Hyperbolic functions: Difference between revisions

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== History ==

The first known calculation of a hyperbolic trigonometry problem is attributed to [[Gerardus Mercator]] when issuing the [[Mercator projection|Mercator map projection]] circa 1566. It requires tabulating solutions to a [[transcendental equation]] involving hyperbolic functions.<ref name=":3">{{Cite book |last=George F. Becker |url=https://archive.org/details/hyperbolicfuncti027682mbp/page/n49/mode/2up?q=mercator |title=Hyperbolic Functions |last2=C. E. Van Orstrand |date=1909 |publisher=The Smithsonian Institution |others=Universal Digital Library}}</ref>

The first to suggest a similarity between the sector of the circle and that of the hyperbola was [[Isaac Newton]] in his 1687 [[Philosophiæ Naturalis Principia Mathematica|''Principia Mathematica'']].<ref name=":0">{{Cite book |last=McMahon |first=James |url=https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up |title=Hyperbolic Functions |date=1896 |publisher=John Wiley And Sons |others=Osmania University, Digital Library Of India}}</ref>

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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>

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Since the [[Circular sector#Area|area of a circular sector]] with radius {{mvar|r}} and angle {{mvar|u}} (in radians) is {{math|1=''r''²''u''/2}}, it will be equal to {{mvar|u}} when {{math|1=''r'' = {{sqrt|2}}}}. In the diagram, such a circle is tangent to the hyperbola ''xy'' = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red regions together depict a [[hyperbolic sector]] with area corresponding to hyperbolic angle magnitude.

The legs of the two [[right triangle]]s with [[hypotenuse]] on the ray defining the angles are of length {{radic|2}} times the circular and hyperbolic functions.

The hyperbolic angle is an [[invariant measure]] with respect to the [[squeeze mapping]], just as the circular angle is invariant under rotation.<ref>[[Mellen W. Haskell|Haskell, Mellen W.]], "On the introduction of the notion of hyperbolic functions", [[Bulletin of the American Mathematical Society]] '''1''':6:155–9, [https://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf full text]</ref>

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The [[Gudermannian function]] gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers.

The graph of the function {{mathtmath|''a'' \cosh (''x''/''a'')}} is the [[catenary]], the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.

==Relationship to the exponential function==

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\cosh(ix) &= \frac{1}{2} \left(e^{i x} + e^{-i x}\right) = \cos x \\

\sinh(ix) &= \frac{1}{2} \left(e^{i x} - e^{-i x}\right) = i \sin x \\

\tanh(ix) &= i \tan x \\

\cosh(x+iy) &= \cosh(x) \cos(y) + i \sinh(x) \sin(y) \\

\sinh(x+iy) &= \sinh(x) \cos(y) + i \cosh(x) \sin(y) \\

\tanh(ixx+iy) &= \frac{\tanh(x) + i \tan(y)}{1 + i \tanh(x) \tan(y)} \\

\cosh x &= \cos(ix) \\

\sinh x &= - i \sin(ix) \\