## The Special Functions and Their ApproximationsA detailed and self-contained and unified treatment of many mathematical functions which arise in applied problems, as well as the attendant mathematical theory for their approximations. many common features of the Bessel functions, Legendre functions, incomplete gamma functions, confluent hypergeometric functions, as well as of otherw, can be derived. Hitherto, many of the material upon which the volumes are based has been available only in papers scattered throughout the literature. |

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### Contents

1 | |

8 | |

Chapter III Hypergeometric Functions | 38 |

Chapter IV Confluent Hypergeometric Functions | 115 |

Chapter V The Generalized Hypergeometric Function and the GFunction | 136 |

Chapter VI Identification of the pFq and GFunctions with the Special Functions of Mathematical Physics | 209 |

Chapter VII Asymptotic Expansions of pFq for Large Parameters | 235 |

Chapter VIII Orthogonal Polynomials | 267 |

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### Common terms and phrases

Abramowitz and Stegun analytic continuation applications arbitrary integer arg(l argz asymptotic expansion Bernoulli polynomials Bessel functions change of notation Chapter Chebyshev polynomials coefﬁcients coeﬂicients contour convenient converges deduced deﬁned deﬁnition denominator parameter derived differential equation Erdélyi evaluation expansions in series exponential ﬁnd ﬁnite ﬁrst kind ﬁxed fundamental solutions further G-function given hypergeometric functions hypergeometric series incomplete gamma functions independent solutions inﬁnity integer or zero integrand Jacobi polynomial latter logarithmic solutions Multiply both sides negative integer Note numbers numerator parameter omit orthogonal polynomials path of integration poles proof proved Q m Q radius rational approximations recursion formula relations replaced respectively restriction right-hand side s a positive integer satisﬁes series of Chebyshev special functions suﬂicient suppose Theorem unity valid values Watson write