The book begins with the so-called Eulerian functions, namely the well-known Gamma and Beta functions, followed by some special functions related to them. It introduces a series of functional forms attributed to the Gamma or Euler function in an attempt to find an analytical continuation of the factorial function where analytical tools employed are consistent with the multiplicative Fourier transform or the so-called Mellin transform. The Gamma function is the cornerstone of the theory of special functions and is discussed first because of its appearance in almost every integral or series representation of other advanced mathematical functions.
One of the central notions in mathematical and quantum statistical physics as well as quantum mechanics is the complete orthogonal set of functions. This is why emphasis is placed on a uniform viewpoint of a generalized theory which includes hypergeometric functions with real and complex variables and orthogonal polynomials (Jacobi, ultra-spherical, Gegenbauer, Tschebycheff, Legendre, Laguerre and Hermite polynomials then spherical functions). It also covers parabolic cylinder, elliptic, Lame and Mathieu functions in detail. The series and integral approaches are examined for cylindrical, Gauss hypergeometric and orthogonal polynomials. Attention is devoted to the relation between orthogonal polynomials and confluent hypergeometric functions.
The textbook covers special functions at the undergraduate and graduate levels in detail, with an elegant mathematical treatment, contemporary style and rejuvenated approach. It is a comprehensive-student-friendly and self-contained structured textbook aimed at stimulating students' interests and curiosity on special functions. It rewards hard work in learning due to the elegant mathematical treatment with contemporary style and rejuvenated approach. The balance between theory and pertinent worked examples which is an integral part of each chapter will reinforce the readers' understanding of the fundamental concepts. The analytical methods employed are written with mathematical rigor and clarity emphasizing a thorough knowledge of special functions with useful and helpful techniques for students pursuing graduate and undergraduate studies in mathematics, statistical and condensed matter physics, quantum mechanics, modern engineering and electronics. The comprehensive coverage of special functions and their applications constitutes an unusually valuable reference and handbook. The book is organized in a logical and simple manner in order of complexity which aids in easy understanding of subsequent chapters of the special functions.
Key Features:
- Organized in a logical and simple manner in order of complexity which aids in easy understanding.
- Many worked examples which form an integral part of each chapter.
- Comprehensive-student-friendly and self-contained structure.
- Rigorous mathematical treatment.
- Balance between theory and worked examples.
- Includes applications in several areas of physics and engineering.
