1st Edition
On a Class of Incomplete Gamma Functions with Applications
The subject of special functions is rich and expanding continuously with the emergence of new problems encountered in engineering and applied science applications. The development of computational techniques and the rapid growth in computing power have increased the importance of the special functions and their formulae for analytic representations. However, problems remain, particularly in heat conduction, astrophysics, and probability theory, whose solutions seem to defy even the most general classes of special functions.
On a Class of Incomplete Gamma Functions with Applications introduces a class of special functions, developed by the authors, useful in the analytic study of several heat conduction problems. It presents some basic properties of these functions, including their recurrence relations, special cases, asymptotic representations, and integral transform relationships. The authors explore applications of these generalized functions to problems in transient heat conduction, special cases of laser sources, and problems associated with heat transfer in human tissues. They also discuss applications to astrophysics, probability theory, and other problems in theory of functions and present a fundamental solution to time-dependent laser sources with convective-type boundary conditions.
Appendices include an introduction to heat conduction, Fourier conduction, a table of Laplace transforms, and well-known results regarding the improper integrals. Filled with tabular and graphical representations for applications, this monograph offers a unique opportunity to add to your mathematical toolbox a new and useful class of special functions.
The Gamma Fun ction G(a)
Definition of the Generalized Gamma Function
Properties of the Generalized Gamma Function
Mellin and Laplace Transforms
Asymptotic Representations
The Macdonald Function
The Digamma Function y(x)
Generalization of Psi (Digamma) Function
Integral Representations of yb (a)
Properties of the Generalized Psi Function
Graphical and Tabular Representations
THE GENERALIZED INCOMPLETE GAMMA FUNCTIONS
The Incomplete Gamma Functions
Definition of the Generalized Incomplete Gamma Functions
Properties of the Incomplete Generalized Gamma Functions
Convolution Representations and Laplace Transforms
Connection with Other Special Functions
KdF Functions and Incomplete Integrals
Representation in terms of KdF Functions
Reduction Formulas for F0:2;1 2:0;0[x,y]
Integrals of Product of Bessel and Gamma Functions
Asymptotic Representations
Integral Representation for G(a,x;b)
Graphical and Tabular Representations
THE FAMILY OF THE GAMMA FUNCTIONS
The Family of Incomplete Gamma Functions
The Generalized Error Functions
The Generalized Exponential Integral function
The Generalized Fresnel Integrals
The Decomposition Functions
The Extended Decomposition functions
The E(u,v) and F(u,v) Functions
The e(u) and f(u) Functions
Graphical and Tabular Representations
EXTENSION OF GENERALIZED INCOMPLETE GAMMA FUNCTIONS
Introduction
The Decomposition Formula
Recurrence Relations
Laplace and K-Transform Representation
Parametric Differentiation and Integration
Connection with Other Special Functions
Integral Representations
Differential Representations
The Mellin Transform Representation
EXTENDED BETA FUNCTION
The Beta Function
The Incomplete Beta Function
The Beta Probability Distribution
Definition of the Extended Beta Function
Properties of the Extended Beta Function
Integral Representations of the Extended Beta Function
Conncection with Other Special Functions
Representations in Terms of Whittaker functions
Extended Incomplete Beta Function
The Extended Beta Distribution
Graphical and Tabular Representations
EXTENDED INCOMPLETE GAMMA FUNCTIONS
Introduction
Definition of the Extended Incomplete Gamma Functions
The Decomposition Formula
Recurrence Formula
Connection with Other Special Functions
The H-Function
Incomplete Fox H-Functions
EXTENDED RIEMANN ZETA FUNCTIONS
Introduction
Bernoulli's Numbers and Polynomials
The Zeta Function
Zeros of the Zeta Function and the Function p(x)
The Extended Zeta Function z(a)
The Second Extended Zeta Function zb*(a)
The Hurwitz Zeta Function
Extended Hurwitz Zeta Functions
Extended Hurwitz Formulae
Further Remarks and Comments
Graphical and Tabular Representations
PHASE-CHANGE HEAT-TRANSFER
Introduction
Constant Temperature Boundary Conditions
Convective Boundary Conditions
Freezing of Tissues around a Capillary Tube
Freezing of Binary Alloys
Freezing Around an Impurity
Numerical Methods for Phase-Change Problems
HEAT CONDUCTION WITH TIME-DEPENDENT BOUNDARY CONDITIONS
Introduction
Time-Dependent Surface Temperatures
Time-Dependent Surface Heat Fluxes
Illustrative Example
HEAT CONDUCTION DUE TO TIME-DEPENDENT LASER SOURCES
Introduction
Mathematical Formulation
Some Cases of Practical Interest
A UNIFIED APPROACH TO HEAT SOURCE PROBLEMS
Introduction
Thermal Explosions
Continuously Operating Heat Sources
APPENDICES
Heat Conduction
Table of Laplace Transforms
Integrals Dependent of Parameters
REFERENCES
SYMBOLS
INDEX
Biography
Syed M. Zubair, M. Aslam Chaudhry
