The objective of this textbook is twofold-to introduce advanced novel mathematical techniques and then to use them to solve a variety of problems that may arise in various applications. The novel mathematical techniques, covered in Part I of the text, involve the use of generalized hypergeometric functions, including the Meijer G-function. We first present an introduction to asymptotic analysis for both small and large arguments of special functions and then use these asymptotic expressions to develop accurate algebraic approximations for generalized hypergeometric functions. In Part II we apply these novel mathematical techniques to a number of problems in electromagnetic wave propagation, imaging through atmospheric turbulence, and non-Kolmogorov turbulence. The idea is to present the mathematical details that are often missing in the literature in the development of various statistical models in these application areas and to introduce new methods for solution of the challenging equations that result from optical turbulence. In many cases we present exact results followed by accurate algebraic approximations. All required mathematical identities and integrals are presented in five appendices at the end of the text. In writing this text, the authors have tried to present the material in sufficient detail to reach an audience with limited knowledge of the subjects covered. To aid the reader in this regard, the text comprises a large number of worked examples in most chapters.
Preface Glossary of Symbols and Acronyms 1 Review of Complex Variables 1.0 Introduction 1.1 Complex Functions 1.1.1 Analytic functions 1.1.2 Elementary complex functions 1.2 Complex Integration 1.2.1 Deformation of contours 1.3 Taylor Series and Laurent Series 1.3.1 Analytic continuation 1.4 Singularities 1.4.1 Residues 1.5 Applications 1.6 Historical Remarks Reference Bibliography of Further Reading Part I: Special Functions, Asymptotic Analysis, and Approximations 2 Basic Concepts 2.0 Introduction 2.1 The Gamma Function 2.2 The Pochhammer Symbol 2.3 Asymptotic Series and the Cauchy Product 2.3.1 Large arguments of the function 2.3.2 Stirling's formula 2.3.3 The Cauchy product 2.4 The Mellin Transform 2.4.1 Residue theory 2.4.2 Convolution integral 2.5 Mellin-Barnes Integral 2.6 Zernike Polynomials 2.6.1 Application in optics 2.7 Applications 2.7.1 Asymptotic series 2.7.2 The Mellin transform 2.8 Historical Remarks References Generalized Hypergeometric Functions 3.0 Introduction 3.1 Generalized Hypergeometric Series 3.2 The Function 1F0 3.2.1 Large-argument asymptotic series 3.3 The Function 1F1 3.3.1 The Barnes integral 3.3.2 Large-argument asymptotic series 3.3.3 Elementary properties 3.4 The Function 1Fk 3.4.1 Large-argument asymptotic series 3.5 The Function 2F1 3.5.1 The Barnes integral 3.5.2 Large-argument asymptotic series 3.5.3 Elementary properties 3.6 The Function 2F2 3.6.1 Large-argument asymptotic series 3.7 The Function pFq 3.8 The Function U(a; c; z) 3.8.1 The Barnes integral 3.8.2 Large-argument asymptotic series 3.9 Whittaker Functions 3.10 Asymptotic Behavior of Other Special Functions 3.10.1 Error functions 3.10.2 Fresnel integrals 3.10.3 Incomplete gamma functions 3.10.4 The exponential integral 3.10.5 Sine and cosine integrals 3.10.6 Complete elliptic integrals 3.11 Applications 3.11.1 Summing series 3.11.2 Probability and statistics 3.11.3 Elliptic integrals 3.12 Historical Remarks References 4 Bessel Functions 4.0 Introduction 4.1 Standard Bessel Functions 4.1.1 The function Jp(z) 4.1.2 Large-argument asymptotic approximation 4.1.3 The function Yp(z) 4.1.4 Spherical Bessel functions 4.2 Modified Bessel Functions 4.2.1 The function Ip(z) 4.2.2 The function Kp(z) 4.2.3 Modified spherical Bessel functions 4.3 Other Bessel Functions 4.3.1 Hankel functions 4.3.2 Struve functions 4.3.3 Kelvin's functions 4.3.4 Airy functions 4.3.5 Bessel-integral function 4.3.6 Anger and Weber functions 4.3.7 Lommel functions 4.4 Applications 4.4.1 Differential equations related to Bessel's equation 4.4.2 Oscillations of a hanging chain 4.4.3 Other Bessel functions 4.5 Historical Remarks References 5 Meijer G-Function 5.1 The Meijer G-Function Definitions 5.2 Elementary Properties 5.2.1 Reduction formulas 5.2.2 Integral formulas 5.3 Asymptotic Relations for Large Arguments 5.3.1 The function 1F1 5.3.2 The function 1Fk 5.3.3 The function 2F1 5.3.4 The function 3F3 5.4 MacRobert E-Function 5.5 Comments on the G-Function 5.6 Historical Remarks References 6 Approximations 6.0 Introduction 6.1 Approximations for the 2F1 Function 6.2 Approximations for the 1F1 Function 6.3 Other Functions 6.3.1 Error function 6.3.2 Bessel functions 6.3.3 Modified Bessel functions 6.3.4 Modified Struve function 6.3.5 Kelvin's functions 6.4 Discussion Reference Part II: Applications 7 Integrals and Products 7.0 Introduction 7.1 Miscellaneous Integrals 7.1.1 Elementary functions 7.1.2 Bessel functions 7.1.3 Hypergeometric functions 7.2 Products of Special Functions 7.2.1 0F1 functions 7.2.2 1F1 and 2F1 functions 7.2.3 Saalschuetz's theorem 7.3 Products and Integrals of Bessel Functions 7.3.1 Product of two Bessel functions 7.3.2 Product of three Bessel functions 7.4 Laplace Transform 7.4.1 Error function 7.4.2 Bessel functions 7.4.3 Generalized hypergeometric functions 7.4.4 Miscellaneous functions 7.5 Mellin Transform 7.5.1 Elementary functions 7.5.2 Bessel functions 7.6 Hankel Transform 7.6.1 Elementary functions 7.6.2 Bessel functions 7.7 Applications of Integral Transforms 7.7.1 Heat conduction in a long rod 7.7.2 Probability and statistics 7.7.3 Summation of series 7.7.4 Optical wave propagation 7.7.5 Steady-state heat conduction in a cylinder 7.7.6 Fourier transform 7.7.7 Inverse transforms 7.8 Discussion 7.9 Historical Remarks References 8 Electromagnetic Wave Propagation 8.0 Introduction 8.1 Atmospheric Turbulence 8.1.1 Random fields 8.1.2 Power spectrum models 8.1.3 Index-of-refraction structure function 8.2 Optical/IR Wave Propagation 8.2.1 Propagation in free space 8.2.2 Rytov approximation 8.2.3 Comments on the exponential spectrum model 8.3 Mutual Coherence Function 8.3.1 WSF for a plane wave 8.3.2 Outer-scale effect 8.3.3 WSF for a spherical wave 8.3.4 WSF for a Gaussian-beam wave 8.3.5 Long-term beam radius 8.4 Scintillation Index 8.4.1 Plane wave 8.4.2 Spherical wave 8.4.3 Gaussian-beam wave 8.5 Irradiance Covariance Function: Weak Fluctuations 8.5.1 Plane wave 8.5.2 Plane wave: aperture averaging 8.5.3 Spherical wave 8.5.4 Spherical wave: aperture averaging 8.5.5 Gaussian-beam wave 8.6 Irradiance Covariance Function: Strong Fluctuations 8.6.1 Plane wave 8.6.2 Spherical wave 8.7 Temporal Spectrum of Irradiance 8.7.1 Plane wave 8.7.2 Plane wave: aperture averaging 8.7.3 Spherical wave 8.8 Phase Statistics: Plane Wave 8.8.1 Phase structure function 8.8.2 Angle-of-arrival 8.8.3 Phase variance 8.8.4 Phase covariance function 8.8.5 Temporal spectrum of phase 8.9 Phase Statistics: Spherical Wave 8.9.1 Phase structure function 8.9.2 Phase covariance function 8.10 Discussion 8.11 Historical Remarks References 9 Imaging Through Atmospheric Turbulence 9.0 Introduction 9.1 Imaging Performance Measures 9.1.1 Point spread function and modulation transfer function 9.1.2 Short-exposure imaging 9.1.3 Strehl ratio 9.1.4 Strehl ratio: outer-scale effect 9.1.5 Strehl ratio with tilt removed 9.1.6 Long-exposure resolution 9.1.7 Short-exposure resolution 9.2 Zernike Polynomials and Aperture Filter Functions 9.2.1 Zernike-tilt variance 9.2.2 Zernike-tilt variance: outer-scale effect 9.2.3 Gradient-tilt variance 9.2.4 Zernike-tilt power spectral density 9.3 Anisoplanatism 9.3.1 Laser guide star 9.4 Tilt Anisoplanatism 9.4.1 Multi-aperture case 9.4.2 Multi-aperture: outer-scale effect 9.4.3 Multi-source case 9.4.4 Multi-source: outer-scale effect 9.5 Angular and Focal Anisoplanatism 9.5.1 LGS anisoplanatic error: zero NGS offset 9.5.2 LGS anisoplanatic error: nonzero NGS offset 9.5.3 Focus anisoplanatism 9.6 Discussion References 10 Non-Kolmogorov Turbulence 10.0 Introduction 10.1 Non-Kolmogorov Spectral Models 10.1.1 Isotropic models 10.1.2 Anisotropic models 10.1.3 Index-of-refraction structure function 10.2 Generalized Rytov Variance 10.3 Wave Structure Function: Isotropic Turbulence 10.3.1 Plane wave 10.3.2 Spherical wave 10.3.3 Gaussian-beam wave 10.4 Wave Structure Function: Anisotropic Turbulence 10.5 Scintillation Index: Weak Irradiance Fluctuations 10.5.1 Plane wave: isotropic turbulence 10.5.2 Spherical wave: isotropic turbulence 10.5.3 Gaussian-beam wave: anisotropic turbulence 10.6 Scintillation Index: Strong Irradiance Fluctuations 10.6.1 Plane wave: isotropic turbulence 10.6.2 Plane wave: anisotropic turbulence 10.6.3 Spherical wave: isotropic turbulence 10.6.4 Gaussian-beam wave: anisotropic turbulence 10.7 Aperture Averaging: Weak Irradiance Fluctuations 10.7.1 Flux variance: plane wave 10.7.2 Flux variance: spherical wave 10.8 Irradiance Covariance Function 10.8.1 Plane wave 10.8.2 Spherical wave 10.9 Temporal Spectrum of Irradiance 10.9.1 Plane wave 10.9.2 Spherical wave 10.10 Phase Statistics: Plane Wave 10.10.1 Phase structure function 10.10.2 Phase variance 10.11 Imaging Performance Measures 10.11.1 Long-term beam radius: detector plane 10.11.2 Point spread function and modulation transfer function 10.11.3 Strehl ratio 10.12 Tilt Variances 10.12.1 Zernike-tilt variance 10.12.2 Gradient-tilt variance 10.12.3 Zernike-tilt power spectral density 10.13 Tilt Anisoplanatism 10.13.1 Multi-apertures 10.13.2 Multi-source 10.14 Focus Anisoplanatism 10.15 Discussion References 11 Miscellaneous Applications 11.0 Introduction 11.1 Statistical Communication Theory 11.1.1 Nonlinear devices 11.1.2 Ideal bandpass limiter 11.1.3 Joint moments of the envelopes 11.2 Cross Correlators 11.2.1 Cross correlator with limiter in one channel 11.2.2 Polarity coincidence correlator 11.3 Free-Space Optical Communication 11.3.1 Threshold detection 11.3.2 Performance measures 11.4 Fluid Mechanics 11.4.1 Unsteady hydrodynamic flow past an infinite plate 11.4.2 Irrotational flow of an ideal fluid References Appendix A: Special Function Identities Appendix B: Formulas for the Generalized Hypergeometric Functions Appendix C: Formulas for the Meijer G-Function Appendix D: Integral Table Appendix E: Asymptotic Expressions Index
