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Ordinary Differential Equations in the Complex Domain

Einar Hille
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

Einar Hille’s book on ordinary differential equations in the complex plane is one of the few introductory treatments of the subject. It is also quite a good one. It was first published in 1976, and while there have been some significant advances in the field since then, Hille’s treatment provides a strong basis for anyone interested in more advanced work in the field.

Real and complex analysis, as well as at least a good undergraduate course in real ordinary differential equations, would be desirable prerequisites. Hille says in his preface:

The reader is expected to have some knowledge of complex variables, a subject in which our students are frequently weak: they comprehend little and often their knowledge is too abstract and is of the wrong kind. Elementary manipulative skill is too often atrophied.

Strong words to the potential reader. Right from the beginning — the second half of the first chapter — Hille sets to work to remedy those deficiencies before he gets started with differential equations. But it is a short section and much of it is pretty basic, so I have to wonder what his concern really is.

Many of the chapters that follow have much that would be familiar to those who have studied ordinary differential equations over the real numbers. Beginning with \(w' = F[z,w(z)]\), there are two possibilities: one where \(F\) is an analytic function of the two variables \(w\) and \(z\), holomorphic in a given dicylinder in \(\mathbb{C}^2\); or, \(w(z)\) is a vector-valued function in \(\mathbb{C}^n\), \(z\) is a complex variable, and \(F\) is a holomorphic mapping from \(\mathbb{C}^{n+1}\) into \(\mathbb{C}^n\). As in the real case, the latter alternative allows us to treat \(n\)th- order equations as a special case. The expected existence and uniqueness results hold and can be proved in virtually the same ways as with real ODEs.

Power series solutions and the method of undetermined coefficients are discussed next, and with them Hille introduces the Poincaré’s method of majorants for establishing convergence of a formal series and corresponding radius of convergence. Hille gives a good deal of attention to growth questions, so there is a more extensive treatment of dominants, majorants and minorants than in other treatments. Singularities (fixed and movable) are treated next followed by analytic continuation of solutions.

Two chapters on linear differential equations of first, second and \(n\)th order establish the general theory in the complex plane, discuss asymptotics in the plane and on the real line, deal with issues concerning regular- and irregular-singular points, and introduce the monodromy group. Riccati’s equation gets a chapter all its own, and several special second order linear equations (the hypergeometric, Legendre’s, Bessel’s and Laplace’s ) are also afforded treatment in a separate chapter.

This is a clear and well-written text that would be as good an introduction to the subject now as it was almost forty years ago. Hille was clearly devoted to the subject and it shows.

Bill Satzer ( was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.



Chapter 1.
  l. Algebraic and Geometric Structures
    1.1. Vector Spaces
    1.2. Metric Spaces
    1.3. Mappings
    1.4. Linear Transformations on C into Itself ; Matrices
    1.5. Fixed Point Theorems
    1.6. Functional Inequalities
  II. Analytical Structures
    1.7. Holomorphic Functions
    1.8. Power Series
    1.9. Cauchy Integrals
    1.10. Estimates of Growth
    1.11. Analytic Continuation; Permanency of Functional Equations
  Chapter 2.
    Existence and Uniqueness Theorems
    2.1. Equations and Solutions
    2.2. The Fixed Point Method
    2.3. The Method of Successive Approximations
    2.4. Majorants and Majorant Methods
    2.5. The Cauchy Majorant
    2.6. The Lindelöf Majorant
    2.7. The Use of Dominants and Minorants
    2.8. Variation of Parameters
  Chapter 3.
    3.1. Fixed and Movable Singularities
    3.2. Analytic Continuation; Movable Singularities
    3.3. Painlevé's Determinateness Theorem; Singularities
    3.4. Indeterminate Forms
  Chapter 4.
    Riccati's Equation
    4.1. Classical Theory
    4.2. Dependence on Internal Parameters; Cross Ratios
    4.3. Some Geometric Applications
    4.4. "Abstract of the Nevanlinna Theory, I "
    4.5. "Abstract of the Nevanlinna Theory, II "
    4.6. The Malmquist Theorem and Some Generalizations
  Chapter 5.
    Linear Differential Equations: First and Second Order
    5.1. General Theory: First Order Case
    5.2. General Theory: Second Order Case
    5.3. Regular-Singular Points
    5.4. Estimates of Growth
    5.5. Asymptotics on the Real Line
    5.6. Asymptotics in the Plane
    5.7. Analytic Continuation; Group of Monodromy
  Chapter 6.
    Special Second Order Linear Dulerential Equations
    6.1. The Hypergeometric Equation
    6.2. Legendre's Equation
    6.3. Bessel's Equation
    6.4. Laplace's Equation
    6.5. The Laplacian; the Hermite-Weber Equation; Functions of the Parabolic Cylinder
    6.6. The Equation of Mathieu; Functions of the Elliptic Cylinder
    6.7. Some Other Equations
  Chapter 7.
    Representation Theorems
    7.1. Psi Series
    7.2. Integral Representations
    7.3. The Euler Transform
    7.4. Hypergeometric Euler Transforms
    7.5. The Laplace Transform
    7.6. Mellin and Mellin-Barnes Transforms
  Chapter 8.
    Complex Oscillation Theory
    8.1. Stunnian Methods; Green's Transform
    8.2. Zero-free Regions and Lines of Influence
    8.3. Other Comparison Theorems
    8.4. Applications to Special Equations
  Chapter 9.
    Linear nth Order and Matrix Differential Equations
    9.1. Existence and Independence of Solutions
    9.2. Analyticity of Matrix Solutions in a Star
    9.3. Analytic Continuation and the Group of Monodromy
    9.4. Approach to a Singularity
    9.5. Regular-Singular Points
    9.6. The Fuchsian Class; the Riemann Problem
    9.7. Irregular-Singular Points
  Chapter 10.
    The Schwarzian
    10.1. The Schwarzian Derivative
    10.2. Applications to Conformal Mapping
    10.3. Algebraic Solutions of Hypergeometric Equations
    10.4. Univalence and the Schwarzian
    10.5. Uniformization by Modular Functions
  Chapter 11.
    First Order Nonlinear Differential Equations
    11.1. Some Briot-Bouquet Equations
    11.2. Growth Properties
    11.3. Binomial Briot-Bouquet Equations of Elliptic Function Theory
    Appendix. Elliptic Functions
  Chapter 12.
    Second Order Nonlinear Differential Equations and Some Autonomous Systems
    12.1 Generalities; Briot-Bouquet Equations
    12.2 The Painlevé Transcendents
    12.3 The Asymptotics of Boutroux
    12.4 The Emden and the Thomas-Fermi Equations
    12.5 Quadratic Systems
    12.6 Other Autonomous Polynomial Systems