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Operational Calculus

Science & Math

MIKUSINSKI, JAN

Mathematics

Pergamon Pr

Operational Calculus, Vol.2 (International Series in Monographs in Pure and Applied Mathematics)

Boehme, Thomas K.

Pergamon Press / PWN

Based on the lifetime work of leading teacher and researcher Jan Mikusinski, this classroom-tested book provides a thorough grounding in mathematical analysis, calculus and mathematical proofing. It introduces natural numbers through a new mathematical approach; replaces the Riemann integral with the more general Lebesgue integral; and rigorously develops the real number system from four simple axioms of natural numbers. Additional features include a wider range of problems than other texts--including simple and routine as well as problems requiring more in depth creativity, answers to common questions, a new approach to the concept of equivalence relation which simplifies the construction of real numbers, and a large number of computational applications.

Introduction to Analysis

John Wiley&Sons

Aims to provide students with a solid background for further studies in mathematical analysis. The text features a new approach to natural numbers which simplifies the construction of real numbers and develops the concepts of calculus more logically and systematically.

From Number to Integral: Introduction to Analysis

Mikusiński, Piotr

John Wiley and Sons Ltd.

Engineering & Transportation

Engineering

Pergamon

The theory of the Lebesgue integral is still considered as a difficult theory, no matter whether it is based the concept of measure or introduced by other methods. The primary aim of this book is to give an approach which would be as intelligible and lucid as possible. Our definition, produced in Chapter I, requires for its background only a little of the theory of absolutely convergent series so that it is understandable for students of the first undergraduate course. Nevertheless, it yields the Lebesgue integral in its full generality and, moreover, extends automatically to the Bochner integral (by replacing real coefficients of series by elements of a Banach space). It seems that our approach is simple enough as to eliminate the less useful Riemann integration theory from regular mathematics courses. Intuitively, the difference between various approaches to integration may be brought out by the following story on shoemakers. A piece of leather, like in Figure 1, is given. The task consists in measuring its area. There are three shoemakers and each of them solves the task in his own way. A B Fig. 1 The shoemaker R. divides the leather into a finite number of vertical strips and considers the strips approximately as rectangles. The sum of areas of all rectangles is taken for an approximate area of the leather (Figure 2). If he is not satisfied with the obtained exactitude, he repeats the whole procedure, by dividing the leather into thinner strips.

The Bochner Integral (Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften)

Teen & Young Adult

Birkhauser

The book contains a rigorous exposition of calculus of a single real variable. It covers the standard topics of an introductory analysis course, namely, functions, continuity, differentiability, sequences and series of numbers, sequences and series of functions, and integration. A direct treatment of the Lebesgue integral, based solely on the concept of absolutely convergent series, is presented, which is a unique feature of a textbook at this level. The standard material is complemented by topics usually not found in comparable textbooks, for example, elementary functions are rigorously defined and their properties are carefully derived and an introduction to Fourier series is presented as an example of application of the Lebesgue integral.The text is for a post-calculus course for students majoring in mathematics or mathematics education. It will provide students with a solid background for further studies in analysis, deepen their understanding of calculus, and provide sound training in rigorous mathematical proof.

INTRODUCTION TO ANALYSIS, AN

World Scientific Publishing Company

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