This is a book for undergraduates. To be more precise, it is designed
for students who have learned the basic principles of analysis, as taught
to undergraduates in advanced calculus courses, and are prepared to ex-
plore substantial topics in classical analysis. And there is much to explore:
Fourier series, orthogonal polynomials, Stirling’s formula, the gamma func-
tion, Bernoulli numbers, elliptic integrals, Bessel functions, Tauberian the-
orems, etc. Yet the modern undergraduate curriculum typically does not
encompass such topics, except perhaps by way of physical applications. In
effect the student struggles to master abstract concepts and general theo-
rems of analysis, then is left wondering what to do with them.
It was not always so. Around 1950 the typical advanced calculus course
in American colleges contained a selection of concrete topics such as those
just mentioned. However, the development could not be entirely rigorous
because the underlying theory of calculus had been deferred to graduate
courses. To remedy this unsatisfactory state of affairs, the theory of calculus
was moved to the undergraduate level. Textbooks by Walter Rudin and
Creighton Buck helped transform advanced calculus to a theoretical study
of basic principles. Certainly much was gained in the process, but also much
was lost. Various concrete topics, natural sequels to the abstract theory,
were crowded out of the curriculum.
The purpose of this book is to recover the lost topics and introduce
others, making them accessible at the undergraduate level by building on the
theoretical foundation provided in modern advanced calculus courses. My
aim has been to develop the mathematics in a rigorous way while holding
the prerequisites to a minimum. The exposition probes rather deeply into
each topic and is at times intellectually demanding, but every effort has
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