A garden of integrals /

The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy,...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Burk, Frank
Otros Autores: Scully, Terence, 1935-
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Washington, DC : Mathematical Association of America, ©2007.
Colección:Dolciani mathematical expositions ; no. 31.
Temas:
Integrals.
Intégrales.
MATHEMATICS > Calculus.
MATHEMATICS > Mathematical Analysis.
Integrals
Reelle Funktion
Integral
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Foreword
  • An historical overview
  • 1.1. Rearrangements
  • 1.2. The lune of Hippocrates
  • 1.3. Exdoxus and the method of exhaustion
  • 1.4. Archimedes' method
  • 1.5. Gottfried Leibniz and Isaac Newton
  • 1.6. Augustin-Louis Cauchy
  • 1.7. Bernhard Riemann
  • 1.8. Thomas Stieltjes
  • 1.9. Henri Lebesgue
  • 1.10. The Lebesgue-Stieltjes integral
  • 1.11. Ralph Henstock and Jaroslav Kurzweil
  • 1.12. Norbert Wiener
  • 1.13. Richard Feynman
  • 1.14. References
  • 2. The Cauchy integral
  • 2.1. Exploring integration
  • 2.2. Cauchy's integral
  • 2.3. Recovering functions by integration
  • 2.4. Recovering functions by differentiation
  • 2.5. A convergence theorem
  • 2.6. Joseph Fourier
  • 2.7. P.G. Lejeune Dirichlet
  • 2.8. Patrick Billingsley's example
  • 2.9. Summary
  • 2.10. References
  • 3. The Riemann integral
  • 3.1. Riemann's integral
  • 3.2. Criteria for Riemann integrability
  • 3.3. Cauchy and Darboux criteria for Riemann integrability
  • 3.4. Weakening continuity
  • 3.5. Monotonic functions are Riemann integrable
  • 3.6. Lebesgue's criteria
  • 3.7. Evaluating à la Riemann
  • 3.8. Sequences of Riemann integrable functions
  • 3.9. The Cantor set
  • 3.10. A nowhere dense set of positive measure
  • 3.11. Cantor functions
  • 3.12. Volterra's example
  • 3.13. Lengths of graphs and the Cantor function
  • 3.14. Summary
  • 3.15. References.
  • 4. Riemann-Stieltjes integral
  • 4.1. Generalizing the Riemann integral-- 4.2. Discontinuities
  • 4.3. Existence of Riemann-Stieltjes integrals
  • 4.4. Monotonicity of [null]
  • 4.5. Euler's summation formula
  • 4.6. Uniform convergence and R-S integration
  • 4.7. References
  • 5. Lebesgue measure
  • 5.1. Lebesgue's idea
  • 5.2. Measurable sets
  • 5.3. Lebesgue measurable sets and Carathéodory
  • 5.4. Sigma algebras
  • 5.5. Borel sets
  • 5.6. Approximating measurable sets
  • 5.7. Measurable functions
  • 5.8. More measureable functions
  • 5.9. What does monotonicity tell us?
  • 5.10. Lebesgue's differentiation theorem
  • 5.11. References
  • 6. The Lebesgue-Stieltjes integral
  • 6.1. Introduction
  • 6.2. Integrability : Riemann ensures Lebesgue
  • 6.3. Convergence theorems
  • 6.4. Fundamental theorems for the Lebesgue integral
  • 6.5. Spaces
  • 6.6. L²[-pi, pi] and Fourier series
  • 6.7. Lebesgue measure in the plane and Fubini's theorem
  • 6.8. Summary-- References
  • 7. The Lebesgue-Stieltjes integral
  • 7.1. L-S measures and monotone increasing functions
  • 7.2. Carathéodory's measurability criterion
  • 7.3. Avoiding complacency
  • 7.4. L-S measures and nonnegative Lebesgue integrable functions
  • 7.5. L-S measures and random variables
  • 7.6. The Lebesgue-Stieltjes integral
  • 7.7. A fundamental theorem for L-S integrals
  • 7.8. References.
  • 8. The Henstock-Kurzweil integral
  • 8.1. The generalized Riemann integral
  • 8.2. Gauges and [infinity]-fine partitions
  • 8.3. H-K integrable functions
  • 8.4. The Cauchy criterion for H-K integrability
  • 8.5. Henstock's lemma
  • 8.6. Convergence theorems for the H-K integral
  • 8.7. Some properties of the H-K integral
  • 8.8. The second fundamental theorem
  • 8.9. Summary-- 8.10. References
  • 9. The Wiener integral
  • 9.1. Brownian motion
  • 9.2. Construction of the Wiener measure
  • 9.3. Wiener's theorem
  • 9.4. Measurable functionals
  • 9.5. The Wiener integral
  • 9.6. Functionals dependent on a finite number of t values
  • 9.7. Kac's theorem
  • 9.8. References
  • 10. Feynman integral
  • 10.1. Introduction
  • 10.2. Summing probability amplitudes
  • 10.3. A simple example
  • 10.4. The Fourier transform
  • 10.5. The convolution product
  • 10.6. The Schwartz space
  • 10.7. Solving Schrödinger problem A
  • 10.8. An abstract Cauchy problem
  • 10.9. Solving in the Schwartz space
  • 10.10. Solving Schrödinger problem B
  • 10.11. References
  • Index
  • About the author.

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