论文标题
关于逆倾斜函数的复杂奇异性
On the complex singularities of the inverse Langevin function
论文作者
论文摘要
我们研究了反向langevin函数$ \ mathscr {l}^{ - 1}（x）$，因为它在建模有限拉伸弹性的情况下很重要，在这种情况下，压力和应变能成为无限的无限，随着特定的最大应变，在此建模$ x \ to1 $。反向langevin函数$ \ mathscr {l}^{ - 1}（x）$的唯一真正的奇异性是$ x = \ pm1 $时的两个简单的杆，我们看到如何乘坐或添加性地删除其效果。此外，我们发现$ \ mathscr {l}^{ - 1}（x）$具有复杂的奇点。对泰勒系列的检查，内容涉及$ \ mathscr {l}^{ - 1}（x）$的起源，表明最接近原点的四个复杂奇点与原点相距，并且具有相同的强度；我们开发了一种用于查找这四个复杂奇异性的新算法。图形插图似乎指出了这些复杂的奇异性是平方根的性质。然后，精确的分析证明了这些是平方根分支点。
We study the inverse Langevin function $\mathscr{L}^{-1}(x)$ because of its importance in modelling limited-stretch elasticity where the stress and strain energy become infinite as a certain maximum strain is approached, modelled here by $x\to1$. The only real singularities of the inverse Langevin function $\mathscr{L}^{-1}(x)$ are two simple poles at $x=\pm1$ and we see how to remove their effects either multiplicatively or additively. In addition, we find that $\mathscr{L}^{-1}(x)$ has an infinity of complex singularities. Examination of the Taylor series about the origin of $\mathscr{L}^{-1}(x)$ shows that the four complex singularities nearest the origin are equidistant from the origin and have the same strength; we develop a new algorithm for finding these four complex singularities. Graphical illustration seems to point to these complex singularities being of a square root nature. An exact analysis then proves these are square root branch points.
