Showing papers by "Roderick Wong published in 2020"
TL;DR: In this article, the authors studied the asymptotic behavior of the Wilson polynomials Wn(x; a,b,c,d) as their degree tends to infinity.
Abstract: In this paper, we study the asymptotic behavior of the Wilson polynomials Wn(x; a,b,c,d) as their degree tends to infinity. These polynomials lie on the top level of the Askey scheme of hypergeomet...
7 citations
01 Jan 2020
TL;DR: JacJacobi elliptic functions as mentioned in this paper are a realization of one of the simplest cases of elliptic function, as described in Chapter 14: functions with two simple poles in a period parallelogram that are odd around each pole.
Abstract: Jacobi elliptic functions are a realization of one of the simplest cases of elliptic functions, as described in Chapter 14: functions with two simple poles in a period parallelogram that are odd around each pole. These functions come up naturally in certain problems of mechanics, such as the motion of an ideal pendulum. In pure mathematics they arise, for example, in connection with maps from the upper half plane to a parallelogram. In this chapter we begin with the pendulum equation and derive the properties of the functions associated to it. The triple of Jacobi functions \({{\,\mathrm{\mathrm{sn\,}}\,}}\), \({{\,\mathrm{\mathrm{cn\,}}\,}}\), \({{\,\mathrm{\mathrm{dn\,}}\,}}\) is closely analogous to the pair of trigonometric functions sine and cosine, and satisfy similar identities.
3 citations
01 Jan 2020
TL;DR: The Schwarzian derivative of a function f is a rational function of the derivatives of f to order 3, and it can be expressed in terms of the logarithmic derivative f''/f' of f'' of f' as mentioned in this paper.
Abstract: The Schwarzian derivative of a function f is a rational function of the derivatives of f to order 3. In fact it can be expressed in terms of the logarithmic derivative \(f''/f'\) of \(f'\). Here we show that the Schwarzian derivative is a natural object: a measure of the “curvature” of f, the pointwise deviation from a best approximation of f by a linear fractional transformation.
2 citations
1 citations
01 Jan 2020
TL;DR: The trigonometric functions are the basic functions that are periodic with respect to a translation of the plane (see as discussed by the authors for a general theory of doubly periodic trigonometrical functions, and Jacobi's construction via theta functions).
Abstract: The trigonometric functions are the basic functions that are periodic with respect to a translation of the plane \(\mathbb {C}\). An important class of complex functions is doubly periodic: periodic with respect to two sets of translations. This chapter presents the general theory of such functions, and Jacobi’s construction via theta functions. The following two chapters, which are independent of each other, present constructions due to Jacobi and to Weierstrass, respectively.