Singular Differential Equations and Special Functions

359 Pages·2019·9.446 MB·English

by  Luis Manuel Braga da Costa Campos

Singular Differential
Equations and Special
Functions
Mathematics and Physics for Science and Technology
Series Editor: L.M.B.C. Campos
Director of the Center for Aeronautical
and Space Science and Technology
Lisbon University
Volumes in the series:
Topic A – Theory of Functions and Potential Problems
Volume I (Book 1) – Complex Analysis with Applications to Flows
and Fields
L.M.B.C. Campos
Volume II (Book 2) – Elementary Transcendentals with Applications
to Solids and Fluids
L.M.B.C. Campos
Volume III (Book 3) – Generalized Calculus with Applications to Matter
and Forces
L.M.B.C. Campos
Topic B – Boundary and Initial-Value Problems
Volume IV – Ordinary Differential Equations with Applications
to Trajectories and Oscillations
L.M.B.C. Campos
Book 4 – Linear Differential Equations and Oscillators
L.M.B.C. Campos
Book 5 – Non-Linear Differential Equations and Dynamical Systems
L.M.B.C. Campos
Book 6 – Higher-Order Differential Equations and Elasticity
L.M.B.C. Campos
Book 7 – Simultaneous Differential Equations and Multi-Dimensional
Vibrations
L.M.B.C. Campos
Book 8 – Singular Differential Equations and Special Functions
L.M.B.C. Campos
Book 9 – Classification and Examples of Differential Equations and
their Applications
L.M.B.C. Campos
For more information about this series, please visit: https://www.crcpress.
com/Mathematics-and-Physics-for-Science-and-Technology/book-series/
CRCMATPHYSCI
Mathematics and Physics for Science and Technology
Volume IV
Ordinary Differential
Equations with Applications
to Trajectories and Oscillations
Book 8
Singular Differential
Equations and Special
Functions
By
L.M.B.C. Campos
Director of the Center for Aeronautical
and Space Science and Technology
Lisbon University
CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Campos, Luis Manuel Braga da Costa, author.
Title: Singular differential equations and special functions / Luis Manuel
Braga da Campos.
Description: Boca Raton : CRC Press, Taylor & Francis Group, 2018. | Includes
bibliographical references and index.
Identifiers: LCCN 2018050996 | ISBN 9780367137236 (hardback : alk. paper)
Subjects: LCSH: Functions, Special. | Differential equations. | Mathematical analysis.
Classification: LCC QA351 .C2525 2018 | DDC 515/.5–dc23
LC record available at https://lccn.loc.gov/2018050996
Visit the Taylor & Francis Web site at
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to Leonor Campos

Contents
Contents .................................................................................................................vii
Diagrams, List, Notes, and Tables ....................................................................xiii
Preface ..................................................................................................................xvii
Acknowledgments ..............................................................................................xix
About the Author ................................................................................................xxi
Notation for Functions .....................................................................................xxiii
9. Existence Theorems and Special Functions ..............................................1
9.1 Existence, Unicity, Robustness, and Uniformity of Solutions ........1
9.1.1 Existence of Solution and the Lipschitz Condition .............2
9.1.2 Transformation of a Differential into an Integral
Equation (Picard 1893) .............................................................3
9.1.3 Convergence of the Successive Approximations .................4
9.1.4 Contraction Mapping and Fixed Point .................................5
9.1.5 Lipschitz (1864) or Contraction Condition,
Continuity, and Differentiability ...........................................6
9.1.6 Continuity (Contraction) Condition on the
Independent (Dependent) Variable .......................................7
9.1.7 Uniform Convergence to a Continuous Solution ................9
9.1.8 Theorem of Existence of Solution of a Differential
Equation ..................................................................................10
9.1.9 Rectangle as the Domain of Successive
Approximations .....................................................................11
9.1.10 Theorem of Unicity of Solution of a Differential
Equation ..................................................................................11
9.1.11 Robustness Relative to Perturbed Initial Conditions .......12
9.1.12 Comparison of Successive Approximations to the
Original and Perturbed Problems .......................................14
9.1.13 Theorem of Robustness with Regard to the Initial
Condition.................................................................................15
9.1.14 Theorem on Uniformity with Regard to a Parameter
in a Differential Equation .....................................................16
9.1.15 Simultaneous System of First-Order Differential
Equations .................................................................................16
9.1.16 Contraction or Lipschitz Condition for Several
Variables ..................................................................................18
9.1.17 Combined Theorem of Existence, Unicity,
Robustness, and Uniformity ................................................19
9.1.18 Ordinary Differential Equation of Any Order ..................20
vii
viii Contents
9.1.19 Linear Ordinary Differential Equation ..............................21
9.1.20 Combined Theorem for an Ordinary Differential
Equation ..................................................................................22
9.2 Autonomous Systems and Stability of Equilibria
(Lyapunov 1954) ..................................................................................23
9.2.1 Local/Asymptotic Stability/Instability
and Indifference .....................................................................25
9.2.2 Positive/Negative Definite/Semi-Definite and
Indefinite Function ................................................................28
9.2.3 Derivative along an Unsteady/Autonomous
Differential System ................................................................29
9.2.4 Lyapunov (1966) Function and First Theorem ...................31
9.2.5 Proof of the Lyapunov Stability Theorem ..........................34
9.2.6 Proof of the Theorem on Asymptotic Stability ..................35
9.2.7 Proof of the Theorem on Instability ....................................36
9.2.8 Damped Oscillator with Parameters Dependent
in Position................................................................................37
9.2.9 Damped Oscillator with Parameters Depending on
Time .........................................................................................38
9.2.10 Stability of Undamped Oscillator with Parametric
Resonance ...............................................................................40
9.2.11 Exponential Asymptotic Growth or Decay ........................40
9.2.12 Proof of the Second Lyapunov Theorem (1892) .................41
9.2.13 Existence of at Least One Eigenvalue of an
Autonomous System ..............................................................43
9.2.14 Linear Differential Equation with Bounded
Coefficients .............................................................................44
9.2.15 Linear Autonomous System with Constant
Coefficients .............................................................................45
9.2.16 Linearized Differential System Near an Equilibrium
Point .........................................................................................46
9.2.17 Fundamental Solution of an Autonomous
Differential System ................................................................48
9.2.18 Properties of Positive-Definite Quadratic Forms ..............50
9.2.19 Stability Function and Its Derivative Following the
Differential System ................................................................51
9.2.20 Third Lyapunov Theorem on Autonomous Systems .......52
9.2.21 Stability of Solutions of an Autonomous Differential
Equation ..................................................................................53
9.3 Linear Differential Equations with Periodic Coefficients
(Floquet 1883; Lyapunov 1907) ..........................................................54
9.3.1 Conditions for the Existence of Periodic Solutions
(Floquet 1883) ..........................................................................54
9.3.2 Distinct and Coincident Eigenvalues and Exponents ......55
9.3.3 Natural Integrals and Asymptotic Stability ......................56
Contents ix
9.3.4 Diagonal Matrices and Jordan Blocks .................................57
9.3.5 Invariant Second-Order Differential Equation ..................59
9.3.6 Growing/Decaying and Monotonic/Oscillatory
Solutions ..................................................................................60
9.3.7 Fundamental Solution of a Linear Differential
Equation ..................................................................................61
9.3.8 Eigenvalues of the Invariant Second Order Equation ......62
9.3.9 Proof of the Fourth Lyapunov (1907) Theorem .................63
9.4 Analytic Coefficients and Generalized Circular/Hyperbolic
Functions ..............................................................................................64
9.4.1 Singularities of Single-Valued Complex Functions ..........65
9.4.2 Regular Points, Poles, and Essential Singularities ............68
9.4.3 Sheets of the Riemann Surface of a Multi-Valued
Function...................................................................................70
9.4.4 Principal Branch, Branch-Point, and Branch-Cut ..............72
9.4.5 Regular Points and Regular/Irregular Singularities ........74
9.4.6 Analytic, Regular, and Irregular Integrals .........................75
9.4.7 Singularities and Integrals at the Point-at-Infinity ...........78
9.4.8 Linear Autonomous System with Analytic
Coefficients .............................................................................79
9.4.9 Linear Differential Equation with Analytic
Coefficients .............................................................................81
9.4.10 Calculation of the Coefficients of the Analytic
Solution ....................................................................................82
9.4.11 Two Methods to Obtain the Recurrence Relation
for the Coefficients .................................................................84
9.4.12 Generalized Hyperbolic Differential Equation .................85
9.4.13 Generalized Hyperbolic Cosine and Sine ..........................86
9.4.14 Airy (1838) Differential Equation and Functions ..............88
9.4.15 Generalized Circular Cosine and Sine ...............................88
9.4.16 Generalized Circular Differential Equation ......................90
9.4.17 Complex Non-Integer Values of the Parameter .................90
9.4.18 Differentiation of the Generalized Cosine and Sine
Functions .................................................................................92
9.4.19 Inequalities for Generalized Cosines and Sines ................93
9.4.20 Generalized Secant, Cosecant, Tangent, and
Cotangent ................................................................................94
9.5 Regular Singularities and Integrals of Two Kinds
(Fuchs 1860, Frobenius 1873)..............................................................96
9.5.1 Linear Autonomous System with a Regular
Singularity ..............................................................................97
9.5.2 Indices and Coefficients of the Regular Natural
Integrals ...................................................................................98
9.5.3 Two Related Sets of Regular Natural Integrals .................99
9.5.4 Application of Compatibility and Initial Conditions .....101