Arkus sekans |
|
Osnovne osobine |
Parnost | neparna |
Domen | (-∞,-1] i [1,∞) |
Kodomen | [0,π/2) i (π/2,π] |
Specifične vrednosti |
Vrednost u +∞ | π/2 |
Vrednost u -∞ | π/2 |
Vrednost u -1 | π |
Vrednost u 1 | 0 |
Specifične osobine |
Asimptote | y = π/2 |
Arkus sekans je funkcija inverzna funkciji sekansa.
Slede neke od formula koje se vezuju za arkus sekans:
![{\displaystyle \operatorname {arcsec}(-x)=\pi -\operatorname {arcsec} x\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe26c12ec7a899df3d946a682cc1a8cb867a7bd2)
![{\displaystyle \operatorname {arcsec} {\frac {1}{x}}=\arccos x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3625049e60febc12c2ca6d9e364d1c855de183)
Izvod:
![{\displaystyle {\frac {d}{dx}}\operatorname {arcsec} x{}={\frac {1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edd8f2c51a11c7052498c6d5a184b464a4e02359)
Predstavljanje u formi integrala:
![{\displaystyle \operatorname {arcsec} x{}=\int _{1}^{x}{\frac {1}{x{\sqrt {x^{2}-1}}}}\,dx,\qquad x\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f814039ec0d6cc597d0eea90ec838d32ea44a2b1)
Predstavljanje u formi beskonačne sume:
![{\displaystyle {\begin{aligned}\operatorname {arcsec} x&{}=\arccos \left(x^{-1}\right)\\&{}={\frac {\pi }{2}}-(x^{-1}+\left({\frac {1}{2}}\right){\frac {x^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-7}}{7}}+\cdots )\\&{}={\frac {\pi }{2}}-\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-(2n+1)}}{(2n+1)}};\qquad \left|x\right|\geq 1\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de9af5c3bcad334bf955957f26567d7800857d6f)
Vanjske veze
- Funkcija arcsec na wolfram.com