Fungsi hiperbolik
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 dan e −x
cosh x adalah rerata ex dan 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}
, maka[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 teori bilangan Fungsi trigonometri Fungsi berdasarkan huruf Yunani
Fungsi beta
Fungsi chi
Fungsi delta
Fungsi eta
Fungsi gamma
Fungsi digamma
Barnes
Meijer
banyak
eliptik
Hadamard
multivariabel
p-adik
q
taklengkap
Fungsi poligamma
Fungsi trigamma
Fungsi lambda
Dirchlet
modular
von Mangoldt
Fungsi mu
Fungsi phi
Fungsi pi
Fungsi sigma
Fungsi theta
Fungsi zeta
Fungsi berdasarkan nama matematikawan
Airy
Ackermann
Bessel
Bessel–Clifford
Bottcher
Chebyshev
Clausen
Dawson
Dirichlet
Faddeeva
Fermi–Dirac
Fresnel
Fox
Gudermann
Hermite
Fungsi Jacob
Kelvin
Fungsi Kummer
Fungsi Lambert
Lamé
Laguerre
Legendre
Liouville
Mathieu
Meijer
Mittag-Leffler
Painlevé
Riemann
Riesz
Scorer
Spence
von Mangoldt
Weierstrass
Fungsi khusus Fungsi lainnya
Aritmetik-geometrik
eliptik
Fungsi hiperbolik
K
sinkrotron
tabung parabolik
tanda tanya Minkowski
Pentasi
Student
Tetrasi
Perpustakaan nasional Lain-lain
The article is a derivative under the Creative Commons Attribution-ShareAlike License .
A link to the original article can be found here and attribution parties here
By using this site, you agree to the Terms of Use . Gpedia ® is a registered trademark of the Cyberajah Pty Ltd