# 反正割

（重定向自Arcsec

 反正割 性質 奇偶性 非奇非偶 定義域 ${\displaystyle \left\{x\in \mathbb {R} :\left|x\right|\geq 1\right\}}$[1] 到達域 ${\displaystyle \left\{y\in \mathbb {R} :0\leq y\leq \pi \land y\neq {\frac {\pi }{2}}\right\}}$ 周期 N/A 特定值 當x=0 不存在[註 1] 當x=+∞ ${\displaystyle {\frac {\pi }{2}}}$ 當x=-∞ ${\displaystyle {\frac {\pi }{2}}}$ 當x=1 0 當x=-1 ${\displaystyle \pi }$ 其他性質 渐近线 ${\displaystyle y={\frac {\pi }{2}}}$

## 定義

${\displaystyle \operatorname {arcsec} x=-{\mathrm {i} }\ln \left({\tfrac {1}{x}}+{\sqrt {1-{\tfrac {\mathrm {i} }{x^{2}}}}}\right)\,}$

### 直角三角形中

${\displaystyle \sec ^{-1}{\frac {\mathrm {Hypotenuse} }{\mathrm {Adjacent} }}=\theta \,\!}$

${\displaystyle \sec ^{-1}x=\operatorname {arcsec} x=\theta \,\!}$

${\displaystyle \sin(\operatorname {arcsec} x)={\frac {\sqrt {x^{2}-1}}{x}}}$
${\displaystyle \cos(\operatorname {arcsec} x)={\frac {1}{x}}}$
${\displaystyle \tan(\operatorname {arcsec} x)={\sqrt {x^{2}-1}}}$

### 直角坐標系中

${\displaystyle \alpha }$是平面直角坐标系xOy中的一個未知的象限角${\displaystyle P\left({x,y}\right)}$是角的终边上一点，${\displaystyle r={\sqrt {x^{2}+y^{2}}}>0}$是P到原点O的距离，若已知${\displaystyle {\frac {r}{x}}\,\!}$，則可利用反正割求得未知的象限角${\displaystyle \alpha }$

${\displaystyle \sec ^{-1}{\frac {r}{x}}=\operatorname {arcsec} {\frac {r}{x}}=\alpha \,\!}$

### 級數定義

{\displaystyle {\begin{aligned}\operatorname {arcsec} z&{}=\arccos \left({\frac {1}{z}}\right)\\&{}={\frac {\pi }{2}}-[z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{-7}}{7}}+\cdots ]\\&{}={\frac {\pi }{2}}-\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]{\frac {z^{-(2n+1)}}{(2n+1)}};\qquad \left|z\right|\geq 1\end{aligned}}}

${\displaystyle \operatorname {arcsec} x={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(2n-1)!!}{(2n+1)(2n)!!}}x^{-2n-1}={\frac {\pi }{2}}-{\frac {1}{x}}-{\frac {1}{6x^{3}}}-{\frac {3}{40x^{5}}}-{\frac {5}{112x^{7}}}-\cdots }$

## 註釋

1. ^ 由於反正割在x=0未定義，因此考慮複變反正割函數，[2]但由在x=0時於左極限不等於右極限，因此也不存在極限因此Arcsec 0不存在。

## 參考文獻

1. ^ Weisstein, Eric W. "Inverse Secant." From MathWorld--A Wolfram Web Resource.
2. ^ 反正割在x=0的極限 wolframalpha.com [2014-08-08]
3. ^ 反正割arcsecant-學術名詞資訊 互联网档案馆存檔，存档日期2014-08-09. 國家教育研究院 terms.naer.edu.tw [2014-08-07]
4. ^ Gradshtein, I. S., I. M. Ryzhik, et al. (2000). Table of integrals, series, and products, Academic Pr.
5. ^ Zwillinger, D.(Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.
6. Abramowitz, M. and Stegun, I. A.(Eds.). "Inverse Circular Functions." §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.
7. ^ Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 315, 1998.
8. ^ Jeffrey, A. Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, 2000.
9. ^ 《 Exponentielle & logarithme 》, § Fonctions circulaires réciproques, Dictionnaire de mathématiques – algèbre, analyse, géométrie, Encyclopædia Universalis.
10. ^ Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 141-143, 1987.