analisis matematico apostol

Chapter 1 The Real and Complex Number Systems 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33

Introduction The field axioms The order axioms . Geometric representation of real numbers Intervals Integers The unique factorization theorem for integers Rational numbers Irrational numbers Upper bounds, maximum element, least upper bound (supremum) The completeness axiom Some properties of the supremum Properties of the integers deduced from the completeness ax The Archimedean property of the real-number system Rational numbers with finite decimal representation Finite decimal approximations to real numbers Infinite decimal representation of real numbers Absolute values and the triangle inequality The Cauchy-Schwarz inequality Plus and minus infinity and the extended real number syste Complex numbers Geometric representation of complex numbers The imaginary unit Absolute value of a complex number Impossibility of ordering the complex numbers Complex exponentials Further properties, of complex exponentials The argument of a complex number Integral powers and roots of complex numbers Complex logarithms Complex powers • Complex sines and cosines Infinity and the extended complex plane C* Exercises ix

b) Prove that A = {x : x e S and fA(x} = 0}. 4.57 In a metric space (S, d), let y4 and 5 be disjoint closed subsets of S. exist disjoint open subsets U and V of S such that A £ C/ and 5 £ #(X) = /A(X~) ~ /B('X~)> in the notation of Exercise 4.56, and consider g OifzeA. Let B = f(A) denote the image of A under /and prov

a) If J^is an open subset of A, then/(JO is an open subset 'of B. b) B is an open disk of radius 1. c) For each point UQ + iv0 in B, there is only a finite number of p that/(z) = w0 + w0. Extremum problems

13.8 Find and classify the extreme values (if any) of the functions defined equations: a) f(x, y} = y~ + x~y + x4; b) f(x, y) = x2 + y2 + x + y + xy,

c) f(x, y) = (x- I)4 + (x- y)\) f(x, y)=y2 - x\9 Fin

x~ — 4y = 0. Solve this problem using Lagrange's method and als Lagrange's method. 13.10 Solve the following geometric problems by Lagrange's method: a) Find the shortest distance from the point (