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Hyperbolic Functions

Exponential functions related to the hyperbola

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Hyperbolic Functions

Hyperbolic functions are combinations of the exponentials ex and e-x.

sinh x = (ex - e-x) / 2                  (1)

cosh x = (ex + e-x) / 2                   (2)

tanh x = sinh x / cosh x = (ex - e-x) / (ex + e-x)                          (3)

Values for Hyperbolic Functions

xsinh xcosh xtanh x
0 0.00000 1.0000 0.00000
0.1 0.10017 1.00500 0.09967
0.3 0.30452 1.04534 0.29131
0.4 0.41075 1.08107 0.37995
0.5 0.52110 1.12763 0.46212
0.6 0.63665 1.18547 0.53705
0.7 0.75858 1.25517 0.60437
0.8 0.88811 1.33743 0.66404
0.9 1.02652 1.43309 0.71630
1.0 1.17520 1.54308 0.76159
1.1 1.33565 1.66852 0.80050
1.2 1.50946 1.81066 0.83365
1.3 1.69838 1.97091 0.86172
1.4 1.90430 2.15090 0.88535
1.5 2.12928 2.35241 0.90515
1.6 2.37557 2.57746 0.92167
1.7 2.64563 2.82832 0.93541
1.8 2.94217 3.10747 0.94681
1.9 3.26816 3.41773 0.95624
2.0 3.62686 3.76220 0.96403
2.5 6.05020 6.13229 0.98661
3.0 10.0179 10.0677 0.99505
3.5 16.5426 16.5728 0.99818
4.0 27.2899 27.3082 0.99933
4.5 45.0030 45.0141 0.99975
5.0 74.2032 74.2099 0.99991

Inverse Hyperbolic Functions

sinh-1 x = loge(x + (x2 + 1)1/2)                 (4)

cosh-1 x = loge(x ± (x2 + 1)1/2)   x ≥ 1                     (5)

tanh-1 x = 1/2 loge((1 + x) / (1 - x))   x2 ‹ 1                        (6)

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  • en: hyperbolic functions exponential hyberbola
  • es: funciones hiperb√≥licas hyberbola exponencial
  • de: hyperbolischen Funktionen exponentielle hyberbola
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