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. Daftar fungsi matematika
Fungsi polinomial Fungsi konstan (0) Fungsi linear (1) Fungsi kuadrat (2) Fungsi kubik (3) Fungsi kuartik (4) Fungsi kuintik (5) Fungsi aljabar Fungsi rasional Fungsi eksponensial Fungsi hiperbolik Fungsi logaritma Berdasarkan basis teriterasi Superlogaritma 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
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