Certain functions, capable of expansion only as a divergent series, may nevertheless be calculated with great accuracy by taking the sum of a suitable number of terms. The theory of such asymptotic expansions is of great importance in many branches of pure and applied mathematics and in theoretical physics. Solutions of ordinary differential equations are frequently obtained in the form of a definite integral or contour integral, and this tract is concerned with the asymptotic representation of a function of a real or complex variable defined in this way. After a preliminary account of the properties of asymptotic series, the standard methods of deriving the asymptotic expansion of an integral are explained in detail and illustrated by the expansions of various special functions. These methods include integration by parts, Laplace's approximation, Watson's lemma on Laplace transforms, the method of steepest descents, and the saddle-point method. The last two chapters deal with Airy's integral and uniform asymptotic expansions.
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The theory of asymptotic expansions is of great importance in many branches of pure and applied mathematics and in theoretical physics. Solutions of ordinary differential equations are frequently obtained in the form of a definite integral or contour integral, and this tract is concerned with the asymptotic representation of a function of a real or complex variable defined in this way.
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Although the subject of 'Modern Analysis' had its beginnings in the seventeenth century, the mathematicians of the eighteenth century were often more interested in the formal use of infinite processes than in their rigorous proofs. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
asymptotic power series expansion, steepest curves, uniform asymptotic expansions, complete asymptotic expansion, steepest path, asymptotic sequence, smallest term, analytical continuation New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Although the subject of 'Modern Analysis' had its beginnings in the seventeenth century, the mathematicians of the eighteenth century were often more interested in the formal use of infinite processes than in their rigorous proofs. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
asymptotic power series expansion, steepest curves, uniform asymptotic expansions, complete asymptotic expansion, steepest path, asymptotic sequence, smallest term, analytical continuation New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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This book cites 3
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66
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cite this book:
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- A Treatise on the Theory of Bessel Functions (Cambridge Mathematical Library) by G. N. Watson in Back Matter
- Laplace Transform (Mathematics Series) by D. V. Widder in Back Matter
- Asymptotic Methods in Analysis (Dover Books on Mathematics) by N. G. de Bruijn in Back Matter
- Ramanujan's Notebooks: Part II (Pt. 2) by Bruce C. Berndt on page 203, page 307, and Back Matter
- Asymptotic and Computational Analysis (Lecture Notes in Pure and Applied Mathematics) by R. Wong on page 440, and page 637
- Advanced Calculus with Applications in Statistics (Wiley Series in Probability and Statistics) by André I. Khuri on page 550, and Back Matter
- Asymptotics and Special Functions (AKP Classics) by Frank Olver on page 22, and page 116
- Asymptotics and Mellin-Barnes Integrals (Encyclopedia of Mathematics and its Applications) by R. B. Paris in Front Matter, and Back Matter
See all 66 books citing this book
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