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 (bsatzer@gmail.com) 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.