Introduction to integration provides a unified account of integration theory, giving a practical guide to the Lebesgue integral and its uses, with a wealth of illustrative examples and exercises. The book begins with a simplified Lebesgue-style integral (in lieu of the more traditional Riemann integral), intended for a first course in integration. This suffices for elementary applications, and serves as an introduction to the core of the book. The final chapters present selected applications, mostly drawn from Fourier analysis. The emphasis throughout is on integrable functions rather than on measure. The book is designed primarily as an undergraduate or introductory graduate textbook. It is similar in style and level to Priestley's Introduction to complex analysis, for which it provides a companion volume, and is aimed at both pure and applied mathematicians. Prerequisites are the rudiments of integral calculus and a first course in real analysis.
Rezensionen / Stimmen
Delightful book. Those who know Hilary Priestley will recognise at once the impish sense of fun which permeates this book (even down to the selection of notation): she has a real gift for the memorable phrase and the agonies oand ecstasies of teaching 25 years worth of Oxford undergraduates are etched in the motivational and orientational remarks, helpful reiterations of key points, local stock-taking' susummaries and tight internal sign-posting. By its very nature integration theory cannot be made easy, but Professor Priestley will rapidly earn the gratitude of a new generation of students for making it as pleasantly palatable as one could wish for. This is a very readable and well-planned book, most suitable for all mathematics graduates. The emphasis is on practice with many applications in the later chapters.
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Illustrationen
Maße
Höhe: 234 mm
Breite: 156 mm
Dicke: 18 mm
Gewicht
ISBN-13
978-0-19-850123-7 (9780198501237)
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Schweitzer Klassifikation
Autor*in
Reader in MathematicsReader in Mathematics, University of Oxford
1. Setting the scene ; 2. Preliminaries ; 3. Intervals and step functions ; 4. Integrals of step functions ; 5. Continuous functions on compact intervals ; 6. Techniques of Integration I ; 7. Approximations ; 8. Uniform convergence and power series ; 9. Building foundations ; 10. Null sets ; 11. Linc functions ; 12. The space L of integrable functions ; 13. Non-integrable functions ; 14. Convergence Theorems: MCT and DCT ; 15. Recognizing integrable functions I ; 16. Techniques of integration II ; 17. Sums and integrals ; 18. Recognizing integrable functions II ; 19. The Continuous DCT ; 20. Differentiation of integrals ; 21. Measurable functions ; 22. Measurable sets ; 23. The character of integrable functions ; 24. Integration VS. differentiation ; 25. Integrable functions of Rk ; 26. Fubini's Theorem and Tonelli's Theorem ; 27. Transformations of Rk ; 28. The spaces L1, L2 and Lp ; 29. Fourier series: pointwise convergence ; 30. Fourier series: convergence re-assessed ; 31. L2-spaces: orthogonal sequences ; 32. L2-spaces as Hilbert spaces ; 33. The Fourier transform ; 34. Integration in probability theory ; Appendix I ; Appendix II ; Bibliography ; Notation index ; Subject index
