Fungsi hiperbolik Fungsi hiperbolik adalah salah satu hasil kombinasi dari fungsi-fungsi eksponen . Fungsi hiperbolik memiliki rumus. Selain itu memiliki invers serta turunan dan anti turunan fungsi hiperbolik dan inversnya.[ 1]
Definisi
sinh , cosh dan tanh csch , sech dan coth
Definisi Eksponen
sinh x adalah separuh selisih ex e −x cosh x adalah rerata ex e −x Dalam istilah dari fungsi eksponensial :
Hiperbolik sinus:
sinh
-->
x
=
e
x
− − -->
e
− − -->
x
2
=
e
2
x
− − -->
1
2
e
x
=
1
− − -->
e
− − -->
2
x
2
e
− − -->
x
.
{\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.}
Hiperbolik kosinus:
cosh
-->
x
=
e
x
+
e
− − -->
x
2
=
e
2
x
+
1
2
e
x
=
1
+
e
− − -->
2
x
2
e
− − -->
x
.
{\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.}
Hiperbolik tangen:
tanh
-->
x
=
sinh
-->
x
cosh
-->
x
=
e
x
− − -->
e
− − -->
x
e
x
+
e
− − -->
x
=
e
2
x
− − -->
1
e
2
x
+
1
{\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}}
Hiperbolik kotangen: untuk x ≠ 0
coth
-->
x
=
cosh
-->
x
sinh
-->
x
=
e
x
+
e
− − -->
x
e
x
− − -->
e
− − -->
x
=
e
2
x
+
1
e
2
x
− − -->
1
{\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}}
Hiperbolik sekan:
sech
-->
x
=
1
cosh
-->
x
=
2
e
x
+
e
− − -->
x
=
2
e
x
e
2
x
+
1
{\displaystyle \operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}}
Hiperbolik kosekan: untuk x ≠ 0
csch
-->
x
=
1
sinh
-->
x
=
2
e
x
− − -->
e
− − -->
x
=
2
e
x
e
2
x
− − -->
1
{\displaystyle \operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}}
Definisi persamaan diferensial
- Dalam pengembangan -
Definisi kompleks trigonometri
-Dalam pengembangan -
Sifat karakteristik
- Dalam pengembangan -
Penambahan
sinh
-->
(
x
+
y
)
=
sinh
-->
x
cosh
-->
y
+
cosh
-->
x
sinh
-->
y
cosh
-->
(
x
+
y
)
=
cosh
-->
x
cosh
-->
y
+
sinh
-->
x
sinh
-->
y
tanh
-->
(
x
+
y
)
=
tanh
-->
x
+
tanh
-->
y
1
+
tanh
-->
x
tanh
-->
y
{\displaystyle {\begin{aligned}\sinh(x+y)&=\sinh x\cosh y+\cosh x\sinh y\\\cosh(x+y)&=\cosh x\cosh y+\sinh x\sinh y\\[6px]\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\end{aligned}}}
terutama
cosh
-->
(
2
x
)
=
sinh
2
-->
x
+
cosh
2
-->
x
=
2
sinh
2
-->
x
+
1
=
2
cosh
2
-->
x
− − -->
1
sinh
-->
(
2
x
)
=
2
sinh
-->
x
cosh
-->
x
tanh
-->
(
2
x
)
=
2
tanh
-->
x
1
+
tanh
2
-->
x
{\displaystyle {\begin{aligned}\cosh(2x)&=\sinh ^{2}{x}+\cosh ^{2}{x}=2\sinh ^{2}x+1=2\cosh ^{2}x-1\\\sinh(2x)&=2\sinh x\cosh x\\\tanh(2x)&={\frac {2\tanh x}{1+\tanh ^{2}x}}\\\end{aligned}}}
Lihat:
sinh
-->
x
+
sinh
-->
y
=
2
sinh
-->
(
x
+
y
2
)
cosh
-->
(
x
− − -->
y
2
)
cosh
-->
x
+
cosh
-->
y
=
2
cosh
-->
(
x
+
y
2
)
cosh
-->
(
x
− − -->
y
2
)
{\displaystyle {\begin{aligned}\sinh x+\sinh y&=2\sinh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\cosh x+\cosh y&=2\cosh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\end{aligned}}}
Pengurangan
sinh
-->
(
x
− − -->
y
)
=
sinh
-->
x
cosh
-->
y
− − -->
cosh
-->
x
sinh
-->
y
cosh
-->
(
x
− − -->
y
)
=
cosh
-->
x
cosh
-->
y
− − -->
sinh
-->
x
sinh
-->
y
tanh
-->
(
x
− − -->
y
)
=
tanh
-->
x
− − -->
tanh
-->
y
1
− − -->
tanh
-->
x
tanh
-->
y
{\displaystyle {\begin{aligned}\sinh(x-y)&=\sinh x\cosh y-\cosh x\sinh y\\\cosh(x-y)&=\cosh x\cosh y-\sinh x\sinh y\\\tanh(x-y)&={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\\\end{aligned}}}
Dan juga:[ 2]
sinh
-->
x
− − -->
sinh
-->
y
=
2
cosh
-->
(
x
+
y
2
)
sinh
-->
(
x
− − -->
y
2
)
cosh
-->
x
− − -->
cosh
-->
y
=
2
sinh
-->
(
x
+
y
2
)
sinh
-->
(
x
− − -->
y
2
)
{\displaystyle {\begin{aligned}\sinh x-\sinh y&=2\cosh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\cosh x-\cosh y&=2\sinh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\end{aligned}}}
Rumus setengah argumen
sinh
-->
(
x
2
)
=
sinh
-->
x
2
(
cosh
-->
x
+
1
)
=
sgn
-->
x
cosh
-->
x
− − -->
1
2
cosh
-->
(
x
2
)
=
cosh
-->
x
+
1
2
tanh
-->
(
x
2
)
=
sinh
-->
x
cosh
-->
x
+
1
=
sgn
-->
x
cosh
-->
x
− − -->
1
cosh
-->
x
+
1
=
e
x
− − -->
1
e
x
+
1
{\displaystyle {\begin{aligned}\sinh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\sqrt {2(\cosh x+1)}}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{2}}}\\[6px]\cosh \left({\frac {x}{2}}\right)&={\sqrt {\frac {\cosh x+1}{2}}}\\[6px]\tanh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\cosh x+1}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{\cosh x+1}}}={\frac {e^{x}-1}{e^{x}+1}}\end{aligned}}}
di mana sgn adalah fungsi tanda .
Jika
x
≠ ≠ -->
0
{\displaystyle x\neq 0}
[ 3]
tanh
-->
(
x
2
)
=
cosh
-->
x
− − -->
1
sinh
-->
x
=
coth
-->
x
− − -->
csch
-->
x
{\displaystyle \tanh \left({\frac {x}{2}}\right)={\frac {\cosh x-1}{\sinh x}}=\coth x-\operatorname {csch} x}
Rumus kuadrat
sinh
2
-->
x
=
1
2
(
cosh
-->
2
x
− − -->
1
)
cosh
2
-->
x
=
1
2
(
cosh
-->
2
x
+
1
)
{\displaystyle {\begin{aligned}\sinh ^{2}x&={\frac {1}{2}}(\cosh 2x-1)\\\cosh ^{2}x&={\frac {1}{2}}(\cosh 2x+1)\end{aligned}}}
Pertidaksamaan
Pertidaksamaan berikut sangat berguna dalam statistik, yaitu
cosh
-->
(
t
)
≤ ≤ -->
e
t
2
/
2
{\displaystyle \operatorname {cosh} (t)\leq e^{t^{2}/2}}
[ 4]
Fungsi invers sebagai logaritma
arsinh
-->
(
x
)
=
ln
-->
(
x
+
x
2
+
1
)
arcosh
-->
(
x
)
=
ln
-->
(
x
+
x
2
− − -->
1
)
x
⩾ ⩾ -->
1
artanh
-->
(
x
)
=
1
2
ln
-->
(
1
+
x
1
− − -->
x
)
|
x
|
<
1
arcoth
-->
(
x
)
=
1
2
ln
-->
(
x
+
1
x
− − -->
1
)
|
x
|
>
1
arsech
-->
(
x
)
=
ln
-->
(
1
x
+
1
x
2
− − -->
1
)
=
ln
-->
(
1
+
1
− − -->
x
2
x
)
0
<
x
⩽ ⩽ -->
1
arcsch
-->
(
x
)
=
ln
-->
(
1
x
+
1
x
2
+
1
)
=
ln
-->
(
1
+
1
+
x
2
x
)
x
≠ ≠ -->
0
{\displaystyle {\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&&x\geqslant 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)&&|x|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)&&|x|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)&&0<x\leqslant 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)=\ln \left({\frac {1+{\sqrt {1+x^{2}}}}{x}}\right)&&x\neq 0\end{aligned}}}
Turunan
d
d
x
sinh
-->
x
=
cosh
-->
x
d
d
x
cosh
-->
x
=
sinh
-->
x
d
d
x
tanh
-->
x
=
1
− − -->
tanh
2
-->
x
=
sech
2
-->
x
=
1
cosh
2
-->
x
d
d
x
coth
-->
x
=
1
− − -->
coth
2
-->
x
=
− − -->
csch
2
-->
x
=
− − -->
1
sinh
2
-->
x
x
≠ ≠ -->
0
d
d
x
sech
-->
x
=
− − -->
tanh
-->
x
sech
-->
x
d
d
x
csch
-->
x
=
− − -->
coth
-->
x
csch
-->
x
x
≠ ≠ -->
0
d
d
x
arsinh
-->
x
=
1
x
2
+
1
d
d
x
arcosh
-->
x
=
1
x
2
− − -->
1
1
<
x
d
d
x
artanh
-->
x
=
1
1
− − -->
x
2
|
x
|
<
1
d
d
x
arcoth
-->
x
=
1
1
− − -->
x
2
1
<
|
x
|
d
d
x
arsech
-->
x
=
− − -->
1
x
1
− − -->
x
2
0
<
x
<
1
d
d
x
arcsch
-->
x
=
− − -->
1
|
x
|
1
+
x
2
x
≠ ≠ -->
0
{\displaystyle {\begin{aligned}{\frac {d}{dx}}\sinh x&=\cosh x\\{\frac {d}{dx}}\cosh x&=\sinh x\\{\frac {d}{dx}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {d}{dx}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}&&x\neq 0\\{\frac {d}{dx}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\\{\frac {d}{dx}}\operatorname {csch} x&=-\coth x\operatorname {csch} x&&x\neq 0\\{\frac {d}{dx}}\operatorname {arsinh} x&={\frac {1}{\sqrt {x^{2}+1}}}\\{\frac {d}{dx}}\operatorname {arcosh} x&={\frac {1}{\sqrt {x^{2}-1}}}&&1<x\\{\frac {d}{dx}}\operatorname {artanh} x&={\frac {1}{1-x^{2}}}&&|x|<1\\{\frac {d}{dx}}\operatorname {arcoth} x&={\frac {1}{1-x^{2}}}&&1<|x|\\{\frac {d}{dx}}\operatorname {arsech} x&=-{\frac {1}{x{\sqrt {1-x^{2}}}}}&&0<x<1\\{\frac {d}{dx}}\operatorname {arcsch} x&=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}&&x\neq 0\end{aligned}}}
Turunan detik
- Dalam pengembangan -
Standar integral
∫ ∫ -->
sinh
-->
(
a
x
)
d
x
=
a
− − -->
1
cosh
-->
(
a
x
)
+
C
∫ ∫ -->
cosh
-->
(
a
x
)
d
x
=
a
− − -->
1
sinh
-->
(
a
x
)
+
C
∫ ∫ -->
tanh
-->
(
a
x
)
d
x
=
a
− − -->
1
ln
-->
(
cosh
-->
(
a
x
)
)
+
C
∫ ∫ -->
coth
-->
(
a
x
)
d
x
=
a
− − -->
1
ln
-->
(
sinh
-->
(
a
x
)
)
+
C
∫ ∫ -->
sech
-->
(
a
x
)
d
x
=
a
− − -->
1
arctan
-->
(
sinh
-->
(
a
x
)
)
+
C
∫ ∫ -->
csch
-->
(
a
x
)
d
x
=
a
− − -->
1
ln
-->
(
tanh
-->
(
a
x
2
)
)
+
C
=
a
− − -->
1
ln
-->
|
csch
-->
(
a
x
)
− − -->
coth
-->
(
a
x
)
|
+
C
{\displaystyle {\begin{aligned}\int \sinh(ax)\,dx&=a^{-1}\cosh(ax)+C\\\int \cosh(ax)\,dx&=a^{-1}\sinh(ax)+C\\\int \tanh(ax)\,dx&=a^{-1}\ln(\cosh(ax))+C\\\int \coth(ax)\,dx&=a^{-1}\ln(\sinh(ax))+C\\\int \operatorname {sech} (ax)\,dx&=a^{-1}\arctan(\sinh(ax))+C\\\int \operatorname {csch} (ax)\,dx&=a^{-1}\ln \left(\tanh \left({\frac {ax}{2}}\right)\right)+C=a^{-1}\ln \left|\operatorname {csch} (ax)-\coth(ax)\right|+C\end{aligned}}}
∫ ∫ -->
1
a
2
+
u
2
d
u
=
arsinh
-->
(
u
a
)
+
C
∫ ∫ -->
1
u
2
− − -->
a
2
d
u
=
arcosh
-->
(
u
a
)
+
C
∫ ∫ -->
1
a
2
− − -->
u
2
d
u
=
a
− − -->
1
artanh
-->
(
u
a
)
+
C
u
2
<
a
2
∫ ∫ -->
1
a
2
− − -->
u
2
d
u
=
a
− − -->
1
arcoth
-->
(
u
a
)
+
C
u
2
>
a
2
∫ ∫ -->
1
u
a
2
− − -->
u
2
d
u
=
− − -->
a
− − -->
1
arsech
-->
(
u
a
)
+
C
∫ ∫ -->
1
u
a
2
+
u
2
d
u
=
− − -->
a
− − -->
1
arcsch
-->
|
u
a
|
+
C
{\displaystyle {\begin{aligned}\int {{\frac {1}{\sqrt {a^{2}+u^{2}}}}\,du}&=\operatorname {arsinh} \left({\frac {u}{a}}\right)+C\\\int {{\frac {1}{\sqrt {u^{2}-a^{2}}}}\,du}&=\operatorname {arcosh} \left({\frac {u}{a}}\right)+C\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {artanh} \left({\frac {u}{a}}\right)+C&&u^{2}<a^{2}\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {arcoth} \left({\frac {u}{a}}\right)+C&&u^{2}>a^{2}\\\int {{\frac {1}{u{\sqrt {a^{2}-u^{2}}}}}\,du}&=-a^{-1}\operatorname {arsech} \left({\frac {u}{a}}\right)+C\\\int {{\frac {1}{u{\sqrt {a^{2}+u^{2}}}}}\,du}&=-a^{-1}\operatorname {arcsch} \left|{\frac {u}{a}}\right|+C\end{aligned}}}
Referensi
^ "FUNGSI HIPERBOLIK DAN INVERSNYA" . DIGILIB UNNES. Diarsipkan dari versi asli tanggal 2019-08-15. Diakses tanggal 2014-05-28 . ^ Martin, George E. (1986). The foundations of geometry and the non-euclidean plane (edisi ke-1st corr.). New York: Springer-Verlag. hlm. 416. ISBN 3-540-90694-0 . ^ "Prove the identity" . StackExchange (mathematics)Diarsipkan dari versi asli tanggal 2023-07-26. Diakses tanggal 24 January 2016 . ^ Audibert, Jean-Yves (2009). "Fast learning rates in statistical inference through aggregation". The Annals of Statistics. hlm. 1627. [1] Diarsipkan 2023-07-26 di Wayback Machine .
Fungsi polinomial Fungsi aljabar Fungsi dalam Fungsi trigonometri Fungsi berdasarkan Fungsi berdasarkan Fungsi khusus Fungsi lainnya
Perpustakaan nasional Lain-lain
![{\displaystyle {\begin{aligned}\sinh(x+y)&=\sinh x\cosh y+\cosh x\sinh y\\\cosh(x+y)&=\cosh x\cosh y+\sinh x\sinh y\\[6px]\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a343fcb86c5fa15cceb217f4e92410c70a6e200)
![{\displaystyle {\begin{aligned}\sinh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\sqrt {2(\cosh x+1)}}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{2}}}\\[6px]\cosh \left({\frac {x}{2}}\right)&={\sqrt {\frac {\cosh x+1}{2}}}\\[6px]\tanh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\cosh x+1}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{\cosh x+1}}}={\frac {e^{x}-1}{e^{x}+1}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/412a4ffd109486f684e515634b33447b13444954)
