Textbook in PDF format This textbook provides in-depth, clearly written, and comprehensive coverage and offers a renewed approach with applications used to help solve many of the most difficult problems in physics, engineering, and mathematics. It is suitable as a reference volume for scientists and engineers. This textbook extends the work of many existing textbooks in that it has elaborate explanations and the problems are designed to explain the theory. The number of functions covered are exhaustive compared to many existing textbooks. Special functions have a respectable history with great names, including Gauss, Euler, Fourier, Legendre, Bessel, Riemann, Whittaker, and Watson, where a good portion of their work was inspired by physics and the resulting differential equations. Traditionally, special functions arise as solutions to certain linear second-order differential equations with variable coefficients — equations having applications in physics, chemistry, engineering, etc. This book introduces these differential equations, their solutions, and their applications in science and engineering. It also introduces the theory of complex analysis and techniques for solving differential equations. Topics include functions of complex variables, operational calculus, the solution of second-order differential equations in terms of power series, asymptotic methods, hyper-geometric equations, Fuchsian class equation, Euler and Riemann equations, gamma and beta functions, probability integral and related functions, exponential integral and related functions, Bessel functions, orthogonal polynomials with consideration of Legendre, Hermite, Laguerre and Tschebycheff polynomials (with exceptional treatment of the technique of series expansion of functions through the Hermite and Laguerre polynomials), the Airy functions, Gegenbauer and Jacobi polynomials, and hypergeometric functions, the Whittaker functions, the parabolic cylinder functions, and other special functions