A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis
This concise, well-written handbook provides a distillation of real variable theory with a particular focus on the subject's significant applications to differential equations and Fourier analysis. Ample examples and brief explanations---with very few proofs and little axiomatic machinery---are used to highlight all the major results of real analysis, from the basics of sequences and series to the more advanced concepts of Taylor and Fourier series, Baire Category, and the Weierstrass Approximation Theorem. Replete with realistic, meaningful applications to differential equations, boundary value problems, and Fourier analysis, this unique work is a practical, hands-on manual of real analysis that is ideal for physicists, engineers, economists, and others who wish to use the fruits of real analysis but who do not necessarily have the time to appreciate all of the theory. Valuable as a comprehensive reference, a study guide for students, or a quick review, "A Handbook of Real Variables" will benefit a wide audience.
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The Topology of the Real Line
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accumulation point analysis boundary points bounded variation calculate called Cantor set Cauchy criterion Cauchy sequence closed interval closed set coefficients compact set conclude Consider the series continuous function converges uniformly Corollary cosine countable define Definition denote derivative element equal Example exists f f(x)dx Figure finite subcovering fj converge fj(x formula Fourier series function f function with domain geometric series infimum infinitely integrable function interval a,b isolated point Lemma Let f lim f(x limit function Mean Value Theorem natural numbers nonempty open covering open interval open set partial sums partition polynomial power series Proposition Ratio Test rational numbers rb rb real analytic functions real numbers result Riemann integrable Riemann sum Riemann-Stieltjes integral Root Test Section sequence of functions series 00 series converges series of functions solution solve subset supremum theory uncountable uniform convergence uniformly continuous unit interval upper bound Weierstrass write zero