The Location of Critical Points of Analytic and Harmonic Functions (Colloquium Publications)

This book is concerned with the critical points of analytic and harmonic functions. A critical point of an analytic function means a zero of its derivative, and a critical point of a harmonic function means a point where both partial derivatives vanish. The analytic functions considered are largely polynomials, rational functions, and certain periodic, entire, and meromorphic functions. The harmonic functions considered are largely Green's functions, harmonic measures, and various linear combinations of them. The interest in these functions centers around the approximate location of their critical points. The approximation is in the sense of determining minimal regions in which all the critical points lie or maximal regions in which no critical point lies. Throughout the book the author uses the single method of regarding the critical points as equilibrium points in fields of force due to suitable distribution of matter. The exposition is clear, complete, and well-illustrated with many examples.