Muutke küpsiste eelistusi

On a Class of Incomplete Gamma Functions with Applications [Kõva köide]

5.00/5 (2 hinnangut Goodreads-ist)
(King Fahd University of Petroleum and Minerals, Saudi Arabia), (King Fahd University of Pertoleum & Minerals, Saudi Arabia)
  • Formaat: Hardback, 508 pages, kõrgus x laius: 234x156 mm, kaal: 850 g, 66 Tables, black and white
  • Ilmumisaeg: 21-Aug-2001
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1584881437
  • ISBN-13: 9781584881438
Teised raamatud teemal:
On a Class of Incomplete Gamma Functions with ApplicationsSuurem pilt
  • Formaat: Hardback, 508 pages, kõrgus x laius: 234x156 mm, kaal: 850 g, 66 Tables, black and white
  • Ilmumisaeg: 21-Aug-2001
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1584881437
  • ISBN-13: 9781584881438
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781584881438.html
Märksõnad:
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.
Preface ix
Acknowledgments xiii
Generalized Gamma Function
1(36)
The Gamma Function Γ(α)
1(8)
Definition of the Generalized Gamma Function
9(1)
Properties of the Generalized Gamma Function
10(6)
Mellin and Laplace Transforms
16(2)
Asymptotic Representations
18(1)
The Macdonald Probability Function
19(1)
The Digamma Function ψ(x)
20(3)
Generalization of the Psi (Digamma) Function
23(1)
Integral Representations of ψb(α)
24(3)
Properties of the Generalized Psi Function
27(5)
Graphical and Tabular Representations
32(5)
The Generalized Incomplete Gamma Functions
37(86)
The Incomplete Gamma Functions
37(6)
Definition of the Generalized Incomplete Gamma Functions
43(1)
Properties of the Incomplete Generalized Gamma Functions
44(3)
Convolution Representations
47(4)
Connection with Other Special Functions
51(8)
KdF Functions and Incomplete Integrals
59(6)
Representation in Terms of KdF Functions
65(7)
Reduction Formulas for F2:0;0 0:2;1 [ x, y]
72(3)
Integrals of the Product of Bessel and Gamma Functions
75(5)
Asymptotic Representations
80(5)
An Expansion in Terms of Incomplete Gamma Functions
80(1)
An Expansion in Terms of Laguerre Polynomials
81(1)
An Expansion in Terms of Confluent Hypergeometric Functions
81(1)
A Uniform Expansion in Terms of the Error Function
82(3)
Integral Representations for Γ (α, x; b)
85(4)
Graphical and Tabular Representations
89(34)
The Family of the Gamma Functions
123(72)
The Family of Incomplete Gamma Functions
123(1)
The Generalized Error Functions
124(7)
The Generalized Exponential Integral Function
131(3)
The Generalized Fresnel Integrals
134(7)
The Decomposition Functions
141(5)
The Extended Decomposition Functions
146(3)
The E(u, v) and F(u, v) Functions
149(2)
The e(u) and f(u) Functions
151(2)
Graphical and Tabular Representations
153(42)
Extension of Generalized Incomplete Gamma Functions
195(20)
Introduction
195(2)
The Decomposition Formula
197(1)
Recurrence Relation
198(2)
Laplace and K-Transform Representation
200(3)
Parametric Differentiation and Integration
203(2)
Connection with Other Special Functions
205(1)
Integral Representations
206(4)
Differential Representations
210(2)
The Mellin Transform Representation
212(3)
Extended Beta Function
215(50)
The Beta Function
215(2)
The Incomplete Beta Function
217(3)
The Beta Probability Distribution
220(1)
Definition of the Extended Beta Function
221(1)
Properties of the Extended Beta Function
221(4)
Integral Representations of the Extended Beta Function
225(2)
Connection with Other Special Functions
227(8)
Representations in Terms of Whittaker Functions
235(5)
Extended Incomplete Beta Function
240(4)
The Extended Beta Distribution
244(4)
Graphical and Tabular Representations
248(17)
Extended Incomplete Gamma Functions
265(22)
Introduction
265(1)
Definition of the Extended Incomplete Gamma Functions
265(3)
The Decomposition Formula
268(2)
Recurrence Formula
270(1)
Connection with Other Special Functions
271(9)
The H-function
280(1)
Incomplete Fox H-functions
281(6)
Extended Riemann Zeta Functions
287(42)
Introduction
287(1)
Bernoulli's Numbers and Polynomials
287(3)
The Zeta Function
290(7)
Zeros of the Zeta Function and the Function π(x)
297(1)
The Extended Zeta Function ζb(α)
298(6)
The Second Extended Zeta Function ζb*(α)
304(2)
The Hurwitz Zeta Function
306(2)
Extended Hurwitz Zeta Functions
308(3)
Extended Hurwitz Formulae
311(5)
Further Remarks and Comments
316(6)
An Identity of the Hurwitz-Lerch Zeta Function
316(2)
The Zeta Function at Integer Arguments
318(2)
Theorem of Christian Goldbach (1690-1764)
320(2)
Graphical and Tabular Representations
322(7)
Phase-Change Heat Transfer
329(28)
Introduction
329(1)
Constant Temperature Boundary Conditions
330(4)
Convective Boundary Conditions
334(5)
Solid at the Solidification Temperature Tf
338(1)
Surface of the Solid Phase Maintained at T∞
338(1)
Solidification from above with Convection at the Interface
338(1)
Freezing of Tissues around a Capillary Tube
339(4)
Freezing of Binary Alloys
343(4)
Freezing around an Impurity
347(7)
Numerical Methods for Phase-Change Problems
354(3)
Transient Heat Conduction Problems
357(28)
Introduction
357(1)
Time-Dependent Surface Temperatures
358(12)
Some Closed-Form Solutions
359(11)
Time-Dependent Surface Heat Fluxes
370(11)
Some Closed-Form Solutions
373(8)
Illustrative Example
381(4)
Heat Conduction Due to Laser Sources
385(30)
Introduction
385(1)
Mathematical Formulation
386(3)
Some Cases of Practical Interest
389(19)
Instantaneous Laser Source
389(5)
Exponential-Type Laser Source
394(8)
Exponential-Type Initial Temperature Distribution
402(6)
Two-Layer System
408(7)
A Unified Approach to Heat Source Problems
415(26)
Introduction
415(1)
Thermal Explosions
416(2)
Continuously Operating Heat Sources
418(23)
A Moving Point-Heat Source
419(8)
A Moving Line-Heat Source
427(6)
A Moving Plane-Heat Source
433(8)
Appendices 441(20)
A Heat Conduction
441(6)
A.1 The Heat Conduction Equation
441(2)
A.2 Initial and Boundary Conditions
443(1)
A.3 Fundamental Solutions
444(3)
B Table of Laplace Transforms
447(8)
B.1 Abelian Theorems
448(1)
B.2 Watson's Lemma
448(1)
B.3 Tauberian Theorem
449(1)
B.4 Analytic Theorem
449(1)
B.5 Initial Value Theorem
449(1)
B.6 Final Value Theorem
450(1)
B.7 Efros' Theorem
450(1)
B.8 Functional Operations
450(1)
B.9 Table of Laplace Transforms
451(4)
C Integrals Dependent on Parameters
455(6)
C.1 Theorem on Continuity of J(y) := Ja, b (y; f; 1)
455(1)
C.1.1 Theorem on the Continuity of Ja,∞(y; f; g)
455(1)
C.2 Theorem on Differentiation of J(y) := Ja,∞(y; f; 1)
456(1)
C.2.1 Theorem on Differentiation J(y) = Ja,∞(y; f; g)
456(1)
C.3 Theorem on the Integration of J(y) := Ja, b(y; f; 1)
456(1)
C.3.1 Theorem on the Integration of J(y) := Ja,∞(y; f; g)
456(1)
C.4 Theorem on Differentiation of the Integral I(y)
456(1)
C.5 Theorem on the Uniform Convergence of J(y) = Ja,∞(y; f; 1)
457(1)
C.6 Theorem on the Continuity of J(y) = Ja,∞(y; f; 1)
457(1)
C.7 Theorem on the Differentiation of Ja,∞(y; f; 1)
457(1)
C.8 Theorem on an Integration of J(y) := Ja,∞(y; f; 1)
457(1)
C.9 Theorem on Reversing the Order of Intergration (I)
457(1)
C.10 Theorem on Reversing the Order of Integration (II)
458(1)
C.11 Theorem (Abel's Test)
458(1)
C.12 Comparison Test in Terms of Order of Infinities
458(1)
C.13 Theorem (Holder's Inequality)
459(1)
C.14 Differentiation of Γ(α, u; τu)
459(1)
C.15 Differentiation of CΓ(α, u; τu)
459(1)
C.16 Differentiation of SΓ(α, u; τu)
459(2)
References 461(18)
Symbols and Abbreviations 479(4)
Index 483
Syed M. Zubair, M. Aslam Chaudhry