Preface xv

Acknowledgments xix

About the Companion Website xxi

**1 Introduction 1**

1.1 Background 2

1.2 The Fractional Integral and Derivative 3

1.3 The Traditional Trigonometry 6

1.4 Previous Efforts 8

1.5 Expectations of a Generalized Trigonometry and Hyperboletry 8

**2 The Fractional Exponential Function via the Fundamental Fractional Differential Equation 9**

2.1 The Fundamental Fractional Differential Equation 9

2.2 The Generalized Impulse Response Function 10

2.3 Relationship of the F-function to the Mittag-Leffler Function 11

2.4 Properties of the F-Function 12

2.5 Behavior of the F-Function as the Parameter a Varies 13

2.6 Example 16

**3 The Generalized Fractional Exponential Function: The R-Function and Other Functions for the Fractional Calculus 19**

3.1 Introduction 19

3.2 Functions for the Fractional Calculus 19

3.3 The R-Function: A Generalized Function 22

3.4 Properties of the Rq,v(a, t)-Function 23

3.5 Relationship of the R-Function to the Elementary Functions 27

3.6 R-Function Identities 29

3.7 Relationship of the R-Function to the Fractional Calculus Functions 31

3.8 Example: Cooling Manifold 32

3.9 Further Generalized Functions: The G-Function and the H-Function 34

3.10 Preliminaries to the Fractional Trigonometry Development 38

3.11 Eigen Character of the R-Function 38

3.12 Fractional Differintegral of the TimeScaled R-Function 39

3.13 R-Function Relationships 39

3.14 Roots of Complex Numbers 40

3.15 Indexed Forms of the R-Function 41

3.16 Term-by-Term Operations 44

3.17 Discussion 46

**4 R-Function Relationships 47**

4.1 R-Function Basics 47

4.2 Relationships for Rm,0 in Terms of R1,0 48

4.3 Relationships for R1¨Mm,0 in Terms of R1,0 50

4.4 Relationships for the Rational Form Rm¨Mp,0 in Terms of R1¨Mp,0 51

4.5 Relationships for R1¨Mp,0 in Terms of Rm¨Mp,0 53

4.6 Relating Rm¨Mp,0 to the Exponential Function R1,0(b, t) = ebt 54

4.7 Inverse Relationships–Relationships for R1,0 in Terms of Rm,k 56

4.8 Inverse Relationships–Relationships for R1,0 in Terms of R1¨Mm,0 57

4.9 Inverse Relationships–Relationships for eat = R1,0(a, t) in Terms of Rm¨Mp,0 59

4.10 Discussion 61

**5 The Fractional Hyperboletry 63**

5.1 The Fractional R1-Hyperbolic Functions 63

5.2 R1-Hyperbolic Function Relationship 72

5.3 Fractional Calculus Operations on the R1-Hyperbolic Functions 72

5.4 Laplace Transforms of the R1-Hyperbolic Functions 73

5.5 Complexity-Based Hyperbolic Functions 73

5.6 Fractional Hyperbolic Differential Equations 74

5.7 Example 76

5.8 Discussions 77

**6 The R1-Fractional Trigonometry 79**

6.1 R1-Trigonometric Functions 79

6.2 R1-Trigonometric Function Interrelationship 88

6.3 Relationships to R1-Hyperbolic Functions 89

6.4 Fractional Calculus Operations on the R1-Trigonometric Functions 89

6.5 Laplace Transforms of the R1-Trigonometric Functions 90

6.6 Complexity-Based R1-Trigonometric Functions 92

6.7 Fractional Differential Equations 94

**7 The R2-Fractional Trigonometry 97**

7.1 R2-Trigonometric Functions: Based on Real and Imaginary Parts 97

7.2 R2-Trigonometric Functions: Based on Parity 102

7.3 Laplace Transforms of the R2-Trigonometric Functions 111

7.4 R2-Trigonometric Function Relationships 113

7.5 Fractional Calculus Operations on the R2-Trigonometric Functions 119

7.6 Inferred Fractional Differential Equations 127

**8 The R3-Trigonometric Functions 129**

8.1 The R3-Trigonometric Functions: Based on Complexity 129

8.2 The R3-Trigonometric Functions: Based on Parity 134

8.3 Laplace Transforms of the R3-Trigonometric Functions 140

8.4 R3-Trigonometric Function Relationships 141

8.5 Fractional Calculus Operations on the R3-Trigonometric Functions 146

**9 The Fractional Meta-Trigonometry 159**

9.1 The FractionalMeta-Trigonometric Functions: Based on Complexity 160

9.2 The Meta-Fractional Trigonometric Functions: Based on Parity 166

9.3 Commutative Properties of the Complexity and Parity Operations 179

9.4 Laplace Transforms of the FractionalMeta-Trigonometric Functions 188

9.5 R-Function Representation of the FractionalMeta-Trigonometric Functions 192

9.6 Fractional Calculus Operations on the Fractional Meta-Trigonometric Functions 195

9.7 Special Topics in Fractional Differintegration 206

9.8 Meta-Trigonometric Function Relationships 206

9.9 Fractional Poles: Structure of the Laplace Transforms 214

9.10 Comments and Issues Relative to the Meta-Trigonometric Functions 214

9.11 Backward Compatibility to Earlier Fractional Trigonometries 215

9.12 Discussion 215

**10 The Ratio and Reciprocal Functions 217**

10.1 Fractional Complexity Functions 217

10.2 The Parity Reciprocal Functions 219

10.3 The Parity Ratio Functions 221

10.4 R-Function Representation of the Fractional Ratio and Reciprocal Functions 225

10.5 Relationships 226

10.6 Discussion 227

**11 Further Generalized Fractional Trigonometries 229**

11.1 The G-Function-Based Trigonometry 229

11.2 Laplace Transforms for the G-Trigonometric Functions 230

11.3 The H-Function-Based Trigonometry 234

11.4 Laplace Transforms for the H-Trigonometric Functions 235

**12 The Solution of Linear Fractional Differential Equations Based on the Fractional Trigonometry 243**

12.1 Fractional Differential Equations 243

12.2 Fundamental Fractional Differential Equations of the First Kind 245

12.3 Fundamental Fractional Differential Equations of the Second Kind 246

12.4 Preliminaries–Laplace Transforms 246

12.5 Fractional Differential Equations of Higher Order: Unrepeated Roots 250

12.6 Fractional Differential Equations of Higher Order: Containing Repeated Roots 252

12.7 Fractional Differential Equations Containing Repeated Roots 253

12.8 Fractional Differential Equations of Non-Commensurate Order 254

12.9 Indexed Fractional Differential Equations: Multiple Solutions 255

12.10 Discussion 256

**13 Fractional Trigonometric Systems 259**

13.1 The R-Function as a Linear System 259

13.2 R-System Time Responses 260

13.3 R-Function-Based Frequency Responses 260

13.4 Meta-Trigonometric Function-Based Frequency Responses 261

13.5 FractionalMeta-Trigonometry 264

13.6 Elementary Fractional Transfer Functions 266

13.7 Stability Theorem 266

13.8 Stability of Elementary Fractional Transfer Functions 267

13.9 Insights into the Behavior of the Fractional Meta-Trigonometric Functions 268

13.10 Discussion 270

**14 Numerical Issues and Approximations in the Fractional Trigonometry 271**

14.1 R-Function Convergence 271

14.2 The Meta-Trigonometric Function Convergence 272

14.3 Uniform Convergence 273

14.4 Numerical Issues in the Fractional Trigonometry 274

14.5 The R2Cos- and R2Sin-Function Asymptotic Behavior 275

14.6 R-Function Approximations 276

14.7 The Near-Order Effect 279

14.8 High-Precision Software 281

**15 The Fractional Spiral Functions: Further Characterization of the Fractional Trigonometry 283**

15.1 The Fractional Spiral Functions 283

15.2 Analysis of Spirals 288

15.3 Relation to the Classical Spirals 303

15.4 Discussion 307

**16 Fractional Oscillators 309**

16.1 The Space of Linear Fractional Oscillators 309

16.2 Coupled Fractional Oscillators 314

**17 Shell Morphology and Growth 317**

17.1 Nautilus pompilius 317

17.2 Shell 5 329

17.3 Shell 6 330

17.4 Shell 7 332

17.5 Shell 8 332

17.6 Shell 9 336

17.7 Shell 10 336

17.8 Ammonite 339

17.9 Discussion 340

**18 Mathematical Classification of the Spiral and Ring Galaxy Morphologies 341**

18.1 Introduction 341

18.2 Background–Fractional Spirals for Galactic Classification 342

18.3 Classification Process 347

18.4 Mathematical Classification of Selected Galaxies 350

18.5 Analysis 362

18.6 Discussion 367

18.7 Appendix: Carbon Star 370

**19 Hurricanes, Tornados, and Whirlpools 371**

19.1 Hurricane Cloud Patterns 371

19.2 Tornado Classification 373

19.3 Low-Pressure Cloud Pattern 375

19.4 Whirlpool 375

19.5 Order in Physical Systems 379

**20 A Look Forward 381**

20.1 Properties of the R-Function 382

20.2 Inverse Functions 382

20.3 The Generalized Fractional Trigonometries 384

20.4 Extensions to Negative Time, Complementary Trigonometries, and Complex Arguments 384

20.5 Applications: Fractional Field Equations 385

20.6 Fractional Spiral and Nonspiral Properties 387

20.7 Numerical Improvements for Evaluation to Larger Values of atq 387

20.8 Epilog 388

**A Related Works 389**

A.1 Introduction 389

A.2 Miller and Ross 389

A.3 West, Bologna, and Grigolini 390

A.4 Mittag-Leffler-Based Fractional Trigonometric Functions 390

A.5 Relationship to CurrentWork 391

**B Computer Code 393**

B.1 Introduction 393

B.2 MatlabⓇ R-Function 393

B.3 MatlabⓇ R-Function Evaluation Program 394

B.4 MatlabⓇ Meta-Cosine Function 395

B.5 MatlabⓇ Cosine Evaluation Program 395

B.6 MapleⓇ 10 Program Calculates Phase Plane Plot for Fractional Sine versus Cosine 396

**C Tornado Simulation 399**

**D Special Topics in Fractional Differintegration 401**

D.1 Introduction 401

D.2 Fractional Integration of the Segmented tp-Function 401

D.3 Fractional Differentiation of the Segmented tp-Function 404

D.4 Fractional Integration of Segmented Fractional Trigonometric Functions 406

D.5 Fractional Differentiation of Segmented Fractional Trigonometric Functions 408

**E Alternate Forms 413**

E.1 Introduction 413

E.2 Reduced Variable Summation Forms 414

E.3 Natural Quency Simplification 415

References 417

Index 425