An atlas of functions.

*(English)*Zbl 0618.65007
Washington - New York - London: Hemisphere Publishing Corporation, a subsidiary of Harper & Row, Publishers, Inc.; distr. outside North America by Springer-Verlag, Berlin etc. IX, 700 p.; DM 368.00 (1987).

This book is a very nice book with elementary and basic information on functions, that is, the special functions from mathematical physics, engineering, and statistics. From the Preface of the book we quote the following information on the objectives of the authors:

”The majority of engineers and physical scientists must consult reference books containing information on a variety of mathematical functions. This reflects the fact that all but most trivial quantitative work involves relationships that are best described by functions of various complexities. Of course, the need will depend on the user, but all will require information about the general behavior of the function and its values at a number of arguments.

Historically, this latter need has been met primarily by tables of function values. However, the ubiquity of computers and programmable calculators presents an opportunity to provide reliable, fast and accurate function values without the need to interpolate. Computer technology also enables graphical presentations of informations to made with digital accuracy. An Atlas of Functions exploits these opportunities by presenting algorithms for the calculation of most functions to more than seven-digit precision and computer-generated maps that may be read to two or three figures. Of course, the need continues for ready access to many formulas and properties that characterize a special function. This need is met in the Atlas through the display of the most important definitions, relationships, expansions and other properties of the 400 functions covered in this book.

The Atlas is organized into 64 chapters, each of which is devoted to one function or to a number of closely related functions; these appear roughly in order of increasing complexity. A standardized format has been adopted for each chapter to minimize the time required to locate a sought item of information. A description of how the chapters are sectioned is included in Chapter 0. Two appendices, a references/bibliography section and two indexes complete the volume.”

The result of these objectives is a block with 64 chapters with a lot of information. The format is very fancy and I enjoyed scanning the book and reading the nice printing. The Atlas cannot be compared with a text book on special functions. It gives information on expansions, integrals, etc., but no lessons in how to obtain this information. Since the topic is very classic, these lessons can be taken from other books in the field of special functions. The book is intended for students, mathematicians, and engineers, especially for those who are not very familiar with this topic and for researchers from non-mathematical professions, who want a first impression on the underlying functions: formula’s, relation with other functions, and perhaps a numerical scheme. Some parts of the text are very elementary. New research areas, such as the theory of q- functions, are not included.

As in most books on special functions the formulas have to be checked on typing errors, etc. Especially in this book, intended for non- specialists, it is very important to have a text free of errors. My first impression is that the authors have been very careful. I found a few small errors, for example in formulas 35:10:14, 35:12:5 and 45:6:3. A few other points are: The second line of 49:3:5 cannot be true since the left-hand side is not even in x. The claim that 45:6:7 is an asymptotic expansion for the incomplete gamma function is not true; 45:6:2 does have this property (besides, the same mistake occurs in M. Abramowitz and I. Stegun (ed.) [Handbook of mathematical functions, with formulas, graphs, and mathematical tables (1964; Zbl 0171.385), formula 6.5.33]). On page 204 there is an interesting discussion on telescoping, including an algorithm. A few words about Clenshaw’s algorithm to evaluate a Chebyshev series would have been a welcome addition. In the chapter on the sine and cosine functions a discussion on the fast Fourier transform is given, again with algorithm.

”The majority of engineers and physical scientists must consult reference books containing information on a variety of mathematical functions. This reflects the fact that all but most trivial quantitative work involves relationships that are best described by functions of various complexities. Of course, the need will depend on the user, but all will require information about the general behavior of the function and its values at a number of arguments.

Historically, this latter need has been met primarily by tables of function values. However, the ubiquity of computers and programmable calculators presents an opportunity to provide reliable, fast and accurate function values without the need to interpolate. Computer technology also enables graphical presentations of informations to made with digital accuracy. An Atlas of Functions exploits these opportunities by presenting algorithms for the calculation of most functions to more than seven-digit precision and computer-generated maps that may be read to two or three figures. Of course, the need continues for ready access to many formulas and properties that characterize a special function. This need is met in the Atlas through the display of the most important definitions, relationships, expansions and other properties of the 400 functions covered in this book.

The Atlas is organized into 64 chapters, each of which is devoted to one function or to a number of closely related functions; these appear roughly in order of increasing complexity. A standardized format has been adopted for each chapter to minimize the time required to locate a sought item of information. A description of how the chapters are sectioned is included in Chapter 0. Two appendices, a references/bibliography section and two indexes complete the volume.”

The result of these objectives is a block with 64 chapters with a lot of information. The format is very fancy and I enjoyed scanning the book and reading the nice printing. The Atlas cannot be compared with a text book on special functions. It gives information on expansions, integrals, etc., but no lessons in how to obtain this information. Since the topic is very classic, these lessons can be taken from other books in the field of special functions. The book is intended for students, mathematicians, and engineers, especially for those who are not very familiar with this topic and for researchers from non-mathematical professions, who want a first impression on the underlying functions: formula’s, relation with other functions, and perhaps a numerical scheme. Some parts of the text are very elementary. New research areas, such as the theory of q- functions, are not included.

As in most books on special functions the formulas have to be checked on typing errors, etc. Especially in this book, intended for non- specialists, it is very important to have a text free of errors. My first impression is that the authors have been very careful. I found a few small errors, for example in formulas 35:10:14, 35:12:5 and 45:6:3. A few other points are: The second line of 49:3:5 cannot be true since the left-hand side is not even in x. The claim that 45:6:7 is an asymptotic expansion for the incomplete gamma function is not true; 45:6:2 does have this property (besides, the same mistake occurs in M. Abramowitz and I. Stegun (ed.) [Handbook of mathematical functions, with formulas, graphs, and mathematical tables (1964; Zbl 0171.385), formula 6.5.33]). On page 204 there is an interesting discussion on telescoping, including an algorithm. A few words about Clenshaw’s algorithm to evaluate a Chebyshev series would have been a welcome addition. In the chapter on the sine and cosine functions a discussion on the fast Fourier transform is given, again with algorithm.

Reviewer: N.M.Temme

##### MSC:

65D20 | Computation of special functions and constants, construction of tables |

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |

33-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions |

00A22 | Formularies |

00A20 | Dictionaries and other general reference works |