An Engineering Approach to Asymptotics and Approximations

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