Mathematical Analysis: A Very Short Introduction

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Very Short Introductions: Brilliant, sharp, inspiring The 17th-century calculus of Newton and Leibniz was built on shaky foundations, and it wasn't until the 18th and 19th centuries that mathematicians--especially Bolzano, Cauchy, and Weierstrass--began to establish a rigorous basis for the subject. The resulting discipline is now known to mathematicians as analysis. This book, aimed at readers with some grounding in mathematics, describes the nascent evolution of mathematical analysis, its development as a subject in its own right, and its wide-ranging applications in mathematics and science, modelling reality from acoustics to fluid dynamics, from biological systems to quantum theory. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. Cover Mathematical Analysis: A Very Short Introduction Copyright Dedication Contents Acknowledgements List of illustrations Chapter 1. Taming infinity Actual and potential infinities How is π calculated? Defining convergence Countable versus uncountable Axioms and some early results Modern analysis Chapter 2. All change . . . the calculus of Fermat, Newton, and Leibniz Calculus The 17th century Synthetic versus analytic geometry Analytic geometry and the function concept Pierre de Fermat The fundamental theorem of calculus The calculus of Newton and Leibniz Newton’s physics The Newton–Leibniz controversy Chapter 3. To the limit: analysis in the 18th and 19th centuries e and the exponential function Logarithms and powers Power series and Taylor series Radians and trigonometry Euler Differential equations Bolzano and Weierstrass Riemann’s integral Chapter 4. Should I believe my computer? Interpolation and extrapolation Numerical differentiation and integration Numerical stability and error analysis Linearization and stability Chapter 5. Dimensions aplenty Scalars and vectors Directional and partial derivatives Partial differentiation equations The calculus of variations Multivariable integration Surface integrals and flux Line integrals and work Stokes’ theorem and the divergence theorem Chapter 6. I’ll name that tune in . . . The wave equation Derivation of the wave equation Boundary value problems Fourier analysis The impact of Fourier’s work Can you hear the shape of a drum? The spectral theorem Distributions Quantum theory Chapter 7. Putting the i in analysis Complex numbers Cauchy The complex plane Two maps of the complex plane Complex differentiability and the Cauchy–Riemann equations Holomorphic functions Complex trigonometric functions and the exponential function Taylor series and Laurent series Complex integrals Cauchy’s residue theorem Conformal maps and applications Chapter 8. But there’s more . . . Lebesgue integration Measure theory Analytic number theory Hyperreal numbers Epilogue Appendix Chapter 1 Ramanujan’s approximation to π Cantor’s proof that the real numbers are uncountable The axioms of the real numbers Chapter 2 The equation of the cissoid of Diocles Chapter 3 Basic identities of the exponential and logarithmic functions A trigonometric identity Rederiving Madhava’s sum e is irrational Euler’s first solution of the Basel problem Chapter 4 Deriving Newton’s method The stable Lotka–Volterra equilibrium Chapter 5 Minimizing the least-squares error Lines are the shortest curves Stokes’ theorem implies Green’s theorem Chapter 6 Deriving the Fourier coefficients Chapter 7 Deriving the Cauchy–Riemann equations Proving Cauchy’s theorem from Green’s theorem Complex logarithm and powers Evaluating an integral with the residue theorem The real and imaginary parts satisfy Laplace’s equation Chapter 8 A countable set is null There are infinitely many primes Historical timeline References and further reading References Index Numbers Statistics Information Chaos