Hyperbolic functions: Difference between revisions

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[[File:sinh cosh tanh.svg|333x333px|thumb]]

In [[mathematics]], '''hyperbolic functions''' are analogues of the ordinary [[trigonometric function]]s, but defined using the [[hyperbola]] rather than the [[circle]]. Just as the points {{math|(cos ''t'', sin ''t'')}} form a [[unit circle|circle with a unit radius]], the points {{math|(cosh ''t'', sinh ''t'')}} form the right half of the [[unit hyperbola]]. Also, similarly to how the derivatives of {{math|sin(''t'')}} and {{math|cos(''t'')}} are {{math|cos(''t'')}} and {{math|–sin(''t'')}} respectively, the derivatives of {{math|sinh(''t'')}} and {{math|cosh(''t'')}} are {{math|cosh(''t'')}} and {{math|+sinh(''t'')}} respectively.

Hyperbolic functions occurare inused to express the calculations[[angle of angles and distancesparallelism]] in [[hyperbolic geometry]]. They are used to express [[Lorentz boost]]s as [[hyperbolic rotation]]s in [[special relativity]]. They also occur in the solutions of many linear [[differential equation]]s (such as the equation defining a [[catenary]]), [[Cubic equation#Hyperbolic solution for one real root|cubic equations]], and [[Laplace's equation]] in [[Cartesian coordinates]]. [[Laplace's equation]]s are important in many areas of [[physics]], including [[electromagnetic theory]], [[heat transfer]], and [[fluid dynamics]], and [[special relativity]].

The basic hyperbolic functions are:<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|authorlink=Eric W. Weisstein|title=Hyperbolic Functions| url=https://mathworld.wolfram.com/HyperbolicFunctions.html|access-date=2020-08-29|website=mathworld.wolfram.com|language=en}}</ref>

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By [[Lindemann–Weierstrass theorem]], the hyperbolic functions have a [[transcendental number|transcendental value]] for every non-zero [[algebraic number|algebraic value]] of the argument.<ref>{{Cite book | jstor=10.4169/j.ctt5hh8zn| title=Irrational Numbers | volume=11| last1=Niven| first1=Ivan| year=1985| publisher=Mathematical Association of America| isbn=9780883850381}}</ref>

== History ==

Hyperbolic functions were introduced in the 1760s independently by [[Vincenzo Riccati]] and [[Johann Heinrich Lambert]].<ref>Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. ''Euler at 300: an appreciation.'' Mathematical Association of America, 2007. Page 100.</ref> Riccati used {{math|''Sc.''}} and {{math|''Cc.''}} ({{lang|la|sinus/cosinus circulare}}) to refer to circular functions and {{math|''Sh.''}} and {{math|''Ch.''}} ({{lang|la|sinus/cosinus hyperbolico}}) to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today.<ref>Becker, Georg F. ''Hyperbolic functions.'' Read Books, 1931. Page xlviii.</ref> The abbreviations {{math|sh}}, {{math|ch}}, {{math|th}}, {{math|cth}} are also currently used, depending on personal preference.

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>

[[Roger Cotes]] suggested to modify the trigonometric functions using the [[imaginary unit]] <math>i=\sqrt{-1} </math> to obtain an oblate [[spheroid]] from a prolate one.<ref name=":0" />

Hyperbolic functions were formally introduced in 1757 by [[Vincenzo Riccati]].<ref name=":0" /><ref name=":3" /><ref name=":4" /> Riccati used {{math|''Sc.''}} and {{math|''Cc.''}} ({{lang|la|sinus/cosinus circulare}}) to refer to circular functions and {{math|''Sh.''}} and {{math|''Ch.''}} ({{lang|la|sinus/cosinus hyperbolico}}) to refer to hyperbolic functions.<ref name=":0" /> As early as 1759, [[François Daviet de Foncenex|Daviet de Foncenex]] showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended [[de Moivre's formula]] to hyperbolic functions.<ref name=":4" /><ref name=":0" />

During the 1760s, [[Johann Heinrich Lambert]] systematized the use functions and provided exponential expressions in various publications.<ref name=":0" /><ref name=":4">Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. ''Euler at 300: an appreciation.'' Mathematical Association of America, 2007. Page 100.</ref> Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.<ref name=":4" /><ref>Becker, Georg F. ''Hyperbolic functions.'' Read Books, 1931. Page xlviii.</ref>

== Notation ==

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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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==Useful relations==

{{Anchor|Osborn}}

The hyperbolic functions satisfy many identities, all of them similar in form to the [[trigonometric identity|trigonometric identities]]. In fact, '''Osborn's rule'''<ref name="Osborn, 1902" /> states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for <math>\theta</math>, <math>2\theta</math>, <math>3\theta</math> or <math>\theta</math> and <math>\varphi</math> into a hyperbolic identity, by:
# expanding it completely in terms of integral powers of sines and cosines,
# changing sine to sinh and cosine to cosh, and
# switching the sign of every term containing a product of two sinhs.

Odd and even functions:

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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) \\