Special Functions: A Unified Theory Based on Singularities

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Oxford University Press, 2000 - Mathematics - 293 pages
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The subject of this book is the theory of special functions, not considered as a list of functions exhibiting a certain range of properties, but based on the unified study of singularities of second-order ordinary differential equations in the complex domain. The number and characteristics of the singularities serve as a basis for classification of each individual special function. Links between linear special functions (as solutions of linear second-order equations), and non-linear special functions (as solutions of Painleve equations) are presented as a basic and new result. Many applications to different areas of physics are shown and discussed. The book is written from a practical point of view and will address all those scientists whose work involves applications of mathematical methods. Lecturers, graduate students and researchers will find this a useful text and reference work.
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Contents

Linear secondorder ODEs with polynomial coefficients
1
The hypergeometric class of equations
55
The Heun class of equations
97
Application to physical sciences
163
The Painleve class of equations
232
The gamma function and related functions
251
Multipole matrix elements
273
SFTools Database of the special functions
281
Index
291
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About the author (2000)

Wolfgang Lay, Professor of Mathematics at the University of Stuttgart, Germany Sergei Yuryevitsh Slavyanov, Professor of Mathematics at the St. Petersburg State University, Russia

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