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Areasinus hyperbolicus (abgekürzt arsinh {\displaystyle \operatorname {arsinh} } oder asinh {\displaystyle \operatorname {asinh} } ) und Areakosinus hyperbolicus...

von elementareren Funktionen berechnen: arsinh ⁡ x = ln ⁡ ( x + x 2 + 1 )   {\displaystyle \operatorname {arsinh} x=\ln \left(x+{\sqrt {x^{2}+1}}\right)\...

∫ 0 x arsinh ⁡ ( t ) t d t = 1 2 D 1 ( 1 ) [ 2 arsinh ⁡ ( x ) ] + 1 2 arsinh ⁡ ( x ) 2 {\displaystyle \int _{0}^{x}{\frac {\operatorname {arsinh} (t)}{t}}\mathrm...

arsinh ⁡ ( x ) x = d d x { 1 2 Li 2 ⁡ [ 1 − ( x 2 + 1 − x ) 2 ] + 1 2 arsinh ⁡ ( x ) 2 } {\displaystyle {\frac {\operatorname {arsinh} (x)}{x}}={\frac...

0 π ( b ⋅ arsinh ⁡ ( b d ) + d ⋅ arsinh ⁡ ( d b ) + O ( a ) ) {\displaystyle \;\;-{\frac {\mu _{0}}{\pi }}\left(b\cdot \operatorname {arsinh} \left({\frac...

4 n [ arsinh ⁡ ( z ) + z 1 + z 2 ] C 1 C 1 + 2 n = L 4 n [ ( C 1 + 2 n ) 1 + ( C 1 + 2 n ) 2 − C 1 1 + C 1 2 + arsinh ⁡ ( C 1 + 2 n ) − arsinh ⁡ ( C 1...

a 2 [ φ 1 + φ 2 + arsinh ⁡ φ ] φ 1 φ 2 {\displaystyle {\frac {a}{2}}\left[\varphi \,{\sqrt {1+\varphi ^{2}}}+\operatorname {arsinh} \varphi \right]_{\varphi...

]}_{x=0}^{x=1}=} = 1 2 2 arctan ⁡ ( 1 ) + 1 8 2 ln ⁡ ( 2 + 2 2 − 2 ) = 1 8 2 π + 1 4 2 arsinh ⁡ ( 1 ) {\displaystyle ={\frac {1}{2}}{\sqrt {2}}\arctan(1)+{\frac {1}{8}}{\sqrt...

x,} etc. Inverse hyperbolische Funktionen: arsinh ⁡ x ,   arcosh ⁡ x , {\displaystyle \operatorname {arsinh} x,\ \operatorname {arcosh} x,} etc. oder sich...

gefunden werden: Z S { 1 4 2 sinh ⁡ [ ( 2 n − 1 ) arsinh ⁡ ( 2 ) ] + 1 2 } = 3 16 cosh ⁡ [ ( 4 n − 2 ) arsinh ⁡ ( 2 ) ] + 1 16 = {\displaystyle ZS{\bigl \{}{\frac...

}}\right)\\&=\mathop {\rm {artanh}} \left(\sin \varphi \right)\\&=\mathop {\rm {arsinh}} \left(\tan \varphi \right)\end{aligned}}} Die Inverse ist die Gudermannfunktion:...

=\lambda ^{*}(6)} ∏ n = 1 ∞ t a n h [ π 2 6 ( 2 n − 1 ) ] = sech ⁡ { 1 2 arsinh ⁡ [ ( 2 − 1 ) 2 ] } 1 / 4 {\displaystyle \prod _{n=1}^{\infty }\mathrm {tanh}...

c 2 a 2 − c 2 arsinh ⁡ ( a 2 − c 2 c ) ) , {\displaystyle A=2\pi a\left(a+{\frac {c^{2}}{\sqrt {a^{2}-c^{2}}}}\,\operatorname {arsinh} \left({\frac {\sqrt...

c 2 a 2 − c 2 arsinh ⁡ ( a 2 − c 2 c ) )   , {\displaystyle A=2\pi a\left(a+{\frac {c^{2}}{\sqrt {a^{2}-c^{2}}}}\,\operatorname {arsinh} \left({\frac...

1 x arsinh ⁡ ( x ) n {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)^{n}} 1 n arsinh ⁡ ( x ) n D n [ 2 arsinh ⁡ ( x ) ] + 1 n + 1 arsinh ⁡ ( x...

Areasinus-Funktion ausdrücken: Φ ± 1 = e arsinh ⁡ ( ± 1 2 ) {\displaystyle \Phi ^{\pm 1}=e^{\operatorname {arsinh} \left(\pm {\frac {1}{2}}\right)}} Einsetzen...

{\displaystyle {\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x} arsinh ⁡ x {\displaystyle \operatorname {arsinh} x} 1 x 2 + 1 {\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}}...

{\displaystyle t(a)={\frac {2}{3H_{0}{\sqrt {\Omega _{\Lambda }}}}}\;{\rm {arsinh\left({\sqrt {{\frac {\Omega _{\Lambda }}{\Omega _{\rm {M}}}}\left({\frac...

bestimmen und ergibt sich zu f ( ϵ ) = ϵ 2 − 1 ϵ 3 ( 3 ϵ + ( ϵ 2 − 3 ) ⋅ arsinh ⁡ ϵ 1 − ϵ 2 ) {\displaystyle f(\epsilon )={\frac {\epsilon ^{2}-1}{\epsilon...

e n = 1 4 { 11 3 + 16 3 14 sinh ⁡ [ 1 3 arsinh ⁡ ( 65 392 14 ) ] } 2 + 339 + 22 3 + 8 3 14 sinh ⁡ [ 1 3 arsinh ⁡ ( 65 392 14 ) ] + {\displaystyle \lim...

hyperbolicus her: Ti 2 ⁡ ( x 2 + 1 + x ) − Ti 2 ⁡ ( x 2 + 1 − x ) = 1 2 π arsinh ⁡ ( x ) {\displaystyle \operatorname {Ti} _{2}({\sqrt {x^{2}+1}}+x)-\operatorname...

dieses a {\displaystyle a} in w = 2 a ⋅ arsinh ⁡ ( 1 2 l a ) {\displaystyle w=2a\cdot \operatorname {arsinh} \left({\frac {{\frac {1}{2}}l}{a}}\right)}...

ein Maß für das Überschwingen: γ = arsinh ⁡ ( Ripple in dB ) n {\displaystyle \gamma ={\frac {\operatorname {arsinh} ({\text{Ripple in dB}})}{n}}} Die...

)=\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...

Z Q ( 1 2 P 2 n − 1 + 1 2 P 2 n − 2 + 1 2 ) = 1 2 cosh ⁡ [ ( 2 n − 1 ) arsinh ⁡ ( 1 ) ] 2 {\displaystyle (P_{2n-1})^{2}=ZQ({\tfrac {1}{2}}P_{2n-1}+{\tfrac...

Funktionswert: ∑ n = 1 ∞ 1 P ( 2 n − 1 ) = 2 2 π λ ∗ [ 16 π − 2 arsinh ⁡ ( 1 ) 2 ] K { λ ∗ [ 16 π − 2 arsinh ⁡ ( 1 ) 2 ] } {\displaystyle \sum _{n=1}^{\infty }{\frac...

hyperbolicus (tanh) führt zu den Areafunktionen: Areasinus hyperbolicus (arsinh), Areakosinus hyperbolicus (arcosh) und Areatangens hyperbolicus (artanh)...

Li 3 ⁡ ( − x 2 + 1 + x ) − Li 3 ⁡ ( − x 2 + 1 − x ) = 1 6 π 2 arsinh ⁡ ( x ) + 1 6 arsinh ⁡ ( x ) 3 {\displaystyle \operatorname {Li} _{3}(-{\sqrt...

{3b-a^{2}}}\cdot \sinh {\eta }-a\right)} mit η = 1 3 arsinh ⁡ ( Γ ) {\displaystyle \eta ={\frac {1}{3}}\operatorname {arsinh} {(\Gamma )}} Es ergibt sich eine reelle...

Abstandsfunktion wie folgt definiert: d ( u , v ) = arcosh ⁡ ( 1 + δ ( u , v ) ) = 2 ⋅ arsinh ⁡ ( δ ( u , v ) 2 ) = 2 ⋅ ln ⁡ ( ‖ u − v ‖ + ‖ u ‖ 2 ⋅ ‖ v ‖ 2 − 2 ⋅ u ⋅...

1 + x − 1 − x 2 1 + x 2 + 2 = sin ⁡ [ 1 2 arcsin ⁡ ( x ) ] sech ⁡ [ 1 2 arsinh ⁡ ( x ) ] . {\displaystyle \operatorname {sl} [{\frac {1}{2}}\operatorname...

\operatorname {arccot} x} (Arkussinus …), arsinh ⁡ x , arcosh ⁡ x , artanh ⁡ x , arcoth ⁡ x {\displaystyle \operatorname {arsinh} x,\operatorname {arcosh} x,\operatorname...

Punkten gemäß der Metrik der hyperbolischen Ebene: d ( p 1 , p 2 ) = 2 ⋅ arsinh ⁡ ( ‖ p 2 − p 1 ‖ 2 ⋅ y 1 ⋅ y 2 ) = 2 ⋅ artanh ⁡ ( ‖ p 2 − p 1 ‖ ‖ p 2 −...

+15)^{3/2}}}{\biggr ]}{\biggr \}}-} − 2 μ 20 ν + 15 5 ν 2 + 1 4 sinh ⁡ { 1 5 arsinh ⁡ [ 125 ν 2 + 1 4 ( 2 ν 2 + 1 − 2 ν + 1 ) ( 20 ν + 15 ) 3 / 2 ] } {\displaystyle...

7 ( x 2 + 1 ) 2 arsinh ⁡ ( x ) d x {\displaystyle \zeta (3)=\int _{0}^{\infty }{\frac {\pi ^{2}x}{7(x^{2}+1)^{2}\operatorname {arsinh} (x)}}\,\mathrm...

{4+6y-4y^{2}}}}}{\biggr ]}{\biggr \}}-} − 2 5 y − 1 / 4 50 + 75 y − 50 y 2 4 sinh ⁡ { 1 5 arsinh ⁡ [ 5 y 5 + 5 y 2 ( 2 − y ) 4 + 6 y − 4 y 2 ] } {\displaystyle -{\frac...

\left({\frac {1}{x}}\right)} arcsch ( x ) = arsinh ⁡ ( 1 x ) {\displaystyle \operatorname {arcsch} \,(x)=\operatorname {arsinh} \left({\frac {1}{x}}\right)} Trigonometrische...

(0)} Teil eines Integrals: 1 V 0 Z ( E F ) = ∫ 0 ℏ ω D d η η 2 + Δ 2 = arsinh ℏ ω D Δ ( 0 ) {\displaystyle {\frac {1}{V_{0}Z(E_{\text{F}})}}=\int _{0}^{\hbar...

average ARMA-Modell arsech area secans hyperbolicus Areasekans hyperbolicus arsinh area sinus hyperbolicus Areasinus hyperbolicus artanh area tangens hyperbolicus...

Pell-Zahlen: ∑ n = 1 ∞ 1 P ( 2 n − 1 ) = 2 2 π λ ∗ [ 16 π − 2 arsinh ⁡ ( 1 ) 2 ] K { λ ∗ [ 16 π − 2 arsinh ⁡ ( 1 ) 2 ] } {\displaystyle \sum _{n=1}^{\infty }{\frac...

1 2 ln ⁡ 1 + sin ⁡ φ 1 − sin ⁡ φ = artanh ⁡ ( sin ⁡ φ ) (nach Gl. 4) = arsinh ⁡ ( tan ⁡ φ ) (nach Gl. 3) = arcosh ⁡ ( sec ⁡ φ ) {\displaystyle...

. Integrieren liefert arsinh ⁡ ( u ) = k ⋅ ln ⁡ x + C {\displaystyle \operatorname {arsinh} (u)=k\cdot \ln x+C} (siehe arsinh), sowie Rücksubstitution...

\operatorname {arcosh} x} asinh Areasinus hyperbolicus arsinh ⁡ x {\displaystyle \operatorname {arsinh} x} atanh Areatangens hyperbolicus artanh ⁡ x {\displaystyle...

1 3 6 sinh ⁡ [ 1 3 arsinh ⁡ ( 9 2 6 O n ) ] {\displaystyle n={\tfrac {1}{3}}{\sqrt {6}}\sinh[{\tfrac {1}{3}}\operatorname {arsinh} ({\tfrac {9}{2}}{\sqrt...

csch ⁡ [ 1 3 arsinh ⁡ ( 3 2 3 ) ] {\displaystyle \psi ={\tfrac {1}{2}}{\sqrt {3}}\operatorname {csch} [{\tfrac {1}{3}}\operatorname {arsinh} ({\tfrac {3}{2}}{\sqrt...

/ 4 3 − 1 ( 116 + 12 93 3 + 116 − 12 93 3 + 2 ) = 1 2 18 4 csch ⁡ [ 1 3 arsinh ⁡ ( 3 2 3 ) ] = 2 1 / 4 ψ {\displaystyle G(31)=2^{-3/4}3^{-1}({\sqrt[{3}]{116+12{\sqrt...

x-Werte gültig: Ti 2 ⁡ ( x 2 + 1 + x ) − Ti 2 ⁡ ( x 2 + 1 − x ) = 1 2 π arsinh ⁡ ( x ) {\displaystyle \operatorname {Ti} _{2}({\sqrt {x^{2}+1}}+x)-\operatorname...

Hyperbelfunktionen sinh, cosh, tanh mit deren Umkehrfunktionen (Areafunktionen: arsinh, arcosh, artanh) Kreiszahl-Taste ( π ≈ 3,141 592 …   {\displaystyle \pi...

Radius r = ln ⁡ 3 = 1 2 ln ⁡ 3 = artanh ⁡ 1 2 = 2 artanh ⁡ ( 2 − 3 ) = arsinh ⁡ 1 3 3 = arcosh ⁡ 2 3 3 ≈ 0,549 {\displaystyle r=\ln {\sqrt {3}}={\frac...

( x 2 + 1 ) 3 / 2 arsinh ⁡ ( x ) d x {\displaystyle G=\int _{0}^{\infty }{\frac {\pi \,x}{4\,(x^{2}+1)^{3/2}\operatorname {arsinh} (x)}}\,\mathrm {d}...