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Computation of the Schläfli function

Andrey A. Shoom

TL;DR

The computation is based on the Chebyshev approximation of the Schläfli function, related to the Schläfli function via a simple factor of an algebraic expression and regular in the interval.

Abstract

In this work a method for numerical computation of the Schläfli function $f_n(x)$, for $x\in[n-1,n+1]$ and $n\geq4$ is presented. The computation is based on the Chebyshev approximation of the function $q_{n}(x)$, which is related to the Schläfli function and regular in the interval.

Computation of the Schläfli function

TL;DR

The computation is based on the Chebyshev approximation of the Schläfli function, related to the Schläfli function via a simple factor of an algebraic expression and regular in the interval.

Abstract

In this work a method for numerical computation of the Schläfli function , for and is presented. The computation is based on the Chebyshev approximation of the function , which is related to the Schläfli function and regular in the interval.
Paper Structure (6 sections, 1 theorem, 44 equations, 1 table)

This paper contains 6 sections, 1 theorem, 44 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Schläfli function
  3. A historical note
  4. Properties of the Schläfli function
  5. Evaluation of $f_{n}(x)$ in the interval $x\in[n-1, n+1]$
  6. Concluding remarks

Key Result

THEOREM 1

If function $f(z)$ is analytic in the ring ${\cal K}:\,0<|z-a|<r$, where $a\ne\infty$ is algebraic branch point of order $p<\infty$, then it can be expanded as follows: where the series converges in ${\cal K}$.

Theorems & Definitions (1)

  • THEOREM