Hyperbolic functions

Hyperbolic functions are different versions of trigonometric functions. They are defined using a hyperbola instead of a circle. Also, as the derivatives of $sin(t)$ and $cos(t)$ are $cos(t)$ and $-sin(t)$ , the derivatives of $sinh(t)$ and $cosh(t)$ are $cosh(t)$ and $sinh(t)$ .

The basic hyperbolic functions are:

hyperbolic sine " $sinh$ " (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),
hyperbolic cosine " $cosh$ " (/ˈkɒʃ, ˈkoʊʃ/),

Using these, you can get:

hyperbolic tangent " $tanh$ " (/ˈtæŋ, ˈtæntʃ, ˈθæn/),
hyperbolic cotangent " $coth$ " (/ˈkɒθ, ˈkoʊθ/),
hyperbolic secant " $sech$ " (/ˈsɛtʃ, ˈʃɛk/),
hyperbolic cosecant " $csch$ " or " $cosech$ " (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/)

and these have similar logic to the normal trigonometric functions.

The inverse hyperbolic functions are:

inverse hyperbolic sine " $arsinh$ " (also " $sinh -1$ ", " $asinh$ " or sometimes " $arcsinh$ ")
inverse hyperbolic cosine " $arcosh$ " (also " $cosh -1$ ", " $acosh$ " or sometimes " $arccosh$ ")
inverse hyperbolic tangent " $artanh$ " (also " $tanh -1$ ", " $atanh$ " or sometimes " $arctanh$ ")
inverse hyperbolic cotangent " $arcoth$ " (also " $coth -1$ ", " $acoth$ " or sometimes " $arccoth$ ")
inverse hyperbolic secant " $arsech$ " (also " $sech -1$ ", " $asech$ " or sometimes " $arcsech$ ")
inverse hyperbolic cosecant " $arcsch$ " (also " $arcosech$ ", " $csch -1$ ", " $cosech -1$ "," $acsch$ ", " $acosech$ ", or sometimes " $arccsch$ " or " $arccosech$ ")

Definitions

With hyperbolic angle u, the hyperbolic functions sinh and cosh can be defined using the exponential function e^u. In the figure $A = (e^{- u}, e^{u}), B = (e^{u}, e^{- u}), O A + O B = O C$ .

Exponential definitions

Hyperbolic sine: the odd part of the exponential function, that is, $\sinh x = \frac{e^{x} - e^{- x}}{2} = \frac{e^{2 x} - 1}{2 e^{x}} = \frac{1 - e^{- 2 x}}{2 e^{- x}} .$
Hyperbolic cosine: the even part of the exponential function, that is, $\cosh x = \frac{e^{x} + e^{- x}}{2} = \frac{e^{2 x} + 1}{2 e^{x}} = \frac{1 + e^{- 2 x}}{2 e^{- x}} .$

Hyperbolic tangent: $\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} = \frac{e^{2 x} - 1}{e^{2 x} + 1} .$
Hyperbolic cotangent: for $x \neq 0$ , $\coth x = \frac{\cosh x}{\sinh x} = \frac{e^{x} + e^{- x}}{e^{x} - e^{- x}} = \frac{e^{2 x} + 1}{e^{2 x} - 1} .$
Hyperbolic secant: $sech x = \frac{1}{\cosh x} = \frac{2}{e^{x} + e^{- x}} = \frac{2 e^{x}}{e^{2 x} + 1} .$
Hyperbolic cosecant: for $x \neq 0$ , $csch x = \frac{1}{\sinh x} = \frac{2}{e^{x} - e^{- x}} = \frac{2 e^{x}}{e^{2 x} - 1} .$

Related pages

References

1 2 Weisstein, Eric W. "Hyperbolic Functions". mathworld.wolfram.com. Retrieved 2020-08-29.
↑ (1999) Collins Concise Dictionary, 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p. 1386
1 2 "Hyperbolic Functions". www.mathsisfun.com. Retrieved 2020-08-29.
↑ Woodhouse, N. M. J. (2003), Special Relativity, London: Springer, p. 71, ISBN 978-1-85233-426-0
↑ Abramowitz, Milton; Stegun, Irene A., eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, ISBN 978-0-486-61272-0
Osborn, G. (July 1902). "Mnemonic for hyperbolic formulae". The Mathematical Gazette. 2 (34): 189. doi:10.2307/3602492. JSTOR 3602492. S2CID 125866575.

This short article about mathematics can be made longer. You can help Wikipedia by adding to it.

[:1-1] 1 2 Weisstein, Eric W. "Hyperbolic Functions". mathworld.wolfram.com. Retrieved 2020-08-29.

[2] (1999) Collins Concise Dictionary, 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p. 1386

[:2-3] 1 2 "Hyperbolic Functions". www.mathsisfun.com. Retrieved 2020-08-29.

[4] Woodhouse, N. M. J. (2003), Special Relativity, London: Springer, p. 71, ISBN 978-1-85233-426-0

[5] Abramowitz, Milton; Stegun, Irene A., eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, ISBN 978-0-486-61272-0

[Osborn,_1902-6] Osborn, G. (July 1902). "Mnemonic for hyperbolic formulae". The Mathematical Gazette. 2 (34): 189. doi:10.2307/3602492. JSTOR 3602492. S2CID 125866575.

Authority control databases
National	United States France BnF data Japan Czech Republic Israel
Other	Yale LUX