Universal Formulas In Integral And Fractional Differential Calculus

by Khavtgai Namsrai

Overview

This reference book presents unique and traditional analytic calculations, and features more than a hundred universal formulas where one can calculate by hand enormous numbers of definite integrals, fractional derivatives and inverse operators. Despite the great success of numerical calculations due to computer technology, analytical calculations still play a vital role in the study of new, as yet unexplored, areas of mathematics, physics and other branches of sciences. Readers, including non-specialists, can obtain themselves universal formulas and define new special functions in integral and series representations by using the methods expounded in this book. This applies to anyone utilizing analytical calculations in their studies.

Product Details

ISBN-13:	9789814675598
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	02/25/2016
Pages:	296
Product dimensions:	6.69(w) x 9.61(h) x 0.69(d)

Table of Contents

Preface vii

1 Mathematical Preparation 1

1.1 Going to the Complex Number of Integration and the Mellin Representation 1

1.2 Theory of Residues and Gamma-Function Properties 3

1.2.1 Basic Theorem of the Residue Theory 3

1.2.2 The L'Hôpitol Rule 3

1.2.3 Calculation of Residue Encountered in the Mellin Integrals 4

1.2.4 Gamma Function Properties 5

1.2.5 Psi-Function Ψ(x) 6

1.3 Calculation of Integrals in the Form of the Mellin Representation 7

1.4 The Mellin Representation of Functions 8

1.4.1 The Exponential Function 8

1.4.2 The Trigonometric Functions (See Exercises in Section 1.4.8, Chapter 1) 9

1.4.3 The Cylindrical Functions 9

1.4.4 The Struve Function 10

1.4.5 The β(x)-Function 10

1.4.6 The Incomplete Gamma-Function 10

1.4.7 The Probability Integral and Integrals of Frenel 11

1.4.8 Exercises 11

2 Calculation of Integrals Containing Trigonometric and Power Functions 13

2.1 Derivation of General Unified Formulas 13

2.1.1 The First General Formula 13

2.1.2 The Second General Formula 14

2.1.3 The Third General Formula 14

2.1.4 The Fourth General Formula 14

2.2 Calculation of Concrete Particular Integrals Involving x^-γ and Sine Functions 15

2.3 Integrals Involving x^-γ, Sine and Cosine Functions 23

3 Integrals Involving x^γ, (p + tx^-ρ)^-γ, Sine and Cosine Functions 39

3.1 Derivation of General Unified Formulas for this Class of Integrals 39

3.1.1 The Fifth General Formula 39

3.1.2 The Sixth General Formula 40

3.1.3 The Seventh General Formula 40

3.1.4 The Eighth General Formula 41

3.2 Calculation of Concrete Integrals 41

4 Derivation of General Formulas for Integrals Involving Powers of x, (a + bx)-Type Binomials and Trigonometric Functions 59

4.1 Derivation of General Formulas 59

4.1.1 9^th General Formula 59

4.1.2 10^th General Formula 60

4.1.3 11^th General Formula 60

4.1.4 12^th General Formula 60

4.1.5 13^th General Formula 61

4.1.6 14^th General Formula 61

4.1.7 15^th General Formula 62

4.1.8 16^th General Formula 62

4.1.9 17^th General Formula 63

4.1.10 18^th General Formula 63

4.1.11 19^th General Formula 63

4.1.12 20^th General Formula 64

4.1.13 21^st General Formula 64

4.1.14 22^nd General Formula 64

4.1.15 23^rd General Formula 65

4.1.16 24^th General Formula 65

4.1.17 25^th General Formula 65

4.1.18 26^th General Formula 66

4.1.19 27^th General Formula 66

4.1.20 28^th General Formula 66

4.1.21 29^th General Formula 67

4.1.22 30^th General Formula 67

4.1.23 31^st General Formula 67

4.1.24 32^nd General Formula 68

4.2 Calculation of Particular Integrals 68

5 Integrals Involving x^γ, $$$, e^-axν and Trigonometric Functions 87

5.1 Universal Formulas for Integrals Involving Exponential Functions 87

5.1.1 33^rd General Formula 87

5.1.2 34^th General Formula 88

5.1.3 35^th General Formula 88

5.1.4 36^th General Formula 89

5.1.5 37^th General Formula 89

5.1.6 38^th General Formula 89

5.1.7 39^th General Formula 90

5.1.8 40^th General Formula 90

5.2 Calculation of Concrete Integrals 90

5.3 Simple Formulas for Integrals Involving Exponential and Polynomial Functions 94

5.3.1 41^st General Formula 94

5.3.2 42^nd General Formula 95

5.4 Unified Formulas for Integrals Containing Exponential and Trigonometric Functions 99

5.4.1 43^rd General Formula 99

5.4.2 44^th General Formula 99

5.4.3 45^th General Formula 100

5.4.4 46^th General Formula 100

5.5 Calculation of Particular Integrals Arising from the General Formulas in Section 5.4 101

5.6 Universal Formulas for Integrals Involving Exponential, Trigonometric and x^γ [p + tx^ρ]^-λ -Functions 114

5.6.1 47^th General Formula 114

5.6.2 48^th General Formula 115

5.6.3 49^th General Formula 116

5.6.4 50^th General Formula 117

5.6.5 51^st General Formula 118

5.6.6 52^nd General Formula 118

5.7 Some Consequences of the General Formulas Obtained in Section 5.6 119

6 Integrals Containing Bessel Functions 123

6.1 Integrals Involving J_μ(x), x^γ and [p + tx^ρ]^-λ 123

6.1.1 53^rd General Formula 123

6.1.2 54^th General Formula 124

6.1.3 55^th General Formula 124

6.1.4 56^th General Formula 124

6.1.5 57^th General Formula 125

6.1.6 58^th General Formula 125

6.1.7 59^th General Formula 125

6.1.8 60^th General Formula 126

6.1.9 61^st General Formula 126

6.1.10 62^nd General Formula 126

6.1.11 63^rd General Formula 127

6.2 Calculation of Concrete Integrals 127

6.3 Integrals Containing J_μ(x) and Logarithmic Functions 136

6.3.1 64^th General Formula 136

6.3.2 65^th General Formula 137

6.3.3 Examples of Concrete Integrals 137

6.4 Integrals Containing J_σ(x) and Exponential Functions 138

6.4.1 66^th General Formula 138

6.4.2 67^th General Formula 139

6.4.3 68^th General Formula 140

6.4.4 Calculation of Concrete Integrals 140

6.5 Integrals Involving J_σ(x), x^γ and Trigonometric Functions 143

6.5.1 69^th General Formula 143

6.5.2 70^th General Formula 144

6.5.3 71^st General Formula 145

6.5.4 72^nd General Formula 146

6.6 Calculation of Particular Integrals 146

6.7 Integrals Containing Two J_σ(x), x^γ and Trigonometric Functions 152

6.7.1 73^rd General Formula 152

6.7.2 74^th General Formula 153

6.7.3 75^th General Formula 154

6.7.4 76^th General Formula 155

6.7.5 77^th General Formula 156

6.7.6 78^th General Formula 157

6.7.7 79^th General Formula 158

6.7.8 80^th General Formula 159

6.8 Exercises 1 160

6.9 Integrals Containing J_σ(x), x^γ, Trigonometric and Exponential Functions 161

6.9.1 81^st General Formula 161

6.9.2 82^nd General Formula 164

6.9.3 83^rd General Formula 166

6.9.4 84^th General Formula 167

6.10 Exercises 2 169

7 Integrals Involving the Neumann Function N_σ(x) 171

7.1 Definition of the Neumann Function 171

7.2 The Mellin Representation of N_σ(x) 171

7.3 85^th General Formula 172

7.4 86^th General Formula 172

7.5 87^th General Formula 173

7.6 88^th General Formula 174

7.7 89^th General Formula 175

7.8 90^th General Formula 176

7.9 91^st General Formula 177

7.10 Calculation of Concrete Integrals 178

8 Integrals Containing Other Cylindrical and Special Functions 181

8.1 Integrals Involving Modified Bessel Function of the Second Kind 181

8.1.1 Mellin Representations of K_δ(x) and I_λ(x) 181

8.1.2 92^nd General Formula 182

8.1.3 93^rd General Formula 182

8.1.4 94^th General Formula. 183

8.1.5 95^th Genera! Formula 183

8.1.6 96^th General Formula 184

8.1.7 97^th General Formula 185

8.1.8 98^th General Formula 186

8.1.9 99^th General Formula 187

8.1.10 100^th General Formula 188

8.1.11 Calculation of Concrete Integrals 188

8.1.12 101^st General Formula 193

8.1.13 102^nd General Formula 194

8.1.14 103^rd General Formula 195

8.1.15 104^th General Formula 197

8.1.16 Some Examples of Calculation of Integrals 198

8.2 Integrals Involving the Struve Function 199

8.2.1 105^th General Formula 199

8.2.2 106^th General Formula 200

8.2.3 107^th General Formula 200

8.2.4 108^th General Formula 201

8.2.5 Exercises 201

8.3 Integrals Involving Other Special Functions 203

8.3.1 Hypergeometric Functions, Order (1.1) 203

8.3.2 Hypergeometric Function 203

8.3.3 Tomson Function 203

8.3.4 Anger Function 203

8.3.5 Veber Function 204

8.3.6 Legendre's Function of the Second Kind 204

8.3.7 Complete Elliptic Integral of the First Kind 204

8.3.8 Complete Elliptic Integral of the Second Kind 204

8.3.9 Exponential Integral Functions 204

8.3.10 Sine Integral Function 205

8.3.11 Cosine Integral Function 205

8.3.12 Probability Integral 205

8.3.13 Frenel Functions 205

8.3.14 Incomplete Gamma Function 206

8.3.15 Psi-Functions Ψ(x) 206

8.3.16 Euler's Constant 206

8.3.17 Hankel Function 207

8.3.18 Cylindrical Function of Imaginary Arguments 207

8.4 Some Examples 207

9 Integrals Involving Two Trigonometric Functions 213

9.1 109^th General Formula 213

9.2 110^th General Formula 214

9.3 111^th General Formula 215

9.4 112^th General Formula 216

9.5 113^th General Formula 217

9.6 114^th General Formula 218

9.7 115^th General Formula 219

9.8 116^th General Formula 220

9.9 117^th General Formula 221

9.10 118^th General Formula 222

9.11 Exercises 223

9.12 Answers 225

9.13 An Example of a Solution 229

10 Derivation of Universal Formulas for Calculation of Fractional Derivatives and Inverse Operators 231

10.1 Introduction 231

10.2 Derivation of General Formulas for Taking Fractional Derivatives 232

10.2.1 The First General Formula 232

10.2.2 Consequences of the First General Formula 232

10.2.3 Addivity Properties of the Fractional Derivatives 238

10.2.4 Commutativity Properties of the Fractional Derivatives 239

10.2.5 Standard Case 239

10.2.6 Fractional Derivatives for sin ax, cos ax, e^ax and a^x-Functions 239

10.3 Derivation of General Formulas for Calculation of Fractional Derivatives for Some Infinite Differentiable Functions 240

10.3.1 The Second General Formula 240

10.3.2 Fractional Derivatives for the Functions: F(x) = e^-ax, e^ax and a^x 240

10.3.3 Fractional Derivatives for the Hyperbolic Functions F(x) = sinh ax, cosh ax 241

10.3.4 Fractional Derivatives for the 1/x, 1/x², In x, √x, 1/√x-Functions 241

10.3.5 Some Examples 244

10.3.6 Usual Case, When - 1/ν = n 245

10.4 Representation for Inverse Derivatives in the Form of Integer-Order Differentials 246

10.4.1 The Third General Formula 246

10.4.2 4^th General Formula 249

10.4.3 5^th General Formula 250

10.4.4 6^th General Formula 250

10.4.5 7^th General Formula 251

10.4.6 8^th General Formula 252

10.4.7 9^th General Formula 254

10.4.8 10^th General Formula 255

10.4.9 11^th General Formula 255

10.5 Fractional Integrals 256

10.5.1 Fractional Integrals for Sine and Cosine Functions 257

10.5.2 Fractional Integrals for Infinite Differentiable Functions 263

10.6 Final Table for Taking Fractional Derivatives of Some Elementary Functions 266

Appendix: Tables of the Definitions for Fractional Derivatives and Inverse Operators 269

A.1 Taking Fractional Derivatives 269

A.2 Calculation of Inverse Operators 271

Bibliography 277

Index 279