## Pade Approximants: Encyclopedia of Mathematics and It's Applications, Vol. 59 George A. Baker, Jr., Peter Graves-MorrisThe Pade approximant of a given power series is a rational function of numerator degree L and denominator degree M whose power series agrees with the given one up to degree L + M inclusively. A collection of Pade approximants formed by using a suitable set of values of L and M often provides a means of obtaining information about the function outside its circle of convergence, and of more rapidly evaluating the function within its circle of convergence. Applications of these ideas in physics, chemistry, electrical engineering, and other areas have led to a large number of generalizations of Pade approximants that are tailor-made for specific applications. Applications to statistical mechanics and critical phenomena are extensively covered, and there are newly extended sections devoted to circuit design, matrix Pade approximation, computational methods, and integral and algebraic approximants. The book is written with a smooth progression from elementary ideas to some of the frontiers of research in approximation theory. Its main purpose is to make the various techniques described accessible to scientists, engineers, and other researchers who may wish to use them, while also presenting the rigorous mathematical theory. |

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### Contents

1 Introduction and definitions | 1 |

2 Elementary developments | 38 |

3 Pade approximants and numerical methods | 67 |

4 Connection with continued fractions | 122 |

5 Stieltjes series and Pólya series | 193 |

6 Convergence theory | 276 |

7 Extensions of Padé approximants | 335 |

8 Multiseries approximants | 415 |

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### Common terms and phrases

algebraic algorithm analytic application approximant of type assume Baker block bounded calculation called coefficients column compact complex connection consider constant construction continued fraction convergence corresponding defined definition denominator denote derived determined diagonal discussion domain elements equal equations error essentially established example exists expansion expressed fact factor Figure follows formula function further give given Hence identity implies important independent initialization integral interpolation leading least limit linear matrix measure method minimal natural normalization numerator obtain origin Padé approximants poles polynomials positive possible potential power series precisely problem Proof properties prove radius of convergence rational recursion refer region relation represent representation result right-hand satisfy Section sequence shown shows side singularities solution space Stieltjes series Suppose theorem theory unique values vector zeros