The book "Special Functions of Mathematical Physics" by A. F. Nikiforov and V. B. Uvarov, translated from Russian by Ralph P. Boas, provides a unified introduction to special functions used in mathematical physics and quantum mechanics. The authors aim to present the theory in a way that is accessible and practical for students and researchers in physics, engineering, and mathematics. The book is structured into several chapters, each focusing on different aspects of special functions: 1. **Foundations of the Theory of Special Functions**: Discusses differential equations, polynomials of hypergeometric type, integral representations, and recursion relations. 2. **Classical Orthogonal Polynomials**: Explores basic properties, general properties, expansions, asymptotic behavior, and applications in quantum mechanics. 3. **Bessel Functions**: Covers Bessel's differential equation, basic properties, integral representations, special classes, addition theorems, and semiclassical approximation. 4. **Hypergeometric Functions**: Introduces equations of hypergeometric type, basic properties, representations, and definite integrals. 5. **Applications in Mathematical Physics, Quantum Mechanics, and Numerical Analysis**: Solves partial differential equations, boundary value problems, quantum mechanical problems, and numerical analysis using special functions. The authors emphasize a unified approach based on the Rodrigues formula for classical orthogonal polynomials, which allows for explicit integral representations and derivation of fundamental properties. The book also includes appendices on the Gamma function and Laplace integrals, and provides a list of basic formulas and references. The translation by Ralph P. Boas has improved the clarity and precision of the text, particularly in sections that were previously more complex. The book is useful for both undergraduate and graduate students, as well as professionals in mathematical and theoretical physics.The book "Special Functions of Mathematical Physics" by A. F. Nikiforov and V. B. Uvarov, translated from Russian by Ralph P. Boas, provides a unified introduction to special functions used in mathematical physics and quantum mechanics. The authors aim to present the theory in a way that is accessible and practical for students and researchers in physics, engineering, and mathematics. The book is structured into several chapters, each focusing on different aspects of special functions: 1. **Foundations of the Theory of Special Functions**: Discusses differential equations, polynomials of hypergeometric type, integral representations, and recursion relations. 2. **Classical Orthogonal Polynomials**: Explores basic properties, general properties, expansions, asymptotic behavior, and applications in quantum mechanics. 3. **Bessel Functions**: Covers Bessel's differential equation, basic properties, integral representations, special classes, addition theorems, and semiclassical approximation. 4. **Hypergeometric Functions**: Introduces equations of hypergeometric type, basic properties, representations, and definite integrals. 5. **Applications in Mathematical Physics, Quantum Mechanics, and Numerical Analysis**: Solves partial differential equations, boundary value problems, quantum mechanical problems, and numerical analysis using special functions. The authors emphasize a unified approach based on the Rodrigues formula for classical orthogonal polynomials, which allows for explicit integral representations and derivation of fundamental properties. The book also includes appendices on the Gamma function and Laplace integrals, and provides a list of basic formulas and references. The translation by Ralph P. Boas has improved the clarity and precision of the text, particularly in sections that were previously more complex. The book is useful for both undergraduate and graduate students, as well as professionals in mathematical and theoretical physics.