tan的不动点的渐近阶

hbghlyj

$n=1,2,3,4,5,6,7,8,9$时，用Python验证了
$$\tan\left(n\pi + \frac{\pi}{2} - \frac{1}{n\pi}\right)<n\pi + \frac{\pi}{2} - \frac{1}{n\pi}$$
这个对于正整数$n$都成立吗


问题起源：@战巡的回答

业余的业余

$\tan(n\pi+\frac{\pi}2-\frac1{n\pi})=\cot(\frac1{n\pi})$

考察 $f(x)=\cot(\frac1{\pi x})-\pi x+\frac1{\pi x}, x>1$ 的单调性, 不知道可否走通。

睡神

$\tan(n\pi+\dfrac{\pi}{2}-\dfrac{1}{n\pi})=\dfrac{1}{\tan \dfrac{1}{n\pi}}<\dfrac{1}{\dfrac{1}{n\pi}}=n\pi<n\pi+\dfrac{\pi}{2}-\dfrac{1}{n\pi}$？

hbghlyj

$n=1,2,3,4,5,6,7,8,9$时，用Python验证了
$$\tan\left(n\pi + \frac{\pi}{2} - \frac{1}{n\pi+\frac\pi2}\right)>n\pi + \frac{\pi}{2} - \frac{1}{n\pi+\frac\pi2}$$
这个对于正整数$n$都成立吗

hbghlyj

hbghlyj 发表于 2024-3-26 09:32
$$\tan\left(n\pi + \frac{\pi}{2} - \frac{1}{n\pi+\frac\pi2}\right)>n\pi + \frac{\pi}{2} - \frac{1}{n\pi+\frac\pi2}$$

如果成立$\tan(x)<\dfrac1{\frac1x-x},\forall x\in(0,1)$就有
$$\tan(n\pi+\dfrac{\pi}{2}-\dfrac{1}{n\pi+\frac\pi2})=\dfrac{1}{\tan \dfrac{1}{n\pi+\frac\pi2}}>\dfrac{1}{\dfrac{1}{n\pi+\frac\pi2-\dfrac{1}{n\pi+\frac\pi2}}}=n\pi+\dfrac{\pi}{2}-\dfrac{1}{n\pi+\frac\pi2}$$
如何证明$\tan(x)<\dfrac1{\frac1x-x},\forall x\in(0,1)$？

hbghlyj

hbghlyj 发表于 2024-3-26 09:48
如何证明$\tan(x)<\dfrac1{\frac1x-x},\forall x\in(0,1)$？

易得$\cos x+x^2$在$(0,1)$单增，故$\cos x+x^2>\cos0+0^2=1$，即$\dfrac{\cos x}{1-x^2}>1$
$\dfrac{\sin x}{x}<1<\dfrac{\cos x}{1-x^2},\forall x\in(0,1)$
$\implies\tan(x)<\dfrac1{\frac1x-x},\forall x\in(0,1)$

hbghlyj

Lagrange Inversion Formula.

Let the function $f(z)$ be analytic in some neighborhood of the point $z=0$ of the complex plane.  Assuming that $f(0) \neq 0$, we consider the equation $$w = z/f(z),$$ where $z$ is the unknown.  Then there exist positive numbers $a$ and $b$ such that for $|w| < a$ the equation has just one solution in the domain $|z| < b$, and this solution is an analytic function of $w$: $$z = \sum_{k=1}^{\infty} c_k w^k \hspace{1cm} (|w| < a),$$ where the coefficients $c_k$ are given by $$c_k = \frac{1}{k!} \left\{\left(\frac{d}{dz}\right)^{k-1} (f(z))^k\right\}_{z=0}.$$

用Lagrange Inversion Formula得到tan(x)-x的根
\[x_n = \pi n + \frac{\pi}{2} - \left(\pi n + \frac{\pi}{2}\right)^{-1} - \frac{2}{3}\left(\pi n + \frac{\pi}{2}\right)^{-3} - \frac{13}{15}\left(\pi n + \frac{\pi}{2}\right)^{-5} - \frac{146}{105}\left(\pi n + \frac{\pi}{2}\right)^{-7} - \frac{781}{315}\left(\pi n + \frac{\pi}{2}\right)^{-9} - \frac{16328}{3465}\left(\pi n + \frac{\pi}{2}\right)^{-11} + \cdots\]

hbghlyj

2006年初sci.math帖子说Euler曾得到过这个级数：
I finally have some information about the asymptotic series
for the zeros of the solutions to tan(x) = x:
\[
q - \frac{1}{q} - \frac{2}{3} \cdot \frac{1}{q^3} - \frac{13}{15} \cdot \frac{1}{q^5} - \frac{146}{105} \cdot \frac{1}{q^7} - ...
\quad
\text{where } q = \frac{(2k+1)\pi}{2}.
\]
Apparently, this series was independently obtained by
Euler [1] (1748), Cauchy [2] (1827), and Rayleigh [3] (1877).

[1] Leonhard Euler, "Introductio in Analysin Infinitorum",
Volume 2, 1748. [Reprinted in Euler's OPERA OMNIA
Series 1, Volume 9.]

See pp. 318-320.

This also appears on pp. 323-324 of the French
translation "Introduction à l'Analyse Infinitésimale"
that is on the internet at

visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-3885

See two-thirds down p. 323 for the series.

[2] Augustin-Louis Cauchy, "Théorie de la Propagation
des Ondes à la Surface d'un Fluide Pesant d'une
Profondeur Indéfinie", 1827.

Oeuvres complètes, Series 1, Volume 1, pp. 5-318.
math-doc.ujf-grenoble.fr/cgi-bin/oetoc?id=OE_CAUCHY_1_1

According to [5] (p. 273 & 275), the series is developed
on pp. 277-278 (p. 272 of the original 1827 publication).

[3] John William Strutt Rayleigh, "Theory of Sound",
1877-1878. [Reprinted by Dover in 1945 and 1976.]
tinyurl.com/n9n54

See p. 334, which the URL above will take you to.

[4] Raymond Clare Archibald and Henry Bateman, "Roots
of the equation tan x = cx", Note #8, Mathematical
Tables and Other Aids to Computation (= Mathematics
of Computation) 1 #6 (April 1944), 203.

See the follow-ups by Henry Bateman (Vol. 1 #8,
October 1944, p. 336), Raymond Clare Archibald
(Vol. 1 #12, October 1945, p. 459), Abraham P.
Hillman and Herbert E. Salzer (Vol. 2 #14,
April 1946, p. 95), and L. G. Pooler
(Vol. 3 #27, July 1949, pp. 495-496).

[5] Raymond Clare Archibald and Henry Bateman, "A guide
to tables of Bessel functions", Mathematical Tables
and Other Aids to Computation (= Mathematics of
Computation) 1 #7 (July 1944), 205-308.

See Section F: "Series for the zeros of
Bessel functions", pp. 271-275.


Dave L. Renfro