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Problems and Solutions in Real Analysis

Masayoshi Hata
World Scientific
Publication Date: 
Number of Pages: 
Series on Number Theory and Its Applications 4
Problem Book
[Reviewed by
Allen Stenger
, on

This is an impressive collection of theorems from analysis, with a number-theoretic flavor. Despite the title, the subject matter is classical analysis (series, sequences, inequalities, special functions, definite integrals, summability) rather than what is usually called real analysis (rigorous study of integrals and derivatives, measure theory).

Some notable results that are included are: Carleman's inequality (Problem 2.3), extremal values of polynomials with integer coefficients (Problem 3.2), Hardy & Littlewood's theorem that Abelian summability of a non-negative sequence implies Cesaro (C,1) summability (Problem 7.6), a whole chapter of different proofs that \(\zeta(2)=\pi^2/6\) (Chapter 10), the Bohr-Mollerup theorem about convexity and the gamma function (Problem 16.5), and a recursive formula for \(\zeta(2n)\) (Problem 18.7).

It is presented as a problem book. Most of the problems are quite difficult, roughly as difficult as those in the Problems section of American Mathematical Monthly , and many of the results originally appeared in research papers. Problems are collected in chapters by subject area, but usually each is unrelated to its neighbors, and there's no progression from problem to problem. An exception is Chapter 17, an elementary proof of the Prime Number Theorem that is broken into logical steps. Complete solutions are given for all problems, along with extensive bibliographic information.

The most famous analysis problem book is Pólya & Szegö's Problems and Theorems in Analysis . Pólya & Szegö usually present the big theorems as the culmination of a sequence of related, simpler problems. Their sequences can be quite long; in Part V, for example, Descartes's Rule of Signs is the culmination of 41 exercises.

As a problem book, I think Pólya & Szegö is much superior to Hata, particularly because of the care they took in constructing the sequences. Hata is a good supplemental reference book for classical analysis, but I'm not convinced it is useful as a problem book. The problems as presented are too difficult, and I don't know what audience would use it as a problem book.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

  • Sequences and Limits
  • Infinite Series
  • Continuous Functions
  • Differentiation
  • Integration
  • Improper Integrals
  • Series of Functions
  • Approximation by Polynomials
  • Convex Functions
  • Various Proof ζ(2) = π2/6
  • Functions of Several Variables
  • Uniform Distribution
  • Rademacher Functions
  • Legendre Polynomials
  • Chebyshev Polynomials
  • Gamma Function
  • Prime Number Theorem
  • Miscellanies