. The second part introduces a number of algorithms to construct continued fraction representations for functions and discusses error-management and analysis for these representations. The third part is the most interesting, important, and ultimate part of the book. It includes various chapters giving continued fraction representations of mathematical constants, elementary functions and important special functions such as the Gamma function, the Error function, the Exponential integral, Hypergeometric functions, Bessel functions, and Probability functions. As the reader turn pages, lots of very nice formulas for constants and special functions appear.

As the authors mention, only a few of these formulas are available in other sources, either books or web pages. The chapters are full of numerical tables and graphs that display numerically the continued fractions for special functions and their errors.

There is no doubt that this book is very useful for the people who need to work with special functions. It is not a text book, but, because it includes some of the theory, it is more than a handbook including only list of formulas. An expert could use it in a special course or in parallel to a course on special functions, though of course that will require giving detailed proofs of basic theorems and adding some exercises. The book is suitable for researchers in many fields, so that libraries serving scholars in basic sciences need to have it.

Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.

General considerations.- Part 1, Basic Theory.- 1. Basics.- 2. Continued fraction representation of functions.- 3. Convergence criteria.- 4. Pade approximants.- 5. Moment theory and orthogonal functions.- Part 2, Numerics.- 6. Continued fraction construction.- 7. Truncation error bounds.- 8. Continued fraction evaluation.- Part 3, Special Functions.- 9. On tables and graphs.- 10. Mathematical constants.- 11. Elementary functions.- 12. Gamma function and related functions.- 13. Error function and related integrals.- 14. Exponential integrals and related functions.- 15. Hypergeometric functions.- 16. Confluent hypergeometric functions,.- 17. Bessel functions.- 18. Probability functions.- 19. Basic hypergeometric functions.