Problems and Solutions in Real Analysis

This second edition introduces an additional set of new mathematical problems with their detailed solutions in real analysis. It also provides numerous improved solutions to the existing problems from the previous edition, and includes very useful tips and skills for the readers to master successfully. There are three more chapters that expand further on the topics of Bernoulli numbers, differential equations and metric spaces.

Each chapter has a summary of basic points, in which some fundamental definitions and results are prepared. This also contains many brief historical comments for some significant mathematical results in real analysis together with many references.

Problems and Solutions in Real Analysis can be treated as a collection of advanced exercises by undergraduate students during or after their courses of calculus and linear algebra. It is also instructive for graduate students who are interested in analytic number theory. Readers will also be able to completely grasp a simple and elementary proof of the Prime Number Theorem through several exercises. This volume is also suitable for non-experts who wish to understand mathematical analysis.

Sample Chapter(s)
Chapter 1: Sequences and Limits (153 KB)
Chapter 12: Uniform Distribution (89 KB)

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• 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
• Bernoulli Numbers
• Metric Spaces
• Differential Equations