关于第一类完全椭圆积分用反双曲切线逼近的一个猜想
On a Conjecture Concerning the Approximates of Complete Elliptic Integral of the First Kind by Inverse Hyperbolic Tangent
摘要：
设$\K$为第一类完全椭圆积分。本文证明了函数$r\mapsto r^{-2}\{[\log(2\K(r)/π)]/\log((\arth r)/r)-3/4\}$是从$(0,1)$到$(1/320,1/4)$的严格递增函数，使得$[(\arth r)/r]^{3/4+r^2/320}<2\K(r)/π<[(\arth r)/r]^{3/4+r^2/4}$对于$r\in(0,1)$，其中两个界的指数的所有系数都是最佳可能的，从而证明了Alzer和Qiu提出的一个猜想是正确的，并给出了比他们在公开问题中提出的更好的$\K(r)$的界.得到了完全椭圆积分的其它解析性质，包括$\K(r)$的其它近似。
英文摘要：
Let $\K$ be the complete elliptic integral of the first kind. In this paper, the authors prove that the function $r\mapsto r^{-2}\{[\log(2\K(r)/π)]/\log((\arth r)/r)-3/4\}$ is strictly increasing from $(0,1)$ onto $(1/320,1/4)$, so that $[(\arth r)/r]^{3/4+r^2/320}<2\K(r)/π<[(\arth r)/r]^{3/4+r^2/4}$ for $r\in(0,1)$, in which all the coefficients of the exponents of the two bounds are best possible, thus proving a conjecture raised by Alzer and Qiu to be true, and giving better bounds of $\K(r)$ than those they conjectured and put in an open problem. Some other analytic properties of the complete elliptic integrals, including other kind of approximates for $\K(r)$, are obtained, too.
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