Mathematical Analysis: A Modern Approach to Advanced CalculusThe real and complex number systems -- Some basic notions of set theory -- Elements of point set theory -- The limit concept and continuity -- Differentiation of functions of one real variable -- Differentiation of functions of several variables -- Applications of partial differentiation -- Functions of bounded variation, rectifiable curves and connected sets -- Theory of Riemann-Stieltjes integration -- Multiple integrals and line integrals -- vector analysis -- Infinite series and infinite products -- Sequences of functions -- Improper Riemann-Stieltjes integrals -- Fourier series and Fourier integrals -- Cauchy's theorem and the residue calculus. |
Contents
THE REAL AND COMPLEX NUMBER SYSTEMS | 1 |
THE LIMIT CONCEPT AND CONTINUITY | 4 |
SOME BASIC NOTIONS OF SET THEORY | 24 |
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absolutely convergent accumulation point analytic Assume that ƒ bounded variation called closed interval complex numbers complex-valued converges absolutely converges uniformly countable da(x defined and bounded DEFINITION denote described differential diverges E₁ equation example Exercise exists f be defined f is continuous f(xo finite number fn(x following theorem formula Fourier series function defined function f ƒ and g given Green's theorem hence implies improper integrals inequality infinite Jordan content Let f limit limn line integral n-dimensional N(zo neighborhood NOTE obtain one-to-one open interval open set parametric surface partial sums partition positively oriented power series Proof prove real numbers real-valued function rectifiable curve rectifiable Jordan curve Riemann integral Riemann-Stieltjes integral satisfy sequence Show subintervals subset t₁ uniform convergence variable vector field vector-valued function write