## The Asymptotic Developments of Functions Defined by Maclaurin Series |

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### Contents

Preliminary Considerations First General Theorem | 1 |

Specific Problem to be Considered | 4 |

Remarks and Generalizations Illustrations | 9 |

Copyright | |

36 other sections not shown

### Common terms and phrases

analytic function analytic throughout applying the theorem asymptotic developments asymptotic series characteristic equation coefficients complex variable considered for values const convergent convergent series cosh iry defined developable asymptotically difference equation differential equation Divergent Series dominant term equation 9 following theorem function f(z fundamental solutions gamma function hence hi+ki hypothesis inasmuch indicated indicial equation large modulus lying linearly independent solutions loop integral Maclaurin development Maclaurin series Moreover negative integer non-integral normal series obtained path of integration Pi(x po(x pole positive integer present instance problem proceed properties radius of convergence real axis real or complex regular singular point remark replaced residue right half plane right member roots sector single-valued and analytic singular point solution yi solutions ui(x sufficiently large summation takes the form theorem of Chapter tion transformation ut(x vanishes w-plane wherein write zero