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Course In Analysis, A - Volume I: Introductory Calculus, Analysis Of Functions Of One Real Variable (Paperback)
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Course In Analysis, A - Volume I: Introductory Calculus, Analysis Of Functions Of One Real Variable (Paperback)
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Part 1 begins with an overview of properties of the real numbers
and starts to introduce the notions of set theory. The absolute
value and in particular inequalities are considered in great detail
before functions and their basic properties are handled. From this
the authors move to differential and integral calculus. Many
examples are discussed. Proofs not depending on a deeper
understanding of the completeness of the real numbers are provided.
As a typical calculus module, this part is thought as an interface
from school to university analysis.Part 2 returns to the structure
of the real numbers, most of all to the problem of their
completeness which is discussed in great depth. Once the
completeness of the real line is settled the authors revisit the
main results of Part 1 and provide complete proofs. Moreover they
develop differential and integral calculus on a rigorous basis much
further by discussing uniform convergence and the interchanging of
limits, infinite series (including Taylor series) and infinite
products, improper integrals and the gamma function. In addition
they discussed in more detail as usual monotone and convex
functions.Finally, the authors supply a number of Appendices, among
them Appendices on basic mathematical logic, more on set theory,
the Peano axioms and mathematical induction, and on further
discussions of the completeness of the real numbers. Remarkably,
Volume I contains ca. 360 problems with complete, detailed
solutions.
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