## Special Functions: A Unified Theory Based on SingularitiesThe 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 Painlevé 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 |

2 | 236 |

The gamma function and related functions | 251 |

Multipole matrix elements | 273 |

SF Tools Database of the special functions | 281 |

### Common terms and phrases

according applied appropriate arbitrary assumed asymptotic behaviour belonging biconfluent Birkhoff boundary calculation called central characteristic exponents coefficients complex computations condition confluence process confluent connection problem considered constant constructed convergence corresponding curves defined denoted dependent determined difference equation differential equation doubly confluent eigenfunctions eigenvalue elementary Example expansion exponentially expressed factors finite formulae function give given Heun equation Heun's hypergeometric equation important independent infinity initial integral interval irregular leads limit linear located matrix means method namely needed normal obtained operator Painlevé equations parameters particular solutions physical plane polynomials possible potential presented problem proof properties reads recurrence relation reduced regular singularity relevant representation result s-homotopic transformation s-rank satisfy singular point solutions solved standard Stokes studied substitution Suppose Table Theorem theory Thomé tions transformation triconfluent values variable zero