Peter Walker

The measure of solid angles in n-dimensional Euclidean space

A formula in terms of a definite integral for the measure of a polygonal solid angle in a Euclidean space of arbitrary dimension is proved. The formula is applied to the study of the geometry of n-simplices.

The analyticity of Jacobian functions with respect to the parameter k

We show that the 12 Jacobian elliptic functions sn(z,k), etc., are meromorphic throughout the complex plane, as functions of k for fixed z.

The Zeros of the Partial Sums of the Exponential Series

Denote by sn(z) the partial sum of the exponential series: sn(z) = ∑nk=0 zk/k! . Since the zeros of sn grow asymptotically in proportion to n, it is convenient to rescale, and to consider pn(z) = sn(nz). The asymptotic distribution of the zeros of sn was given by G. Szegö [7] and J. Dieudonné [2], who showed that for large n the zeros of pn cluster along the curve K , K = {z : ∣∣ze1−z∣∣ = 1, |z| ≤ 1} , with argument uniformly distributed with respect to arg(ze1−z). It was shown by Bucholtz in [1] that all zeros of pn lie within distance 2e/ √ n of K and that no zero lies either on K or in the bounded component of C − K where |ze1−z| < 1. We strengthen his elementary argument to show that there are no zeros within distance d of K , where

Functions and Continuity

Informally, a function f is described by a formula or a rule which, for a given input (usually a real number x), determines uniquely an output (again typically a real number y). The input is supposed to be an element of a set A called the domain of the function, and the output belongs to a set B called the codomain. When this happens we write y = f (x) and f: A → B. Examples of functions of this sort with A = B = ℝ (the set of all real numbers) are given for instance by (i) y = f(x) = x2 + 1, (ii) y = g(x) = 1 if x ≥ 0, = 0 if x < 0, (iii) y = h(x) = the smallest prime number ≥ x. Many more examples will be given in the first section of this chapter.