id: 06478875
dt: b
an: 2016a.00746
au: Jacob, Niels; Evans, Kristian P.
ti: A course in analysis. Volume I: Introductory calculus, analysis of
functions of one real variable.
so: Hackensack, NJ: World Scientific (ISBN 978-981-4689-08-3/hbk;
978-981-4689-09-0/pbk). xxiii, 744~p. (2016).
py: 2016
pu: Hackensack, NJ: World Scientific
la: EN
cc: I15
ut: analysis; calculus; real numbers; inequalities; intervals; mathematical
induction; functions; mappings; derivatives; exponential functions;
logarithmic functions; trigonometric functions; inverse functions;
integrals; analysis in one dimension; sequences; limits; series;
convergence criteria; continuous functions; differentiation; convex
functions; norms; uniform convergence; interchanging limits; Riemann
integrals; the fundamental theorem of calculus; differential equations;
improper integrals; power series; Taylor series; infinite products;
Gauss integrals; gamma-function
ci:
li: doi:10.1142/9625
ab: In one sentence, this is a very good book for anyone interested in learning
analysis. This almost 750-pages long book is the first of a planned
series of seven volumes which are meant to cover topics from real
numbers to measure theory, (partial) differential equations, functional
analysis and even Lie groups. It is divided into three parts:
introductory calculus, analysis in one dimension and appendices. The
authors prepared the presentation in great detail, besides classical
tools like a comprehensive list of symbols or subject index providing
the interested reader the Greek alphabet and a nice list of
mathematicians who have contributed to analysis containing names from
Pythagoras, Euclid and Archimedes to Banach or Cohen. The presentation
of the material is very good, one can feel the rich experience of the
authors in teaching mathematics, in particular analysis. Their aim
“to provide students and lecturers with a coherent text which can and
should serve entire undergraduate studies in analysis” has been
successfully achieved. The first part of the book is divided into 13
chapters, entitled Real numbers, Inequalities and intervals,
Mathematical induction, Functions and mappings (2 chapters),
Derivatives (2 chapters), Exponential and logarithmic functions,
Trigonometric functions and their inverses, Investigations on
functions, Integrals and rules of integration, while the second one
contains 19 chapters, namely Problems with the real line, Sequences and
their limits, Series, Completeness of the real numbers, Convergence
criteria for series and and $b$-adic functions, Point sets in
$\mathbb{R}$, Continuous functions, Differentiation, Applications of
the derivative, Convex functions and norms in $\mathbb{R}^n$, Uniform
convergence and interchanging limits, Riemann integrals, The
fundamental theorem of calculus, Differential equations, Improper
integrals and the $Γ$-function, Power series and Taylor series,
Infinite products and Gauss integrals, more on the $Γ$-function and
selected topics on functions of a real variable. The appendices are
almost 300 pages long and cover topics like Elementary aspects of
mathematical logic, Sets and mappings, The Peano axioms, Results from
elementary geometry, Trigonometric and hyperbolic functions,
Completeness of $\mathbb{R}$, Limes superior and limes inferior and
Connected sets in $\mathbb{R}$, that are additional to the main topic,
but still of interest, providing moreover solutions to the more than
360 exercises proposed in the book. These problems are carefully
selected in order to help the students to better understand the
theoretical notions. In conclusion, I highly recommend this book to
anyone teaching or studying analysis at an undergraduate level.
rv: Sorin-Mihai Grad (Chemnitz)