Shchepin E.V. Uppsala Lectures on Calculus. On Euler's footsteps

Moscow: Russian Academy of Science (V.A. Steklov Institute of Math.), 2003. — 137 p.This book represents an introductory course of Calculus. The course evolved from the lectures, which the author had given in the Kolmogorov School in years 1986–1998 for the one-year stream. The Kolmogorov School is a special physics-mathematical undergraduate school for gifted children. Most of the graduates of Kolmogorov School continue their education in Moscow University, where they have to learn the Calculus from the beginning. This motivates the author efforts to create a course of Calculus, which on the one hand facilitates to the students the perception of the standard one, but on the other hand misses the maximum possible of the standard material to provide the freshness of perception of the customary course. In the present form the course was given in Uppsala University (Sweden) in the autumn semester of 2001. for a group of advanced first-year students. The material of the course covers the standard Calculus of the first-year, covers the essential part of the standard course of the complex Calculus, in particular, it includes the theory of residues. Moreover it contains an essential part of the theory of finite differences. Such topics presented here as Newton interpolation formula, Bernoulli polynomials, Gamma-function and Euler-Maclaurin summation formula one usually learns only beyond the common programs of a mathematical faculty. And the last lecture of the course is devoted to divergent series—a subject unfamiliar to the most of modern mathematicians.
The presence of a number of material exceeding the bounds of the standard course is accompanied with the absence of some of “inevitable” topics and concepts. There is no a theory of real numbers. There is no theory of the integral neither Riemann nor Lebesgue. The present course even does not contain the Cauchy criterion of convergence. Such achievements of the 19th. century as uniform convergence and uniform continuity are avoided. Nevertheless the level of rigor in the book is modern. In the first chapter, the greek principle of exhaustion works instead of the theory of limits. The order of exposition in the course is far from the standard one. The standard modern course of Calculus starts with sequences and their limits. This course, following to Euler’s Introductio in Analysin Infinitorum, starts with series. The introduction of the concept of the limits is delayed up to tenth lecture. The Newton-Leibniz formula appears after all elementary integrals are already evaluated. And power series for elementary functions are obtained without help of Taylor series. The course demonstrates the unity of real, complex and discrete Calculus. For example, complex numbers immediately after their introduction are applied to evaluate a real series. The main motivation of the author was to present the power and the beauty of the Calculus. The author understand that this course is somewhere difficult, but he believes that it is nowhere tiresome. The course gives a new approach to exposition of Calculus, which may be interesting for students as well as for teachers. Moreover, it may be interesting for mathematicians as a “mathematical roman”.Preface.
The Legend of Euler’s Series.
Series
Autorecursion of Infinite Expressions.
Positive Series.
Unordered Sums.
Infinite Products.
Telescopic Sums.
Complex Series.
Integrals
Natural Logarithm.
Definite Integral.
Stieltjes Integral.
Asymptotics of Sums.
Quadrature of Circle.
Virtually monotone functions.
Derivatives
Newton-Leibniz Formula.
Exponential Functions.
Euler Formula.
Abel’s Theorem.
Residue Theory.
Analytic Functions.
Differences
Newton Series.
Bernoulli Numbers.
Euler-Maclaurin Formula.
Gamma Function.
The Cotangent.
Divergent Series.