Hyperbolic functions: Difference between revisions

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==Definitions==

[[File:Cartesian_hyperbolic_rhombus.svg|thumb|right|250px|Right triangles with legs proportional to sinh and cosh]]

[[File:sinh cosh tanh.svg|thumb|<span style="color:#b30000;">sinh</span>, <span style="color:#00b300;">cosh</span> and <span style="color:#0000b3;">tanh</span>]]

With [[hyperbolic angle]] ''u'', the hyperbolic functions sinh and cosh can defined with the [[exponential function]] e^u.<ref name=":1" /><ref name=":2" /> In the figure

[[File:csch sech coth.svg|thumb|<span style="color:#b30000;">csch</span>, <span style="color:#00b300;">sech</span> and <span style="color:#0000b3;">coth</span>]]

<math>A =(e^{-u}, e^u), \ B=(e^u, \ e^{-u}), \ OA + OB = OC </math> .

There are various equivalent ways to define the hyperbolic functions.

=== Exponential definitions ===

[[File:Hyperbolic and exponential; sinh.svg|thumb|right|{{math|sinh ''x''}} is half the [[Subtraction|difference]] of {{math|''e^x''}} and {{math|''e''^−''x''}}]]

[[File:Hyperbolic and exponential; cosh.svg|thumb|right|{{math|cosh ''x''}} is the [[Arithmetic mean|average]] of {{math|''e^x''}} and {{math|''e''^−''x''}}]]

In terms of the [[exponential function]]:<ref name=":1" /><ref name=":2" />

* Hyperbolic sine: the [[odd part of a function|odd part]] of the exponential function, that is, <math display="block"> \sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}.</math>

* Hyperbolic cosine: the [[even part of a function|even part]] of the exponential function, that is, <math display="block"> \cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}.</math>

[[File:sinh cosh tanh.svg|thumb|<span style="color:#b30000;">sinh</span>, <span style="color:#00b300;">cosh</span> and <span style="color:#0000b3;">tanh</span>]]

[[File:csch sech coth.svg|thumb|<span style="color:#b30000;">csch</span>, <span style="color:#00b300;">sech</span> and <span style="color:#0000b3;">coth</span>]]

* Hyperbolic tangent: <math display="block">\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}}

= \frac{e^{2x} - 1} {e^{2x} + 1}.</math>