Singular Differential Equations and Special Functions

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Diagrams, List, Notes, and Tables
Preface
Acknowledgments
About the Author
Notation for Functions
9. Existence Theorems and Special Functions
	9.1. Existence, Unicity, Robustness, and Uniformity of Solutions
		9.1.1. Existence of Solution and the Lipschitz Condition
		9.1.2. Transformation of a Differential into an Integral Equation (Picard 1893)
		9.1.3. Convergence of the Successive Approximations
		9.1.4. Contraction Mapping and Fixed Point
		9.1.5. Lipschitz (1864) or Contraction Condition, Continuity, and Differentiability
		9.1.6. Continuity (Contraction) Condition on the Independent (Dependent) Variable
		9.1.7. Uniform Convergence to a Continuous Solution
		9.1.8. Theorem of Existence of Solution of a Differential Equation
		9.1.9. Rectangle as the Domain of Successive Approximations
		9.1.10. Theorem of Unicity of Solution of a Differential Equation
		9.1.11. Robustness Relative to Perturbed Initial Conditions
		9.1.12. Comparison of Successive Approximations to the Original and Perturbed Problems
		9.1.13. Theorem of Robustness with Regard to the Initial Condition
		9.1.14. Theorem on Uniformity with Regard to a Parameter in a Differential Equation
		9.1.15. Simultaneous System of First-Order Differential Equations
		9.1.16. Contraction or Lipschitz Condition for Several Variables
		9.1.17. Combined Theorem of Existence, Unicity, Robustness, and Uniformity
		9.1.18. Ordinary Differential Equation of Any Order
		9.1.19. Linear Ordinary Differential Equation
		9.1.20. Combined Theorem for an Ordinary Differential Equation
	9.2. Autonomous Systems and Stability of Equilibria (Lyapunov 1954)
		9.2.1. Local/Asymptotic Stability/Instability and Indifference
		9.2.2. Positive/Negative Definite/Semi-Definite and Indefinite Function
		9.2.3. Derivative along an Unsteady/Autonomous Differential System
		9.2.4. Lyapunov (1966) Function and First Theorem
		9.2.5. Proof of the Lyapunov Stability Theorem
		9.2.6. Proof of the Theorem on Asymptotic Stability
		9.2.7. Proof of the Theorem on Instability
		9.2.8. Damped Oscillator with Parameters Dependent in Position
		9.2.9. Damped Oscillator with Parameters Depending on Time
		9.2.10. Stability of Undamped Oscillator with Parametric Resonance
		9.2.11. Exponential Asymptotic Growth or Decay
		9.2.12. Proof of the Second Lyapunov Theorem (1892)
		9.2.13. Existence of at Least One Eigenvalue of an Autonomous 
System
		9.2.14. Linear Differential Equation with Bounded Coefficients
		9.2.15. Linear Autonomous System with Constant Coefficients
		9.2.16. Linearized Differential System Near an Equilibrium Point
		9.2.17. Fundamental Solution of an Autonomous Differential System
		9.2.18. Properties of Positive-Definite Quadratic Forms
		9.2.19. Stability Function and Its Derivative Following the Differential System
		9.2.20. Third Lyapunov Theorem on Autonomous Systems
		9.2.21. Stability of Solutions of an Autonomous Differential Equation
	9.3. Linear Differential Equations with Periodic Coefficients (Floquet 1883; Lyapunov 1907)
		9.3.1. Conditions for the Existence of Periodic Solutions (Floquet 1883)
		9.3.2. Distinct and Coincident Eigenvalues and Exponents
		9.3.3. Natural Integrals and Asymptotic Stability
		9.3.4. Diagonal Matrices and Jordan Blocks
		9.3.5. Invariant Second-Order Differential Equation
		9.3.6. Growing/Decaying and Monotonic/Oscillatory Solutions
		9.3.7. Fundamental Solution of a Linear Differential Equation
		9.3.8. Eigenvalues of the Invariant Second Order Equation
		9.3.9. Proof of the Fourth Lyapunov (1907) Theorem
	9.4. Analytic Coefficients and Generalized Circular/Hyperbolic Functions
		9.4.1. Singularities of Single-Valued Complex Functions
		9.4.2. Regular Points, Poles, and Essential Singularities
		9.4.3. Sheets of the Riemann Surface of a Multi-Valued Function
		9.4.4. Principal Branch, Branch-Point, and Branch-Cut
		9.4.5. Regular Points and Regular/Irregular Singularities
		9.4.6. Analytic, Regular, and Irregular Integrals
		9.4.7. Singularities and Integrals at the Point-at-Infinity
		9.4.8. Linear Autonomous System with Analytic Coefficients
		9.4.9. Linear Differential Equation with Analytic Coefficients
		9.4.10. Calculation of the Coefficients of the Analytic Solution
		9.4.11. Two Methods to Obtain the Recurrence Relation for the Coefficients
		9.4.12. Generalized Hyperbolic Differential Equation
		9.4.13. Generalized Hyperbolic Cosine and Sine
		9.4.14. Airy (1838) Differential Equation and Functions
		9.4.15. Generalized Circular Cosine and Sine
		9.4.16. Generalized Circular Differential Equation
		9.4.17. Complex Non-Integer Values of the Parameter
		9.4.18. Differentiation of the Generalized Cosine and Sine Functions
		9.4.19. Inequalities for Generalized Cosines and Sines
		9.4.20. Generalized Secant, Cosecant, Tangent, and Cotangent
	9.5. Regular Singularities and Integrals of Two Kinds (Fuchs 1860, Frobenius 1873)
		9.5.1. Linear Autonomous System with a Regular Singularity
		9.5.2. Indices and Coefficients of the Regular Natural Integrals
		9.5.3. Two Related Sets of Regular Natural Integrals
		9.5.4. Application of Compatibility and Initial Conditions
		9.5.5. Linear Differential Equation with Regular Singularities (Fuchs 1868)
		9.5.6. Indicial Equation and Recurrence Formula for the Coefficients
		9.5.7. Regular Integrals of the First and Second Kinds
		9.5.8. Second-Order Differential Equation with a Regular Singularity
		9.5.9. Regular Integral of Second Kind First Type
		9.5.10. Regular Integral of the Second Kind Second Type
		9.5.11. Convergence of the Series in the Regular Integrals
		9.5.12. Wronskian of a Linear Second-Order Differential Equation
		9.5.13. Gamma Function (Weierstrass 1856) as a Generalization of the Factorial
		9.5.14. Generalized Bessel Differential Equation (Campos et al. 2019)
		9.5.15. Order and Degree of the Generalized Bessel Function
		9.5.16. The Digamma Function as the Logarithmic Derivative of the Gamma Function
		9.5.17. Generalized Neumann Function of Integer Order
		9.5.18. Preliminary and Complementary Functions and Logarithmic Term
		9.5.19. General Integral in the Case of Integer Order
		9.5.20. Wronskians of Generalized Bessel and Neumann Functions
		9.5.21. Generalized Neumann Function with Complex Order
		9.5.22. Cylindrical Bessel Differential Equation (Euler 1764, Bessel 1824, Neumann 1867, Hankel 1869)
		9.5.23. Spherical Bessel Differential Equation
		9.5.24. Spherical Bessel and Neumann Functions
		9.5.25. Classification of Six Cases of Regular Integrals
		9.5.26. Relation between Spherical Bessel and Elementary Functions
	9.6. Irregular Singularities and Asymptotic Normal Integrals (Thome 1883, Poincaré 1886, Campos 2001)
		9.6.1. Regular/Irregular Integrals of a Linear Differential Equation
		9.6.2. Regular Integral near an Irregular Singularity
		9.6.3. Existence of a Normal Integral near an Irregular Singularity
		9.6.4. Singularity of a Differential Equation at the Point at Infinity
		9.6.5. Convergent Series versus Asymptotic Expansions
		9.6.6. Spherical and Cylindrical Hankel (1869) Functions
		9.6.7. Asymptotic Integrals of the Cylindrical Bessel Differential Equation (Campos 2001)
		9.6.8. Asymptotic Expansions for Bessel, Neumann, and Hankel Functions
		9.6.9. Generalized Hankel Function of the First Kind
		9.6.10. Generalized Hankel Function of the Second Kind
		9.6.11. Wronskians and Alternative General Integrals
		9.6.12. Generalized versus Original Bessel Differential Equations
		9.6.13. Parametric Discontinuity in the Asymptotic Integrals
		9.6.14. Bifurcation for the Degree at the Point at Infinity
		9.6.15. Numerical Computation for Small and Large Variable
	9.7. Essential Singularities and Infinite Determinants
		9.7.1. Existence of Eigenvalues and of Eigenintegrals
		9.7.2. Irregular Integrals of the First Kind
		9.7.3. Case of Indices Differing by an Integer
		9.7.4. Irregular Integral of Second Kind
		9.7.5. Regular Integrals of the Second Kind First/Second Type
		9.7.6. Irregular, Regular, and Analytic Integrals
		9.7.7. Infinite Linear System for the Coefficients
		9.7.8. Indices as Roots of an Infinite Determinant
		9.7.9. Five Types of Convergence of Infinite Determinants (Poincare 1886)
		9.7.10. Convergence Theorem for Infinite Determinants (Von Koch 1892)
		9.7.11. Parametric Resonance Forced by a Fundamental and Harmonics
		9.7.12. Hill (1892) Differential Equation with Trigonometric Coefficients
		9.7.13. Hill Differential Equation with Power Coefficients
		9.7.14. Infinite Linear System for the Coefficients
		9.7.15. Exact Analytic Evaluation of the Hill Determinant
		9.7.16. Indices as Roots of a Transcendental Equation
	9.8. Kernel and Path of Integral Transforms (Laplace 1812; Bateman 1909)
		9.8.1. Eulerian Integral of the Second Kind (Euler 1772)
		9.8.2. Permutations and Properties of the Beta Function
		9.8.3. Relation between the Gamma and Beta Functions
		9.8.4. Evaluation of Integrals via Gamma and Beta Functions
		9.8.5. Kernel and Path of a General Integral Transform
		9.8.6. Generalized Laplace Transform in the Complex Plane
		9.8.7. Depression of the Degree of the Coefficients of a Differential Equation
		9.8.8. Application to the Generalized Bessel Differential Equation
		9.8.9. Integral Representation for the Generalized Bessel Function
		9.8.10. Application to the Original Bessel Differential Equation
		9.8.11. Integral Representation for the Bessel Function
		9.8.12. Partial Differential Equation for the Bateman (1909) Kernel
		9.8.13. Adjoint Operator and Bilinear Concomitant
		9.8.14. Integral Representations for the Two Hankel Functions
		9.8.15. Integral Representation for the Neumann Function
		9.8.16. Second Integral Representation for the Bessel Function
	9.9. Continued Fractions as Solutions of Differential Equations
		9.9.1. Continued Fraction Associated with a Differential Equation
		9.9.2. Logarithmic Derivative of the Solution of a Differential Equation
		9.9.3. Solution as a Terminating Continued Fraction
		9.9.4. Confluent Hypergeometric Differential Equation (Kummer 1830)
		9.9.5. Convergence of Simple Infinite Continued Fractions
		9.9.6. Continued Fraction for the Confluent Hypergeometric Function
		9.9.7. Generalized Bessel and Confluent Hypergeometric Differential Equations
		9.9.8. Confluent Hypergeometric Series
		9.9.9. Continued Fraction for the Generalized Bessel Function
		9.9.10. Recurrence and Differentiation Relations for Cylinder Functions
		9.9.11. Cylinder Functions with Integer Order Difference
		9.9.12. A Continued Fraction for Cylinder Functions
		9.9.13. Special Limit Periodic Continued Fractions
		9.9.14. Convergence of Complex Continued Fractions
		9.9.15. Convergence of Special Continued Fractions
	Conclusion 9
Bibliography
Index