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(2008). On Goursat's Proof of Cauchy's Integral Theorem. The American Mathematical Monthly: Vol. 115, No. 7, pp. 648-652.

In this note, we give a simple and direct proof for the "Hook Length Formula." The simplicity of our proof relies on the usage of the residue theorem as a short cut. The number of standard Young tableaux for a given Ferrers diagram . is the dimension of the irreducible representation of the symmetric group corresponding to X. The hook length formula (which was first proved in [1]) is a method for calculating this number and is a surprisingly beautiful formula (because the problem looks complex but the formula looks naive). Let us begin by giving the precise definitions of Ferrers diagrams, standard Young tableaux, and the hook length formula. Suppose that N and m are positive integers and that k = (kI, ..., Xm) is a sequence

We examine the development of Cauchy's closed curve theorem, including the early contributions of Clairaut, d'Alembert, Cauchy himself, Goursat, and Pringsheim, as well as more recent approaches du...

As a side note, Jacobson [1] states that his result may fall into the category of "interesting facts we once knew, but have now forgotten." This indeed seems to be the case, as the right-hand side of (2) is derived in [2, p. 78]. We extend Theorem 1 to a considerably larger function class, which includes functions such as f(t) = t and f(t) = 1/ t that do not satisfy the hypotheses of Theorem 1. Additionally, we apply the new results to obtain approximations of integrals appearing in familiar engineering settings. In fact our consideration of this problem was motivated by the desire to approximate, in closed form, integrals arising from a class of central force problems and to do so independently of the particular mathematical form of the force law itself.

Theorem. Let D be a domain in the complex plane, f(z) an analytic function in D, C a closed curve that is described by a continuously differentiable function z(s) (0 < s < 1) with z(0) = z(l) and zf(0) = z'(\). Suppose that C is homotopic in D to a point, meaning that there is a continuous function z(s, t) (0 < s < I, 0 < t < 1) with z(0, t) = z(l, t) for all t such that z(s, 0) = z(s) for all s, and z(s, 1) is independent ofs. Then

We will show that Cauchy's theorem implies the Fundamental Theorem of Algebra in a direct and elementary manner. Furthermore, the argument can be used to give a transcendental version of the theore...

A classical result by Cauchy determines a disk containing all the zeros of a given polynomial. We derive this result using only linear algebra techniques, and, in the process, discover a twin disk ...

We give a simultaneous proof of Cauchy's theorem and Sylow's theorem by reducing to the case of cyclic groups.

(1997). A Note on the Mean Value Theorem for Integrals. The American Mathematical Monthly: Vol. 104, No. 6, pp. 561-562.

(1991). A Note on Dixon's Proof of Cauchy's Integral Theorem. The American Mathematical Monthly: Vol. 98, No. 3, pp. 242-244.

The Jordan curve theorem is one of those frustrating results in topology: it is intuitively clear but quite hard to prove. In this note we will look at two discrete analogs of the Jordan curve theorem that are easy to prove by an induction argument coupled with some geometric intuition. One of the surprises is that when we discretize the plane we get two Jordan curve theorems rather than one, a consequence of the interplay between two natural products in the category of graphs. Topology in this context has been studied by Farmer in [2]. To state the discrete versions, we need to know what the discrete analog of the plane is and what plays the role of a simple closed curve. Since the plane is the topological product of two lines, we take as our discrete analog the product of two discrete lines. We will use undirected graphs for our analogs of spaces, with vertices for points and edges connecting points which are to be thought of as touching.

(1982). On the Mean Value Theorem for Integrals. The American Mathematical Monthly: Vol. 89, No. 5, pp. 300-301.