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angle measure, for example arsinh ⁡ ( sinh ⁡ a ) = a {\displaystyle \operatorname {arsinh} (\sinh a)=a} and sinh ⁡ ( arsinh ⁡ x ) = x . {\displaystyle...

functions. The inverse hyperbolic functions are: inverse hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh") inverse hyperbolic...

α e − δ x − θ arsinh ⁡ ( x ) , {\displaystyle P(x)={\frac {M}{1+\alpha e^{-\delta x-\theta \operatorname {arsinh} (x)}}},} where arsinh {\displaystyle...

here. ∫ arsinh ⁡ ( a x ) d x = x arsinh ⁡ ( a x ) − a 2 x 2 + 1 a + C {\displaystyle \int \operatorname {arsinh} (ax)\,dx=x\operatorname {arsinh} (ax)-{\frac...

gd − 1 ⁡ φ = arsinh ⁡ ( tan ⁡ φ ) , {\displaystyle \operatorname {gd} ^{-1}\varphi =\operatorname {arsinh} (\tan \varphi ),} and arsinh {\displaystyle...

{\pi }{2}}} as ⁠ arsinh ⁡ ( tan ⁡ x ) {\displaystyle \operatorname {arsinh} (\tan x)} ⁠, where arsinh {\displaystyle \operatorname {arsinh} } is the inverse...

{1-V^{2}/c^{2}}}} Phase 3 : c / a   arsinh ( a   T a / c ) {\displaystyle :\quad c/a\ {\text{arsinh}}(a\ T_{a}/c)\,} Phase 4 : c / a   arsinh ( a   T a / c ) {\displaystyle...

equations x = a arsinh ⁡ ( p a ) + T 0 E p + α , y = a 2 + p 2 + T 0 2 E a p 2 + β . {\displaystyle {\begin{aligned}x&=a\operatorname {arsinh} \left({\frac...

κ ⁡ ( x ) = exp ⁡ ( 1 κ arsinh ( κ x ) ) . {\displaystyle \exp _{\kappa }(x)=\exp {\Bigg (}{\frac {1}{\kappa }}{\text{arsinh}}(\kappa x){\Bigg )}.} The...

2 = c tanh ⁡ { arsinh ⁡ ( u 0 γ 0 + α T c ) } X ( T ) = X 0 + c 2 α ( 1 + ( u 0 γ 0 + α T c ) 2 − γ 0 ) = X 0 + c 2 α { cosh ⁡ [ arsinh ⁡ ( u 0 γ 0 + α...

arguments. An alternative expression in terms of the inverse hyperbolic sine arsinh is numerically well behaved for real arguments | ϕ | < 1 2 π {\textstyle...

arsinh ⁡ ( tan ⁡ ϕ ) . {\displaystyle \psi =\operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\operatorname {sec} t\,\mathrm {d} t=\operatorname {arsinh}...

+ r x | = r − a arsinh ⁡ a x {\displaystyle \int {\frac {r\,dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\,\operatorname {arsinh} {\frac {a}{x}}}...

b 2 [ θ 1 + θ 2 + arsinh ⁡ θ ] θ 1 θ 2 . {\displaystyle {\frac {b}{2}}\left[\theta \,{\sqrt {1+\theta ^{2}}}+\operatorname {arsinh} \theta \right]_{\theta...

valid:[citation needed] sl ( 1 2 arcsl ⁡ x ) = sin ( 1 2 arcsin ⁡ x ) sech ( 1 2 arsinh ⁡ x ) sl ( 1 2 arcsl ⁡ x ) 2 = tan ( 1 4 arcsin ⁡ x 2 ) {\displaystyle...

( 1 + 2 ) 2 n − 1 3 ( 2 − 2 ) 1 2 n = 2 7 4 3 sinh ⁡ ( n log ⁡ ( 2 ) + arsinh ⁡ ( 3 2 5 4 ) ) {\displaystyle {\begin{aligned}l_{n}&={\frac {2}{3}}\left(1+{\sqrt...

the distance function is d ( u , v ) = arcosh ⁡ ( 1 + δ ( u , v ) ) = 2 arsinh ⁡ δ ( u , v ) 2 = 2 ln ⁡ ‖ u − v ‖ + ‖ u ‖ 2 ‖ v ‖ 2 − 2 u ⋅ v + 1 ( 1 −...

{\displaystyle \cosh x} , tanh ⁡ x {\displaystyle \tanh x} arsinh ⁡ x {\displaystyle \operatorname {arsinh} x} , arcosh ⁡ x {\displaystyle \operatorname {arcosh}...

⁠ and ⁠ p 2 {\displaystyle p_{2}} ⁠, ) arsinh ⁡ x = {\displaystyle {\vphantom {\Big )}}\operatorname {arsinh} x={}} ln ⁡ ( x + x 2 + 1 ) {\displaystyle...

defined as: M ( a , b ) = a − b 2 arsinh ⁡ ( a − b a + b ) {\displaystyle M(a,b)={\frac {a-b}{2\operatorname {arsinh} \left({\frac {a-b}{a+b}}\right)}}}...

) ! x 2 n − 1 = x − x 3 3 + 2 x 5 15 − 17 x 7 315 + ⋯ for  | x | < π 2 arsinh ⁡ x = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1...

2 3 sinh ⁡ ( 1 3 arsinh ⁡ ( 3 3 2 ) ) . {\displaystyle 1/\psi ={\frac {2}{\sqrt {3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\frac {3{\sqrt...

{\frac {d}{dx}}\sinh x=\cosh x} d d x arsinh ⁡ x = 1 1 + x 2 {\displaystyle {\frac {d}{dx}}\operatorname {arsinh} x={\frac {1}{\sqrt {1+x^{2}}}}} d d x...

hyperbolic function: τ ( t ) = c g ln ⁡ ( g t c + 1 + ( g t c ) 2 ) = c g arsinh ⁡ ( g t c ) {\displaystyle \tau (t)={\frac {c}{g}}\ln \left({\frac {gt}{c}}+{\sqrt...

< 1 is F − 1 ( p ) = − 2 π arsinh [ cot ⁡ ( π p ) ] , {\displaystyle F^{-1}(p)=-{\frac {2}{\pi }}\,\operatorname {arsinh} \!\left[\cot(\pi \,p)\right]\...

mapping is: N   =   2 log 2 ⁡ ( 1 2 Q + 1 4 Q 2 + 1 )   =   2 ln ⁡ ( 2 ) arsinh ⁡ ( 1 2 Q ) . {\displaystyle N\ =\ 2\log _{2}\left({\frac {1}{2Q}}+{\sqrt...

hyperbolic functions arcosh {\displaystyle {\text{arcosh}}} or arsinh {\displaystyle {\text{arsinh}}} (or involving inverse trigonometric functions if the cosmological...

{\displaystyle \cosh x} ⁠, etc. Inverse hyperbolic functions: ⁠ arsinh ⁡ x {\displaystyle \operatorname {arsinh} x} ⁠, ⁠ arcosh ⁡ x {\displaystyle \operatorname {arcosh}...

inverse the area hyperbolic sine: a = arsinh ⁡ y = ln ⁡ ( y + y 2 + 1 ) . {\displaystyle a=\operatorname {arsinh} y=\ln \left(y+{\sqrt {y^{2}+1}}\right)...

arsinh ⁡ ( 1 2 δ h )   . {\displaystyle h\operatorname {D} =-\ln(1-\nabla _{h})\quad {\text{ and }}\quad h\operatorname {D} =2\operatorname {arsinh}...

cases c = ± 1 {\displaystyle c=\pm 1} give the formulas ∫   d x x 2 + 1 = arsinh ⁡ ( x ) + C , ∫   d x x 2 − 1 = arcosh ⁡ ( x ) + C . ( x > 1 ) {\displaystyle...

u sec ⁡ x d x = d u  or  d x = sech ⁡ u d u u = arcosh ⁡ ( sec ⁡ x ) = arsinh ⁡ ( tan ⁡ x ) = ln ⁡ | sec ⁡ x + tan ⁡ x | {\displaystyle {\begin{aligned}\sec...

)=\exp(-{\sqrt {6}}\,\pi )^{2}=} = q ⟨ tanh ⁡ { 1 2 arsinh ⁡ [ ( 2 − 1 ) 2 ] } ⟩ 2 = q ⟨ tanh ⁡ { 1 4 arsinh ⁡ [ ( 2 − 1 ) 2 ] } 2 ⟩ {\displaystyle =q{\biggl...

\sinh(\mathrm {arsinh} (1/\varepsilon )/n)} and an imaginary semi-axis of length of cosh ⁡ ( a r s i n h ( 1 / ε ) / n ) . {\displaystyle \cosh(\mathrm {arsinh} (1/\varepsilon...

functions (arcsin, arccos, arctan, etc.) and inverse hyperbolic functions (arsinh, arcosh, artanh, etc.) can be defined in terms of logarithms and their principal...

x ) ,   arcsec ⁡ ( x ) ,   arsinh ⁡ ( x ) , {\displaystyle \arctan(x),\ \operatorname {arcsec}(x),\ \operatorname {arsinh} (x),} etc. A – algebraic functions...

}{8}}\right),} or the hyperbolic sine σ = exp ⁡ ( arsinh ⁡ ( 1 ) ) . {\displaystyle \sigma =\exp(\operatorname {arsinh} (1)).} ⁠ σ {\displaystyle \sigma } ⁠ and...

using the relation sinh − 1 ⁡ z = arsinh ⁡ z = ln ⁡ ( z + z 2 + 1 ) {\displaystyle \sinh ^{-1}{z}=\operatorname {arsinh} {z}=\ln(z+{\sqrt {z^{2}+1}})} :...

92638 07403 ln ⁡ ( 1 + 2 ) + 2 = arsinh ⁡ ( 1 ) + 2 {\displaystyle \ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arsinh} (1)+{\sqrt {2}}} Before 1891...

sinh ⁡ ( 1 3 arsinh ⁡ ( − 3 4 3 2 ) ) . {\displaystyle 1/\varsigma =-2{\sqrt {\frac {2}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left(-{\frac...

3 arsinh ⁡ ( 13 7 7 ) ) ) ≈ 0.307401 {\displaystyle Z_{\text{c}}={\frac {1}{32}}\left(11-2{\sqrt {7}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\frac...

for the inverse transformation are: σ = arccos ⁡ ( R 2 − a 2 Q ) , τ = arsinh ⁡ ( 2 a z Q ) , ϕ = arctan ⁡ ( y x ) , {\displaystyle {\begin{aligned}\sigma...

{1}{2}}=2\operatorname {artanh} (2-{\sqrt {3}})=} = arsinh ⁡ 1 3 3 = arcosh ⁡ 2 3 3 ≈ 0.549 {\displaystyle =\operatorname {arsinh} {\frac {1}{3}}{\sqrt {3}}=\operatorname...

] if    4 p 3 + 27 q 2 > 0    and    p < 0 , t 0 = − 2 p 3 sinh ⁡ [ 1 3 arsinh ⁡ ( 3 q 2 p 3 p ) ] if    p > 0. {\displaystyle {\begin{aligned}t_{0}&=-2{\frac...

instance, the inverse of the hyperbolic sine function is typically written as arsinh(x). The expressions like sin−1(x) can still be useful to distinguish the...

2 f ∫ − 2 f 2 f 1 + x 2 4 f 2 d x = ∫ − 1 1 1 + t 2 d t ( x = 2 f t ) = arsinh ⁡ ( 1 ) + 2 = ln ⁡ ( 1 + 2 ) + 2 . {\displaystyle {\begin{aligned}P&:={\frac...

hyperbolic plane is equal to 2 arsinh ⁡ ( 2 cos ⁡ ( 2 π / 7 ) − 1 8 cos ⁡ ( π / 7 ) + 7 ) ≃ 0.2629 {\displaystyle 2\operatorname {arsinh} \left({\sqrt {\frac {2\cos(2\pi...

– argument of the minimum. arsech – inverse hyperbolic secant function. arsinh – inverse hyperbolic sine function. artanh – inverse hyperbolic tangent...

2 ) . {\displaystyle {\frac {n+{\sqrt {n^{2}+4}}}{2}}=e^{\operatorname {arsinh(n/2)} }.} A tangent half-angle formula gives cot ⁡ θ = cot 2 ⁡ θ 2 − 1 2...

x=\operatorname {artanh} \,(\tanh r\cos \theta )} y = arsinh ( sinh ⁡ r sin ⁡ θ ) {\displaystyle y=\operatorname {arsinh} \,(\sinh r\sin \theta )} r = arcosh ( cosh...