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A comprehensive-student-friendly and self-contained structured textbook aimed at stimulating students' interests and curiosity on special functions, organized in a logical and simple manner in order of complexity. The balance between theory and pertinent worked examples, which is an integral part of each chapter, reinforces 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 in several areas of physics and engineering constitutes an unusually valuable reference and handbook.
The book starts with a geometrical approach to complex variables while visualizing complex functions as maps involving translations, rotations and scaling transformations enhancing understanding and application of integral transforms. The Laplace transform offers an elegant technique for solving linear differential equations with constant coefficients without the risk of interference besides generating non-periodic solutions as opposed to the Fourier transform. The Eulerian functions follow and subsequently generalized theory of special functions which includes hypergeometric functions , cylindrical functions then orthogonal polynomials (Jacobi, ultra-spherical, Gegenbauer, Tschebycheff, Legendre, Laguerre and Hermite polynomials then spherical functions) with parabolic cylinder, elliptic, Lame, Mathieu functions in addition to relations 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.
Lukong Cornelius Fai is professor of Theoretical Physics at the Department of Physics, Faculty of Sciences, University of Dschang as well as visiting professor at the Department of Physics, Faculty of Science, University of Bamenda, both in Cameroon. He is Head of Condensed Matter and Nanomaterials program at the Department of Physics, Faculty of Sciences, University of Dschang; He also heads the Condensed Matter, Electronics and Signal Processing as well as Mesoscopic and Multilayer Structures Laboratory at the Department of Physics, Faculty of Sciences, University of Dschang, and is Director of the College of Technology, University of Buea in Cameroon. He was formerly senior associate at the Abdus Salam International Centre for Theoretical Physics (ICTP), Italy. He holds a Master of Science degree in Physics and Mathematics (June 1991) as well as a Doctor of Science degree in Physics and Mathematics (February 1997) from the Faculty of Physics, Department of Theoretical Physics, Moldova State University. He has authored two hundred and nine scientific publications and six books.
Preface
Acknowledgements
Author biography
1 Functions of complex variables
2 Boundary value problems
3 Riemann boundary value problem
4 Riemann boundary value problem with discontinuous coefficients and closed contours
5 Mechanical applications
6 Taylor and Laurent series and the residue theory
7 Singular points
8 Residues of regular functions at isolated singularities
9 Operator analysis
10 Multiplication and expansion theorem
11 Differential equations solution by the Laplace transform
12 Gamma function
13 Gamma function contour integral representation
14 Gamma function operator analysis
15 Beta function: first kind Euler integral
16 Asymptotic methods
17 Fuchsian equations-differential equations' singularities
18 Fuchsian class equation
19 Solution of the hypergeometric differential equation
20 Riemann differential equation transformed into the hypergeometric differential equation
21 Representation of elementary functions via hypergeometric functions
22 Hypergeometric-type integrals
23 Some properties of the hypergeometric functions
24 Lame generalized equation
25 Hypergeometric functions
26 Exponential integral and functions it generates
27 Laplace equation solution-hypergeometric and parabolic cylinder functions
28 Cylindrical functions of the first kind
29 Applications of cylindrical functions of the first kind
30 Cylindrical functions of the second kind
31 Cylindrical function of the third kind
32 Modified Bessel function
33 Thomson (Kelvin) functions and their generalizations
34 Contour integral representation of Bessel equation solution
35 Orthogonal cylindrical functions of the first kind
36 General theory of orthogonal polynomials
37 Jacobi polynomials
38 Ultraspherical (Gegenbauer) polynomials
39 Tschebycheff polynomials
40 Legendre polynomials
41 Spherical functions
42 Generalized Laguerre polynomial
43 Hermite polynomials
44 Elliptic functions
45 Lame and Mathieu functions
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