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incomplete generalized gamma function chaudhry
On a Class of Incomplete Gamma Functions
with Applications
M. Aslam Chaudhry
Syed M. Zubair
CRC 2002
DjVu file
Contents
Preface
ix
1 Generalized Gamma Function
1
1.1 The Gamma Function F(a) ................... 1
1.2 Definition of the Generalized Gamma Function ........ 9
1.3 Properties of the Generalized Gamma Function ........ 10
1.4 Mellin and Laplace Transforms ................. 16
1.5 Asymptotic Representations ................... 18
1.6 The Macdonald Probability Function .............. 19
1.7 The Digamma Function b
r) .................. 20
1.8 Generalization of the Psi (Digamma) Function ........ 23
1.9 Integral Representations of bb(a) ................ 24
1.10 Properties of the Generalized Psi Function ........... 27
1.11 Graphical and Tabular Representations ............ 32
The Generalized Incomplete Gamma Functions 37
The Incomplete Gamma Functions ............... 37
Definition of the Generalized Incomplete Gamma Functions . 43
Properties of the Incomplete Generalized Gamma Functions . 44
Convolution Representations 47
Connection with Other Special Functions ........... 51
I(dF Functions and Incomplete Integrals ........... 59
Representation in Terms of KdF Functions .......... 65
ß -,0:2; 1 r
Reduction Formulas for r2:0; o [z, y{ ............... 72
Integrals of the Product of Bessel and Gamma Functions... 75
Asymptotic Representations ................... 80
2.10.1 An Expansion in Terms of Incomplete Gamma Functions 80
2.10.2 An Expansion in Terms of Laguerre Polynomials . . . 81
2.10.3 An Expansion in Terms of Confluent Hypergeometric Functions 81
2.10.4 A Uniform Expansion in Terms of the Error Function 82
Integral Representations for r(a, x; b) ............. 85
Graphical and Tabular Representations ............ 89
The Family of the Gamma Functions 123
3.1 The Family of Incomplete Gamma Functions ......... 123
3.2 The Generalized Error Functions ................ 124
3.3 The Generalized Exponential Integral Function ........ 131
3.4 The Generalized Fresnel Integrals ................ 134
3.5 The Decomposition Functions .................. 141
3.6 The Extended Decomposition Functions ............ 146
3.7 The E(u, v) and F(u,v) Functions ............... 149
3.8 The e(u) and f(u) Functions .................. 151
3.9 Graphical emd Tabular Representations ............ 153
4 Extension of Generalized Incomplete Gamma Functions
195
4.1 Introduction ............................ 195
4.2 The Decomposition Formula ................... 197
4.3 Recurrence Relation ....................... 198
4.4 Laplace and K-Transform Representation ........... 200
4.5 Parametric Differentiation and Integration ........... 203
4.6 Connection with Other Special Functions ........... 205
4.7 Integral Representations ..................... 206
4.8 Differential Representations ................... 210
4.9 The Mellin Transform Representation ............. 212
5 Extended Beta Function
215
5.1 The Beta Function ........................ 215
5.2 The Incomplete Beta Function ................. 217
5.3 The Beta Probability Distribution ............... 220
5.4 Definition of the Extended Beta Function ........... 221
5.5 Properties of the Extended Beta Function ........... 221
5.6 Integral Representations of the Extended Beta Function . . . 225
5.7 Connection with Other Special Functions ........... 227
5.8 Representations in Terms of Whittaker Functions ....... 235
5.9 Extended Incomplete Beta Function .............. 240
5.10 The Extended Beta Distribution ................ 244
5.11 Graphical and Tabular Representations ............ 248
6 Extended Incomplete Gamma Functions 265
6.1 Introduction ............................ 265
6.2 Definition of the Extended Incomplete Gamma Functions . . 265
6.3 The Decomposition Formula ................... 268
6.4 Recurrence Formula ....................... 270
6.5 Connection with Other Special Functions ........... 271
6.6 The H-function .......................... 280
6.7 Incomplete Fox H-functions ................... 281
7 Extended Riemann Zeta Functions 287
7.1 Introduction ............................ 287
7.2 Bernoulli's Numbers and Polynomials ............. 287
7.3 The Zeta Function ........................ 290
7.4 Zeros of the Zeta Function and the Function r(x) ....... 297
7.5 The Extended Zeta Function b(a) ............... 298
7.6 The Second Extended Zeta Function (a) .......... 304
7.7 The Hurwitz Zeta Function ................... 306
7.8 Extended Hurwitz Zeta Functions ............... 308
7.9 Extended Hurwitz Formulae ................... 311
7.10 Further Remarks and Comments ................ 316
7.10.1 An Identity of the Hurwitz-Lerch Zeta Function . . . 316
7.10.2 The Zeta Function at Integer Arguments ....... 318
7.10.3 Theorem of Christian Goldbach (1690- 1764) ..... 320
7.11 Graphical and Tabular Representations ............ 322
8 Phase-Change Heat Transfer 329
8.1 Introduction ............................ 329
8.2 Constant Temperature Boundary Conditions ......... 330
8.3 Convective Boundary Conditions ................ 334
8.3.1 Solid at the Solidification Temperature T; ....... 338
8.3.2 Surface of the Solid Phase Maintained at To ..... 338
8.3.3 Solidification from above with Convection at the Inter-
face ............................ 338
8.4 Freezing of Tissues around a Capillary Tube .......... 339
8.5 Freezing of Binary Alloys .................... 343
8.6 Freezing around an Impurity .................. 347
8.7 Numerical Methods for Phase-Change Problems ........ 354
9 Transient Heat Conduction Problems 357
9.1 Introduction ............................ 357
9.2 Time-Dependent Surface Temperatures ............. 358
9.2.1 Some Closed-Form Solutions .............. 359
9.3 Time-Dependent Surface Heat Fluxes ............. 370
9.3.1 Some Closed-Form Solutions .............. 373
9.4 Illustrative Example ....................... 381
10 Heat Conduction Due to Laser Sources 385
10.1 Introduction ............................ 385
10.2 Mathematical Formulation .................... 386
10.3 Some Cases of Practical Interest ................ 389
10.3.1 Instantaneous Laser Source ............... 389
10.3.2 Exponential-Type Laser Source ............. 394
10.3.3 Exponenti-Type Initial Temperature Distribution . . 402
10.4 Two-Layer System ........................ 408
11 A Unified Approach to Heat Source Problems 415
11.1 Introduction ............................ 415
11.2 Thermal Explosions ....................... 416
11.3 Continuously Operating Heat Sources ............. 418
11.3.1 A Moving Point-Heat Source ..............
11.3.2 A Moving Line-Heat Source ............... 427
11.3.3 A Moving Plane-Heat Source .............. 433
Appendices 441
A
Heat Conduction 441
A.1 The Heat Conduction Equation ................. 441
A.2 Initial and Boundary Conditions ................ 443
A.3 Fundamental Solutions ...................... 444
B Table of Laplace Transforms 447
B.1 Abelian Theorems ........................ 448
B.2 Watson's Lemma ......................... 448
B.3 Tauberian Theorem ....................... 449
B.4 Analytic Theorem ........................ 449
B.5 Initial Value Theorem ...................... 449
B.6 Final Value Theorem ....................... 450
B.7 Efros' Theorem .......................... 450
B.8 Functional Operations ...................... 450
B.9 Table of Laplace Transforms ................... 451
C
Integrals Dependent on Parameters 455
C.1 Theorem on Continuity of J := Ja,b(Y;.f; 1) ......... 455
C.I.1 Theorem on the Continuity of J,(y; );g) ...... 455
C.2 Theorem on Differentiation of J := Ja,(y; f; 1) ...... 456
C.2.1 Theorem on Differentiation J(Yl = Jo,o(Y;J;g) .... 456
C.3 Theorem on the Integration of J := J,v(y;.f; 1) ...... 456
C.3.1 Theorem on the Integration of J := J,o(y; f;g) 456
C.4 Theorem on Differentiation of the Integral I ........ 456
C.5 Theorem on the Uniform Convergence of J -- Ja,cc(Y;.f; 1) 457
C.6 Theorem on the Continuity of J = Ja,o(y;.f; 1) ...... 457
C.7 Theorem on the Differentiation of Ja,c(y; f; 1) ........ 457
C.8 Theorem on an Integration of J := Ja,oo(Y; f; 1) ...... 457
C.9 Theorem on Reversing the Order of Integration (I) ...... 457
C.10 Theorem on Reversing the Order of Integration (II) ..... 458
C.11 Theorem (Abel's Test) ...................... 458
C.12 Comparison Test in Terms of Order of Infinities ........ 458
C.13 Theorem (HSlder's Inequality) ................. 459
C.14 Differentiation of F(, u; -u) ................... 459
C. 15 Differentiation of Cr(a, u; u) .................. 45 .ø
C.16 Differentiation of Sr(a, u; ru)
References
Symbols and Abbreviations
Index
On a Class of Incomplete Gamma Functions
with Applications
M. Aslam Chaudhry
Syed M. Zubair
CRC 2002
DjVu file
Contents
Preface
ix
1 Generalized Gamma Function
1
1.1 The Gamma Function F(a) ................... 1
1.2 Definition of the Generalized Gamma Function ........ 9
1.3 Properties of the Generalized Gamma Function ........ 10
1.4 Mellin and Laplace Transforms ................. 16
1.5 Asymptotic Representations ................... 18
1.6 The Macdonald Probability Function .............. 19
1.7 The Digamma Function b
r) .................. 20 1.8 Generalization of the Psi (Digamma) Function ........ 23
1.9 Integral Representations of bb(a) ................ 24
1.10 Properties of the Generalized Psi Function ........... 27
1.11 Graphical and Tabular Representations ............ 32
The Generalized Incomplete Gamma Functions 37
The Incomplete Gamma Functions ............... 37
Definition of the Generalized Incomplete Gamma Functions . 43
Properties of the Incomplete Generalized Gamma Functions . 44
Convolution Representations 47
Connection with Other Special Functions ........... 51
I(dF Functions and Incomplete Integrals ........... 59
Representation in Terms of KdF Functions .......... 65
ß -,0:2; 1 r
Reduction Formulas for r2:0; o [z, y{ ............... 72
Integrals of the Product of Bessel and Gamma Functions... 75
Asymptotic Representations ................... 80
2.10.1 An Expansion in Terms of Incomplete Gamma Functions 80
2.10.2 An Expansion in Terms of Laguerre Polynomials . . . 81
2.10.3 An Expansion in Terms of Confluent Hypergeometric Functions 81
2.10.4 A Uniform Expansion in Terms of the Error Function 82
Integral Representations for r(a, x; b) ............. 85
Graphical and Tabular Representations ............ 89
The Family of the Gamma Functions 123
3.1 The Family of Incomplete Gamma Functions ......... 123
3.2 The Generalized Error Functions ................ 124
3.3 The Generalized Exponential Integral Function ........ 131
3.4 The Generalized Fresnel Integrals ................ 134
3.5 The Decomposition Functions .................. 141
3.6 The Extended Decomposition Functions ............ 146
3.7 The E(u, v) and F(u,v) Functions ............... 149
3.8 The e(u) and f(u) Functions .................. 151
3.9 Graphical emd Tabular Representations ............ 153
4 Extension of Generalized Incomplete Gamma Functions
195
4.1 Introduction ............................ 195
4.2 The Decomposition Formula ................... 197
4.3 Recurrence Relation ....................... 198
4.4 Laplace and K-Transform Representation ........... 200
4.5 Parametric Differentiation and Integration ........... 203
4.6 Connection with Other Special Functions ........... 205
4.7 Integral Representations ..................... 206
4.8 Differential Representations ................... 210
4.9 The Mellin Transform Representation ............. 212
5 Extended Beta Function
215
5.1 The Beta Function ........................ 215
5.2 The Incomplete Beta Function ................. 217
5.3 The Beta Probability Distribution ............... 220
5.4 Definition of the Extended Beta Function ........... 221
5.5 Properties of the Extended Beta Function ........... 221
5.6 Integral Representations of the Extended Beta Function . . . 225
5.7 Connection with Other Special Functions ........... 227
5.8 Representations in Terms of Whittaker Functions ....... 235
5.9 Extended Incomplete Beta Function .............. 240
5.10 The Extended Beta Distribution ................ 244
5.11 Graphical and Tabular Representations ............ 248
6 Extended Incomplete Gamma Functions 265
6.1 Introduction ............................ 265
6.2 Definition of the Extended Incomplete Gamma Functions . . 265
6.3 The Decomposition Formula ................... 268
6.4 Recurrence Formula ....................... 270
6.5 Connection with Other Special Functions ........... 271
6.6 The H-function .......................... 280
6.7 Incomplete Fox H-functions ................... 281
7 Extended Riemann Zeta Functions 287
7.1 Introduction ............................ 287
7.2 Bernoulli's Numbers and Polynomials ............. 287
7.3 The Zeta Function ........................ 290
7.4 Zeros of the Zeta Function and the Function r(x) ....... 297
7.5 The Extended Zeta Function b(a) ............... 298
7.6 The Second Extended Zeta Function (a) .......... 304
7.7 The Hurwitz Zeta Function ................... 306
7.8 Extended Hurwitz Zeta Functions ............... 308
7.9 Extended Hurwitz Formulae ................... 311
7.10 Further Remarks and Comments ................ 316
7.10.1 An Identity of the Hurwitz-Lerch Zeta Function . . . 316
7.10.2 The Zeta Function at Integer Arguments ....... 318
7.10.3 Theorem of Christian Goldbach (1690- 1764) ..... 320
7.11 Graphical and Tabular Representations ............ 322
8 Phase-Change Heat Transfer 329
8.1 Introduction ............................ 329
8.2 Constant Temperature Boundary Conditions ......... 330
8.3 Convective Boundary Conditions ................ 334
8.3.1 Solid at the Solidification Temperature T; ....... 338
8.3.2 Surface of the Solid Phase Maintained at To ..... 338
8.3.3 Solidification from above with Convection at the Inter-
face ............................ 338
8.4 Freezing of Tissues around a Capillary Tube .......... 339
8.5 Freezing of Binary Alloys .................... 343
8.6 Freezing around an Impurity .................. 347
8.7 Numerical Methods for Phase-Change Problems ........ 354
9 Transient Heat Conduction Problems 357
9.1 Introduction ............................ 357
9.2 Time-Dependent Surface Temperatures ............. 358
9.2.1 Some Closed-Form Solutions .............. 359
9.3 Time-Dependent Surface Heat Fluxes ............. 370
9.3.1 Some Closed-Form Solutions .............. 373
9.4 Illustrative Example ....................... 381
10 Heat Conduction Due to Laser Sources 385
10.1 Introduction ............................ 385
10.2 Mathematical Formulation .................... 386
10.3 Some Cases of Practical Interest ................ 389
10.3.1 Instantaneous Laser Source ............... 389
10.3.2 Exponential-Type Laser Source ............. 394
10.3.3 Exponenti-Type Initial Temperature Distribution . . 402
10.4 Two-Layer System ........................ 408
11 A Unified Approach to Heat Source Problems 415
11.1 Introduction ............................ 415
11.2 Thermal Explosions ....................... 416
11.3 Continuously Operating Heat Sources ............. 418
11.3.1 A Moving Point-Heat Source ..............
11.3.2 A Moving Line-Heat Source ............... 427
11.3.3 A Moving Plane-Heat Source .............. 433
Appendices 441
A
Heat Conduction 441
A.1 The Heat Conduction Equation ................. 441
A.2 Initial and Boundary Conditions ................ 443
A.3 Fundamental Solutions ...................... 444
B Table of Laplace Transforms 447
B.1 Abelian Theorems ........................ 448
B.2 Watson's Lemma ......................... 448
B.3 Tauberian Theorem ....................... 449
B.4 Analytic Theorem ........................ 449
B.5 Initial Value Theorem ...................... 449
B.6 Final Value Theorem ....................... 450
B.7 Efros' Theorem .......................... 450
B.8 Functional Operations ...................... 450
B.9 Table of Laplace Transforms ................... 451
C
Integrals Dependent on Parameters 455
C.1 Theorem on Continuity of J
:= Ja,b(Y;.f; 1) ......... 455 C.I.1 Theorem on the Continuity of J,(y; );g) ...... 455
C.2 Theorem on Differentiation of J
:= Ja,(y; f; 1) ...... 456 C.2.1 Theorem on Differentiation J(Yl = Jo,o(Y;J;g) .... 456
C.3 Theorem on the Integration of J
:= J,v(y;.f; 1) ...... 456 C.3.1 Theorem on the Integration of J
:= J,o(y; f;g) 456 C.4 Theorem on Differentiation of the Integral I
........ 456 C.5 Theorem on the Uniform Convergence of J
-- Ja,cc(Y;.f; 1) 457 C.6 Theorem on the Continuity of J
= Ja,o(y;.f; 1) ...... 457 C.7 Theorem on the Differentiation of Ja,c(y; f; 1) ........ 457
C.8 Theorem on an Integration of J
:= Ja,oo(Y; f; 1) ...... 457 C.9 Theorem on Reversing the Order of Integration (I) ...... 457
C.10 Theorem on Reversing the Order of Integration (II) ..... 458
C.11 Theorem (Abel's Test) ...................... 458
C.12 Comparison Test in Terms of Order of Infinities ........ 458
C.13 Theorem (HSlder's Inequality) ................. 459
C.14 Differentiation of F(, u; -u) ................... 459
C. 15 Differentiation of Cr(a, u; u) .................. 45 .ø
C.16 Differentiation of Sr(a, u; ru)
References
Symbols and Abbreviations
Index
