On a Class of Incomplete Gamma Functions with Applications

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.

Contents

GENERALIZED GAMMA FUNCTION

• 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 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 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

• 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

• 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

• 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

• 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

• 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

• Introduction
• Time-Dependent Surface Temperatures
• Time-Dependent Surface Heat Fluxes
• Illustrative Example

• Introduction
• Mathematical Formulation
• Some Cases of Practical Interest

• Introduction
• Thermal Explosions
• Continuously Operating Heat Sources

• Heat Conduction
• Table of Laplace Transforms
• Integrals Dependent of Parameters

• REFERENCES
• SYMBOLS
• INDEX