- Source: Fungsi hiperbolik invers
Dalam matematika, fungsi hiperbolik invers merupakan fungsi invers dari fungsi hiperbolik.
Notasi
Asal-usul prefiks ar- berasal dari singkatan dari notasi fungsi hiperbolik yang serupa (seperti, arsinh dan arcosh) berdasarkan ISO 80000-2. Prefiks arc- yang berasal dari fungsi hiperbolik yang serupa (seperti, arcsinh dan arccosh) juga seringkali dipakai berdasarkan penamaan fungsi invers trigonometri. Namun sayangnya, pemakaian kedua prefiks tersebut keliru sebab prefiks arc merupakan singkatan dari arcus, sedangkan prefiks ar merupakan singkatan dari area (bahasa Indonesia: luas, daerah). Karena itu, fungsi hiperbolik secara tidak langsung dikaitkan dengan busur.
Notasi seperti sinh−1(x), cosh−1(x), dst. juga dipakai sebagai penggantinya. Namun sayangnya, superskrip −1 membingungkan para pembaca karena dapat diartikan sebagai perpangkatan atau fungsi invers (sebagai contoh, bandingkan cosh−1(x) dengan cosh(x)−1).
Definisi fungsi invers hiperbolik dalam logaritma
Karena fungsi hiperbolik merupakan fungsi rasional dari ex, dengan derajat pada pembilang maupun penyebut setidaknya bernilai dua, fungsi-fungsi tersebut dapat diselesaikan dalam bentuk ex dengan menggunakan rumus kuadratik. Maka, dengan mengambil logaritma alami akan memberikan ekspresi berikut untuk fungsi hiperbolik invers.
Rumus penambahan
arsinh
u
±
arsinh
v
=
arsinh
(
u
1
+
v
2
±
v
1
+
u
2
)
{\displaystyle \operatorname {arsinh} u\pm \operatorname {arsinh} v=\operatorname {arsinh} \left(u{\sqrt {1+v^{2}}}\pm v{\sqrt {1+u^{2}}}\right)}
arcosh
u
±
arcosh
v
=
arcosh
(
u
v
±
(
u
2
−
1
)
(
v
2
−
1
)
)
{\displaystyle \operatorname {arcosh} u\pm \operatorname {arcosh} v=\operatorname {arcosh} \left(uv\pm {\sqrt {(u^{2}-1)(v^{2}-1)}}\right)}
artanh
u
±
artanh
v
=
artanh
(
u
±
v
1
±
u
v
)
{\displaystyle \operatorname {artanh} u\pm \operatorname {artanh} v=\operatorname {artanh} \left({\frac {u\pm v}{1\pm uv}}\right)}
arcoth
u
±
arcoth
v
=
arcoth
(
1
±
u
v
u
±
v
)
{\displaystyle \operatorname {arcoth} u\pm \operatorname {arcoth} v=\operatorname {arcoth} \left({\frac {1\pm uv}{u\pm v}}\right)}
arsinh
u
+
arcosh
v
=
arsinh
(
u
v
+
(
1
+
u
2
)
(
v
2
−
1
)
)
=
arcosh
(
v
1
+
u
2
+
u
v
2
−
1
)
{\displaystyle {\begin{aligned}\operatorname {arsinh} u+\operatorname {arcosh} v&=\operatorname {arsinh} \left(uv+{\sqrt {(1+u^{2})(v^{2}-1)}}\right)\\&=\operatorname {arcosh} \left(v{\sqrt {1+u^{2}}}+u{\sqrt {v^{2}-1}}\right)\end{aligned}}}
Identitas lainnya
2
arcosh
x
=
arcosh
(
2
x
2
−
1
)
for
x
≥
1
4
arcosh
x
=
arcosh
(
8
x
4
−
8
x
2
+
1
)
for
x
≥
1
2
arsinh
x
=
arcosh
(
2
x
2
+
1
)
for
x
≥
0
4
arsinh
x
=
arcosh
(
8
x
4
+
8
x
2
+
1
)
for
x
≥
0
{\displaystyle {\begin{aligned}2\operatorname {arcosh} x&=\operatorname {arcosh} (2x^{2}-1)&\quad {\hbox{ for }}x\geq 1\\4\operatorname {arcosh} x&=\operatorname {arcosh} (8x^{4}-8x^{2}+1)&\quad {\hbox{ for }}x\geq 1\\2\operatorname {arsinh} x&=\operatorname {arcosh} (2x^{2}+1)&\quad {\hbox{ for }}x\geq 0\\4\operatorname {arsinh} x&=\operatorname {arcosh} (8x^{4}+8x^{2}+1)&\quad {\hbox{ for }}x\geq 0\end{aligned}}}
ln
(
x
)
=
arcosh
(
x
2
+
1
2
x
)
=
arsinh
(
x
2
−
1
2
x
)
=
artanh
(
x
2
−
1
x
2
+
1
)
{\displaystyle \ln(x)=\operatorname {arcosh} \left({\frac {x^{2}+1}{2x}}\right)=\operatorname {arsinh} \left({\frac {x^{2}-1}{2x}}\right)=\operatorname {artanh} \left({\frac {x^{2}-1}{x^{2}+1}}\right)}
Komposisi dari fungsi hiperbolik dan fungsi hiperbolik invers
sinh
(
arcosh
x
)
=
x
2
−
1
untuk
|
x
|
>
1
sinh
(
artanh
x
)
=
x
1
−
x
2
untuk
−
1
<
x
<
1
cosh
(
arsinh
x
)
=
1
+
x
2
cosh
(
artanh
x
)
=
1
1
−
x
2
untuk
−
1
<
x
<
1
tanh
(
arsinh
x
)
=
x
1
+
x
2
tanh
(
arcosh
x
)
=
x
2
−
1
x
untuk
|
x
|
>
1
{\displaystyle {\begin{aligned}&\sinh(\operatorname {arcosh} x)={\sqrt {x^{2}-1}}\quad {\text{untuk}}\quad |x|>1\\&\sinh(\operatorname {artanh} x)={\frac {x}{\sqrt {1-x^{2}}}}\quad {\text{untuk}}\quad -1
Komposisi dari fungsi invers hiperbolik dan fungsi trigonometri
arsinh
(
tan
α
)
=
artanh
(
sin
α
)
=
ln
(
1
+
sin
α
cos
α
)
=
±
arcosh
(
1
cos
α
)
{\displaystyle \operatorname {arsinh} \left(\tan \alpha \right)=\operatorname {artanh} \left(\sin \alpha \right)=\ln \left({\frac {1+\sin \alpha }{\cos \alpha }}\right)=\pm \operatorname {arcosh} \left({\frac {1}{\cos \alpha }}\right)}
ln
(
|
tan
α
|
)
=
−
artanh
(
cos
2
α
)
{\displaystyle \ln \left(\left|\tan \alpha \right|\right)=-\operatorname {artanh} \left(\cos 2\alpha \right)}
Konversi
ln
x
=
artanh
(
x
2
−
1
x
2
+
1
)
=
arsinh
(
x
2
−
1
2
x
)
=
±
arcosh
(
x
2
+
1
2
x
)
{\displaystyle \ln x=\operatorname {artanh} \left({\frac {x^{2}-1}{x^{2}+1}}\right)=\operatorname {arsinh} \left({\frac {x^{2}-1}{2x}}\right)=\pm \operatorname {arcosh} \left({\frac {x^{2}+1}{2x}}\right)}
artanh
x
=
arsinh
(
x
1
−
x
2
)
=
±
arcosh
(
1
1
−
x
2
)
{\displaystyle \operatorname {artanh} x=\operatorname {arsinh} \left({\frac {x}{\sqrt {1-x^{2}}}}\right)=\pm \operatorname {arcosh} \left({\frac {1}{\sqrt {1-x^{2}}}}\right)}
arsinh
x
=
artanh
(
x
1
+
x
2
)
=
±
arcosh
(
1
+
x
2
)
{\displaystyle \operatorname {arsinh} x=\operatorname {artanh} \left({\frac {x}{\sqrt {1+x^{2}}}}\right)=\pm \operatorname {arcosh} \left({\sqrt {1+x^{2}}}\right)}
arcosh
x
=
|
arsinh
(
x
2
−
1
)
|
=
|
artanh
(
x
2
−
1
x
)
|
{\displaystyle \operatorname {arcosh} x=\left|\operatorname {arsinh} \left({\sqrt {x^{2}-1}}\right)\right|=\left|\operatorname {artanh} \left({\frac {\sqrt {x^{2}-1}}{x}}\right)\right|}
Turunan
d
d
x
arsinh
x
=
1
x
2
+
1
,
untuk semua bilangan real
x
d
d
x
arcosh
x
=
1
x
2
−
1
,
untuk semua bilangan real
x
>
1
d
d
x
artanh
x
=
1
1
−
x
2
,
untuk semua bilangan real
|
x
|
<
1
d
d
x
arcoth
x
=
1
1
−
x
2
,
untuk semua bilangan real
|
x
|
>
1
d
d
x
arsech
x
=
−
1
x
1
−
x
2
,
untuk semua bilangan real
x
∈
(
0
,
1
)
d
d
x
arcsch
x
=
−
1
|
x
|
1
+
x
2
,
untuk semua bilangan real
x
, kecuali
0
{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} x&{}={\frac {1}{\sqrt {x^{2}+1}}},{\text{ untuk semua bilangan real }}x\\{\frac {d}{dx}}\operatorname {arcosh} x&{}={\frac {1}{\sqrt {x^{2}-1}}},{\text{ untuk semua bilangan real }}x>1\\{\frac {d}{dx}}\operatorname {artanh} x&{}={\frac {1}{1-x^{2}}},{\text{ untuk semua bilangan real }}|x|<1\\{\frac {d}{dx}}\operatorname {arcoth} x&{}={\frac {1}{1-x^{2}}},{\text{ untuk semua bilangan real }}|x|>1\\{\frac {d}{dx}}\operatorname {arsech} x&{}={\frac {-1}{x{\sqrt {1-x^{2}}}}},{\text{ untuk semua bilangan real }}x\in (0,1)\\{\frac {d}{dx}}\operatorname {arcsch} x&{}={\frac {-1}{|x|{\sqrt {1+x^{2}}}}},{\text{ untuk semua bilangan real }}x{\text{, kecuali }}0\\\end{aligned}}}
Sebagai contoh, misalkan
θ
=
arsinh
x
{\displaystyle \theta =\operatorname {arsinh} x}
, maka
d
arsinh
x
d
x
=
d
θ
d
sinh
θ
=
1
cosh
θ
=
1
1
+
sinh
2
θ
=
1
1
+
x
2
.
{\displaystyle {\frac {d\,\operatorname {arsinh} x}{dx}}={\frac {d\theta }{d\sinh \theta }}={\frac {1}{\cosh \theta }}={\frac {1}{\sqrt {1+\sinh ^{2}\theta }}}={\frac {1}{\sqrt {1+x^{2}}}}.}
dengan
sinh
2
θ
=
(
sinh
θ
)
2
{\displaystyle \sinh ^{2}\theta =(\sinh \theta )^{2}}
.
Ekspansi deret
Ekspansi deret dapat diperoleh untuk fungsi-fungsi di atas:
arsinh
x
=
x
−
(
1
2
)
x
3
3
+
(
1
⋅
3
2
⋅
4
)
x
5
5
−
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
7
7
±
⋯
=
∑
n
=
0
∞
(
(
−
1
)
n
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
2
n
+
1
2
n
+
1
,
|
x
|
<
1
{\displaystyle {\begin{aligned}\operatorname {arsinh} x&=x-\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}}\pm \cdots \\&=\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n+1}}{2n+1}},\qquad \left|x\right|<1\end{aligned}}}
arcosh
x
=
ln
(
2
x
)
−
(
(
1
2
)
x
−
2
2
+
(
1
⋅
3
2
⋅
4
)
x
−
4
4
+
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
−
6
6
+
⋯
)
=
ln
(
2
x
)
−
∑
n
=
1
∞
(
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
−
2
n
2
n
,
|
x
|
>
1
{\displaystyle {\begin{aligned}\operatorname {arcosh} x&=\ln(2x)-\left(\left({\frac {1}{2}}\right){\frac {x^{-2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-6}}{6}}+\cdots \right)\\&=\ln(2x)-\sum _{n=1}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-2n}}{2n}},\qquad \left|x\right|>1\end{aligned}}}
artanh
x
=
x
+
x
3
3
+
x
5
5
+
x
7
7
+
⋯
=
∑
n
=
0
∞
x
2
n
+
1
2
n
+
1
,
|
x
|
<
1
{\displaystyle {\begin{aligned}\operatorname {artanh} x&=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+{\frac {x^{7}}{7}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}},\qquad \left|x\right|<1\end{aligned}}}
arcsch
x
=
arsinh
1
x
=
x
−
1
−
(
1
2
)
x
−
3
3
+
(
1
⋅
3
2
⋅
4
)
x
−
5
5
−
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
−
7
7
±
⋯
=
∑
n
=
0
∞
(
(
−
1
)
n
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
−
(
2
n
+
1
)
2
n
+
1
,
|
x
|
>
1
{\displaystyle {\begin{aligned}\operatorname {arcsch} x=\operatorname {arsinh} {\frac {1}{x}}&=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}}\pm \cdots \\&=\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-(2n+1)}}{2n+1}},\qquad \left|x\right|>1\end{aligned}}}
arsech
x
=
arcosh
1
x
=
ln
2
x
−
(
(
1
2
)
x
2
2
+
(
1
⋅
3
2
⋅
4
)
x
4
4
+
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
x
6
6
+
⋯
)
=
ln
2
x
−
∑
n
=
1
∞
(
(
2
n
)
!
2
2
n
(
n
!
)
2
)
x
2
n
2
n
,
0
<
x
≤
1
{\displaystyle {\begin{aligned}\operatorname {arsech} x=\operatorname {arcosh} {\frac {1}{x}}&=\ln {\frac {2}{x}}-\left(\left({\frac {1}{2}}\right){\frac {x^{2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{6}}{6}}+\cdots \right)\\&=\ln {\frac {2}{x}}-\sum _{n=1}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n}}{2n}},\qquad 0
arcoth
x
=
artanh
1
x
=
x
−
1
+
x
−
3
3
+
x
−
5
5
+
x
−
7
7
+
⋯
=
∑
n
=
0
∞
x
−
(
2
n
+
1
)
2
n
+
1
,
|
x
|
>
1
{\displaystyle {\begin{aligned}\operatorname {arcoth} x=\operatorname {artanh} {\frac {1}{x}}&=x^{-1}+{\frac {x^{-3}}{3}}+{\frac {x^{-5}}{5}}+{\frac {x^{-7}}{7}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{-(2n+1)}}{2n+1}},\qquad \left|x\right|>1\end{aligned}}}
Ekspansi asimtotik untuk fungsi
arsinh
x
{\displaystyle \operatorname {arsinh} x}
dinyatakan dengan
arsinh
x
=
ln
(
2
x
)
+
∑
n
=
1
∞
(
−
1
)
n
−
1
(
2
n
−
1
)
!
!
2
n
(
2
n
)
!
!
1
x
2
n
{\displaystyle \operatorname {arsinh} x=\ln(2x)+\sum \limits _{n=1}^{\infty }{\left({-1}\right)^{n-1}{\frac {\left({2n-1}\right)!!}{2n\left({2n}\right)!!}}}{\frac {1}{x^{2n}}}}
Referensi
Bibilografi
Herbert Busemann and Paul J. Kelly (1953) Projective Geometry and Projective Metrics, page 207, Academic Press.
Pranala luar
Hazewinkel, Michiel, ed. (2001) [1994], "Inverse hyperbolic functions", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Kata Kunci Pencarian:
- Fungsi invers
- Fungsi hiperbolik invers
- Fungsi trigonometri
- Fungsi invers trigonometri
- Fungsi hiperbolik
- Daftar integral dari fungsi invers hiperbolik
- Daftar integral dari fungsi invers trigonometri
- Logaritma
- Ekspresi bentuk tertutup
- Analisis kompleks
