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Hyperbolic functions
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<h2 id="history">History</h2> <p>The first known calculation of a hyperbolic trigonometry problem is attributed to <a href="/facts/Gerardus_Mercator/PXBcFetG">Gerardus Mercator</a> when issuing the <a href="/facts/Mercator_projection/bYVL5bP9">Mercator map projection</a> circa 1566. It requires tabulating solutions to a transcendental equation involving hyperbolic functions.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>The first to suggest a similarity between the sector of the circle and that of the hyperbola was <a href="/facts/Isaac_Newton/4L7gTncN">Isaac Newton</a> in his 1687 <a href="/facts/Philosophi%25C3%25A6_Naturalis_Principia_Mathematica/DRdvI3yx"><i>Principia Mathematica</i></a>.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p><p><a href="/facts/Roger_Cotes/oIsrDoHu">Roger Cotes</a> suggested to modify the trigonometric functions using the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a> i = − 1 {\displaystyle i={\sqrt {-1}}} to obtain an oblate <a href="/facts/Spheroid/GJyyLxQ7">spheroid</a> from a prolate one.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p><p>Hyperbolic functions were formally introduced in 1757 by <a href="/facts/Vincenzo_Riccati/P42H0cFb">Vincenzo Riccati</a>.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> Riccati used <i>Sc.</i> and <i>Cc.</i> (<i>sinus/cosinus circulare</i>) to refer to circular functions and <i>Sh.</i> and <i>Ch.</i> (<i>sinus/cosinus hyperbolico</i>) to refer to hyperbolic functions.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> As early as 1759, <a href="/facts/Fran%25C3%25A7ois_Daviet_de_Foncenex/RvCIacq8">Daviet de Foncenex</a> showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended <a href="/facts/De_Moivre%2527s_formula/uyG4k2uC">de Moivre's formula</a> to hyperbolic functions.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a><a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> </p><p>During the 1760s, <a href="/facts/Johann_Heinrich_Lambert/CG3IHVB9">Johann Heinrich Lambert</a> systematized the use functions and provided exponential expressions in various publications.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a><a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a><a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> </p> <h2 id="notation">Notation</h2> <p class="note">Main article: <a href="/facts/Trigonometric_functions/czej7ur0">Trigonometric functions § Notation</a></p> <h2 id="definitions">Definitions</h2> <p>There are various equivalent ways to define the hyperbolic functions. </p> <h3>Exponential definitions</h3> <p>In terms of the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>:<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a><a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> </p> <ul><li>Hyperbolic sine: the <a href="/facts/Odd_part_of_a_function/tGHWJqAa">odd part</a> of the exponential function, that is, sinh x = e x − e − x 2 = e 2 x − 1 2 e x = 1 − e − 2 x 2 e − x . {\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.} </li> <li>Hyperbolic cosine: the <a href="/facts/Even_part_of_a_function/tGHWJqAa">even part</a> of the exponential function, that is, cosh x = e x + e − x 2 = e 2 x + 1 2 e x = 1 + e − 2 x 2 e − x . {\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.} </li> <li>Hyperbolic tangent: tanh x = sinh x cosh x = e x − e − x e x + e − x = e 2 x − 1 e 2 x + 1 . {\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}.} </li> <li>Hyperbolic cotangent: for <i>x</i> ≠ 0, coth x = cosh x sinh x = e x + e − x e x − e − x = e 2 x + 1 e 2 x − 1 . {\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}.} </li> <li>Hyperbolic secant: sech x = 1 cosh x = 2 e x + e − x = 2 e x e 2 x + 1 . {\displaystyle \operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}.} </li> <li>Hyperbolic cosecant: for <i>x</i> ≠ 0, csch x = 1 sinh x = 2 e x − e − x = 2 e x e 2 x − 1 . {\displaystyle \operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}.} </li></ul> <h3>Differential equation definitions</h3> <p>The hyperbolic functions may be defined as solutions of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>: The hyperbolic sine and cosine are the solution (<i>s</i>, <i>c</i>) of the system c ′ ( x ) = s ( x ) , s ′ ( x ) = c ( x ) , {\displaystyle {\begin{aligned}c'(x)&=s(x),\\s'(x)&=c(x),\\\end{aligned}}} with the initial conditions s ( 0 ) = 0 , c ( 0 ) = 1. {\displaystyle s(0)=0,c(0)=1.} The initial conditions make the solution unique; without them any pair of functions ( a e x + b e − x , a e x − b e − x ) {\displaystyle (ae^{x}+be^{-x},ae^{x}-be^{-x})} would be a solution. </p><p>sinh(<i>x</i>) and cosh(<i>x</i>) are also the unique solution of the equation <i>f</i> ″(<i>x</i>) = <i>f</i> (<i>x</i>), such that <i>f</i> (0) = 1, <i>f</i> ′(0) = 0 for the hyperbolic cosine, and <i>f</i> (0) = 0, <i>f</i> ′(0) = 1 for the hyperbolic sine. </p> <h3>Complex trigonometric definitions</h3> <p>Hyperbolic functions may also be deduced from <a href="/facts/Trigonometric_function/czej7ur0">trigonometric functions</a> with <a href="/facts/Complex_number/2w7mqI7e">complex</a> arguments: </p> <ul><li>Hyperbolic sine:<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> sinh x = − i sin ( i x ) . {\displaystyle \sinh x=-i\sin(ix).} </li> <li>Hyperbolic cosine:<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> cosh x = cos ( i x ) . {\displaystyle \cosh x=\cos(ix).} </li> <li>Hyperbolic tangent: tanh x = − i tan ( i x ) . {\displaystyle \tanh x=-i\tan(ix).} </li> <li>Hyperbolic cotangent: coth x = i cot ( i x ) . {\displaystyle \coth x=i\cot(ix).} </li> <li>Hyperbolic secant: sech x = sec ( i x ) . {\displaystyle \operatorname {sech} x=\sec(ix).} </li> <li>Hyperbolic cosecant: csch x = i csc ( i x ) . {\displaystyle \operatorname {csch} x=i\csc(ix).} </li></ul> <p>where i is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a> with <i>i</i>2 = −1. </p><p>The above definitions are related to the exponential definitions via <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a> (See § Hyperbolic functions for complex numbers below). </p> <h2 id="characterizing-properties">Characterizing properties</h2> <h3>Hyperbolic cosine</h3> <p>It can be shown that the <a href="/facts/Area_under_the_curve/V7lyV997">area under the curve</a> of the hyperbolic cosine (over a finite interval) is always equal to the <a href="/facts/Arc_length/QgTrmLxY">arc length</a> corresponding to that interval:<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> area = ∫ a b cosh x d x = ∫ a b 1 + ( d d x cosh x ) 2 d x = arc length. {\displaystyle {\text{area}}=\int _{a}^{b}\cosh x\,dx=\int _{a}^{b}{\sqrt {1+\left({\frac {d}{dx}}\cosh x\right)^{2}}}\,dx={\text{arc length.}}} </p> <h3>Hyperbolic tangent</h3> <p>The hyperbolic tangent is the (unique) solution to the <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> <i>f</i> ′ = 1 − <i>f</i> 2, with <i>f</i> (0) = 0.<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a><a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> </p> <h2 id="useful-relations">Useful relations</h2> <p> The hyperbolic functions satisfy many identities, all of them similar in form to the <a href="/facts/Trigonometric_identity/Xw9Jx8Vz">trigonometric identities</a>. In fact, Osborn's rule<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for θ {\displaystyle \theta } , 2 θ {\displaystyle 2\theta } , 3 θ {\displaystyle 3\theta } or θ {\displaystyle \theta } and φ {\displaystyle \varphi } into a hyperbolic identity, by: </p> <ol><li>expanding it completely in terms of integral powers of sines and cosines,</li> <li>changing sine to sinh and cosine to cosh, and</li> <li>switching the sign of every term containing a product of two sinhs.</li></ol> <p>Odd and even functions: sinh ( − x ) = − sinh x cosh ( − x ) = cosh x {\displaystyle {\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\end{aligned}}} </p><p>Hence: tanh ( − x ) = − tanh x coth ( − x ) = − coth x sech ( − x ) = sech x csch ( − x ) = − csch x {\displaystyle {\begin{aligned}\tanh(-x)&=-\tanh x\\\coth(-x)&=-\coth x\\\operatorname {sech} (-x)&=\operatorname {sech} x\\\operatorname {csch} (-x)&=-\operatorname {csch} x\end{aligned}}} </p><p>Thus, cosh <i>x</i> and sech <i>x</i> are <a href="/facts/Even_function/tGHWJqAa">even functions</a>; the others are <a href="/facts/Odd_functions/tGHWJqAa">odd functions</a>. </p><p> arsech x = arcosh ( 1 x ) arcsch x = arsinh ( 1 x ) arcoth x = artanh ( 1 x ) {\displaystyle {\begin{aligned}\operatorname {arsech} x&=\operatorname {arcosh} \left({\frac {1}{x}}\right)\\\operatorname {arcsch} x&=\operatorname {arsinh} \left({\frac {1}{x}}\right)\\\operatorname {arcoth} x&=\operatorname {artanh} \left({\frac {1}{x}}\right)\end{aligned}}} </p><p>Hyperbolic sine and cosine satisfy: cosh x + sinh x = e x cosh x − sinh x = e − x {\displaystyle {\begin{aligned}\cosh x+\sinh x&=e^{x}\\\cosh x-\sinh x&=e^{-x}\end{aligned}}} </p><p>which are analogous to <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a>, and </p><p> cosh 2 x − sinh 2 x = 1 {\displaystyle \cosh ^{2}x-\sinh ^{2}x=1} </p><p>which is analogous to the <a href="/facts/Pythagorean_trigonometric_identity/MQsE3Qhc">Pythagorean trigonometric identity</a>. </p><p>One also has sech 2 x = 1 − tanh 2 x csch 2 x = coth 2 x − 1 {\displaystyle {\begin{aligned}\operatorname {sech} ^{2}x&=1-\tanh ^{2}x\\\operatorname {csch} ^{2}x&=\coth ^{2}x-1\end{aligned}}} </p><p>for the other functions. </p> <h3>Sums of arguments</h3> <p> sinh ( x + y ) = sinh x cosh y + cosh x sinh y cosh ( x + y ) = cosh x cosh y + sinh x sinh y tanh ( x + y ) = tanh x + tanh y 1 + tanh x tanh y {\displaystyle {\begin{aligned}\sinh(x+y)&=\sinh x\cosh y+\cosh x\sinh y\\\cosh(x+y)&=\cosh x\cosh y+\sinh x\sinh y\\\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\end{aligned}}} particularly cosh ( 2 x ) = sinh 2 x + cosh 2 x = 2 sinh 2 x + 1 = 2 cosh 2 x − 1 sinh ( 2 x ) = 2 sinh x cosh x tanh ( 2 x ) = 2 tanh x 1 + tanh 2 x {\displaystyle {\begin{aligned}\cosh(2x)&=\sinh ^{2}{x}+\cosh ^{2}{x}=2\sinh ^{2}x+1=2\cosh ^{2}x-1\\\sinh(2x)&=2\sinh x\cosh x\\\tanh(2x)&={\frac {2\tanh x}{1+\tanh ^{2}x}}\\\end{aligned}}} </p><p>Also: sinh x + sinh y = 2 sinh ( x + y 2 ) cosh ( x − y 2 ) cosh x + cosh y = 2 cosh ( x + y 2 ) cosh ( x − y 2 ) {\displaystyle {\begin{aligned}\sinh x+\sinh y&=2\sinh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\cosh x+\cosh y&=2\cosh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\end{aligned}}} </p> <h3>Subtraction formulas</h3> <p> sinh ( x − y ) = sinh x cosh y − cosh x sinh y cosh ( x − y ) = cosh x cosh y − sinh x sinh y tanh ( x − y ) = tanh x − tanh y 1 − tanh x tanh y {\displaystyle {\begin{aligned}\sinh(x-y)&=\sinh x\cosh y-\cosh x\sinh y\\\cosh(x-y)&=\cosh x\cosh y-\sinh x\sinh y\\\tanh(x-y)&={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\\\end{aligned}}} </p><p>Also:<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> sinh x − sinh y = 2 cosh ( x + y 2 ) sinh ( x − y 2 ) cosh x − cosh y = 2 sinh ( x + y 2 ) sinh ( x − y 2 ) {\displaystyle {\begin{aligned}\sinh x-\sinh y&=2\cosh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\cosh x-\cosh y&=2\sinh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\end{aligned}}} </p> <h3>Half argument formulas</h3> <p> sinh ( x 2 ) = sinh x 2 ( cosh x + 1 ) = sgn x cosh x − 1 2 cosh ( x 2 ) = cosh x + 1 2 tanh ( x 2 ) = sinh x cosh x + 1 = sgn x cosh x − 1 cosh x + 1 = e x − 1 e x + 1 {\displaystyle {\begin{aligned}\sinh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\sqrt {2(\cosh x+1)}}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{2}}}\\[6px]\cosh \left({\frac {x}{2}}\right)&={\sqrt {\frac {\cosh x+1}{2}}}\\[6px]\tanh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\cosh x+1}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{\cosh x+1}}}={\frac {e^{x}-1}{e^{x}+1}}\end{aligned}}} </p><p>where sgn is the <a href="/facts/Sign_function/VADy9zQq">sign function</a>. </p><p>If <i>x</i> ≠ 0, then<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> </p><p> tanh ( x 2 ) = cosh x − 1 sinh x = coth x − csch x {\displaystyle \tanh \left({\frac {x}{2}}\right)={\frac {\cosh x-1}{\sinh x}}=\coth x-\operatorname {csch} x} </p> <h3>Square formulas</h3> <p> sinh 2 x = 1 2 ( cosh 2 x − 1 ) cosh 2 x = 1 2 ( cosh 2 x + 1 ) {\displaystyle {\begin{aligned}\sinh ^{2}x&={\tfrac {1}{2}}(\cosh 2x-1)\\\cosh ^{2}x&={\tfrac {1}{2}}(\cosh 2x+1)\end{aligned}}} </p> <h3>Inequalities</h3> <p>The following inequality is useful in statistics:<a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> cosh ( t ) ≤ e t 2 / 2 . {\displaystyle \operatorname {cosh} (t)\leq e^{t^{2}/2}.} </p><p>It can be proved by comparing the Taylor series of the two functions term by term. </p> <h2 id="inverse-functions-as-logarithms">Inverse functions as logarithms</h2> <p class="note">Main article: <a href="/facts/Inverse_hyperbolic_function/myk0wJty">Inverse hyperbolic function</a></p> <p> arsinh ( x ) = ln ( x + x 2 + 1 ) arcosh ( x ) = ln ( x + x 2 − 1 ) x ≥ 1 artanh ( x ) = 1 2 ln ( 1 + x 1 − x ) | x | < 1 arcoth ( x ) = 1 2 ln ( x + 1 x − 1 ) | x | > 1 arsech ( x ) = ln ( 1 x + 1 x 2 − 1 ) = ln ( 1 + 1 − x 2 x ) 0 < x ≤ 1 arcsch ( x ) = ln ( 1 x + 1 x 2 + 1 ) x ≠ 0 {\displaystyle {\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&&x\geq 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)&&|x|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)&&|x|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)&&0<x\leq 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)&&x\neq 0\end{aligned}}} </p> <h2 id="derivatives">Derivatives</h2> <p> d d x sinh x = cosh x d d x cosh x = sinh x d d x tanh x = 1 − tanh 2 x = sech 2 x = 1 cosh 2 x d d x coth x = 1 − coth 2 x = − csch 2 x = − 1 sinh 2 x x ≠ 0 d d x sech x = − tanh x sech x d d x csch x = − coth x csch x x ≠ 0 {\displaystyle {\begin{aligned}{\frac {d}{dx}}\sinh x&=\cosh x\\{\frac {d}{dx}}\cosh x&=\sinh x\\{\frac {d}{dx}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {d}{dx}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}&&x\neq 0\\{\frac {d}{dx}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\\{\frac {d}{dx}}\operatorname {csch} x&=-\coth x\operatorname {csch} x&&x\neq 0\end{aligned}}} d d x arsinh x = 1 x 2 + 1 d d x arcosh x = 1 x 2 − 1 1 < x d d x artanh x = 1 1 − x 2 | x | < 1 d d x arcoth x = 1 1 − x 2 1 < | x | d d x arsech x = − 1 x 1 − x 2 0 < x < 1 d d x arcsch x = − 1 | x | 1 + x 2 x ≠ 0 {\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} x&={\frac {1}{\sqrt {x^{2}+1}}}\\{\frac {d}{dx}}\operatorname {arcosh} x&={\frac {1}{\sqrt {x^{2}-1}}}&&1<x\\{\frac {d}{dx}}\operatorname {artanh} x&={\frac {1}{1-x^{2}}}&&|x|<1\\{\frac {d}{dx}}\operatorname {arcoth} x&={\frac {1}{1-x^{2}}}&&1<|x|\\{\frac {d}{dx}}\operatorname {arsech} x&=-{\frac {1}{x{\sqrt {1-x^{2}}}}}&&0<x<1\\{\frac {d}{dx}}\operatorname {arcsch} x&=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}&&x\neq 0\end{aligned}}} </p> <h2 id="second-derivatives">Second derivatives</h2> <p>Each of the functions sinh and cosh is equal to its <a href="/facts/Second_derivative/3aA6j4EY">second derivative</a>, that is: d 2 d x 2 sinh x = sinh x {\displaystyle {\frac {d^{2}}{dx^{2}}}\sinh x=\sinh x} d 2 d x 2 cosh x = cosh x . {\displaystyle {\frac {d^{2}}{dx^{2}}}\cosh x=\cosh x\,.} </p><p>All functions with this property are <a href="/facts/Linear_combination/CVjL6kuN">linear combinations</a> of sinh and cosh, in particular the <a href="/facts/Exponential_function/ewlYxvR2">exponential functions</a> e x {\displaystyle e^{x}} and e − x {\displaystyle e^{-x}} .<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> </p> <h2 id="standard-integrals">Standard integrals</h2> <p class="note">For a full list, see <a href="/facts/List_of_integrals_of_hyperbolic_functions/UIPA88e1">list of integrals of hyperbolic functions</a>.</p> <p> ∫ sinh ( a x ) d x = a − 1 cosh ( a x ) + C ∫ cosh ( a x ) d x = a − 1 sinh ( a x ) + C ∫ tanh ( a x ) d x = a − 1 ln ( cosh ( a x ) ) + C ∫ coth ( a x ) d x = a − 1 ln | sinh ( a x ) | + C ∫ sech ( a x ) d x = a − 1 arctan ( sinh ( a x ) ) + C ∫ csch ( a x ) d x = a − 1 ln | tanh ( a x 2 ) | + C = a − 1 ln | coth ( a x ) − csch ( a x ) | + C = − a − 1 arcoth ( cosh ( a x ) ) + C {\displaystyle {\begin{aligned}\int \sinh(ax)\,dx&=a^{-1}\cosh(ax)+C\\\int \cosh(ax)\,dx&=a^{-1}\sinh(ax)+C\\\int \tanh(ax)\,dx&=a^{-1}\ln(\cosh(ax))+C\\\int \coth(ax)\,dx&=a^{-1}\ln \left|\sinh(ax)\right|+C\\\int \operatorname {sech} (ax)\,dx&=a^{-1}\arctan(\sinh(ax))+C\\\int \operatorname {csch} (ax)\,dx&=a^{-1}\ln \left|\tanh \left({\frac {ax}{2}}\right)\right|+C=a^{-1}\ln \left|\coth \left(ax\right)-\operatorname {csch} \left(ax\right)\right|+C=-a^{-1}\operatorname {arcoth} \left(\cosh \left(ax\right)\right)+C\end{aligned}}} </p><p>The following integrals can be proved using <a href="/facts/Hyperbolic_substitution/U6LBLAYU">hyperbolic substitution</a>: ∫ 1 a 2 + u 2 d u = arsinh ( u a ) + C ∫ 1 u 2 − a 2 d u = sgn u arcosh | u a | + C ∫ 1 a 2 − u 2 d u = a − 1 artanh ( u a ) + C u 2 < a 2 ∫ 1 a 2 − u 2 d u = a − 1 arcoth ( u a ) + C u 2 > a 2 ∫ 1 u a 2 − u 2 d u = − a − 1 arsech | u a | + C ∫ 1 u a 2 + u 2 d u = − a − 1 arcsch | u a | + C {\displaystyle {\begin{aligned}\int {{\frac {1}{\sqrt {a^{2}+u^{2}}}}\,du}&=\operatorname {arsinh} \left({\frac {u}{a}}\right)+C\\\int {{\frac {1}{\sqrt {u^{2}-a^{2}}}}\,du}&=\operatorname {sgn} {u}\operatorname {arcosh} \left|{\frac {u}{a}}\right|+C\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {artanh} \left({\frac {u}{a}}\right)+C&&u^{2}<a^{2}\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {arcoth} \left({\frac {u}{a}}\right)+C&&u^{2}>a^{2}\\\int {{\frac {1}{u{\sqrt {a^{2}-u^{2}}}}}\,du}&=-a^{-1}\operatorname {arsech} \left|{\frac {u}{a}}\right|+C\\\int {{\frac {1}{u{\sqrt {a^{2}+u^{2}}}}}\,du}&=-a^{-1}\operatorname {arcsch} \left|{\frac {u}{a}}\right|+C\end{aligned}}} </p><p>where <i>C</i> is the <a href="/facts/Constant_of_integration/dFzDzFhl">constant of integration</a>. </p> <h2 id="taylor-series-expressions">Taylor series expressions</h2> <p>It is possible to express explicitly the <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> at zero (or the <a href="/facts/Laurent_series/sSMevFU8">Laurent series</a>, if the function is not defined at zero) of the above functions. </p><p> sinh x = x + x 3 3 ! + x 5 5 ! + x 7 7 ! + ⋯ = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) ! {\displaystyle \sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}} This series is <a href="/facts/Convergent_series/DTok170z">convergent</a> for every <a href="/facts/Complex_number/2w7mqI7e">complex</a> value of x. Since the function sinh <i>x</i> is <a href="/facts/Odd_function/tGHWJqAa">odd</a>, only odd exponents for <i>x</i> occur in its Taylor series. </p><p> cosh x = 1 + x 2 2 ! + x 4 4 ! + x 6 6 ! + ⋯ = ∑ n = 0 ∞ x 2 n ( 2 n ) ! {\displaystyle \cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}} This series is <a href="/facts/Convergent_series/DTok170z">convergent</a> for every <a href="/facts/Complex_number/2w7mqI7e">complex</a> value of x. Since the function cosh <i>x</i> is <a href="/facts/Even_function/tGHWJqAa">even</a>, only even exponents for x occur in its Taylor series. </p><p>The sum of the sinh and cosh series is the <a href="/facts/Infinite_series/yDnlsgIe">infinite series</a> expression of the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>. </p><p>The following series are followed by a description of a subset of their <a href="/facts/Domain_of_convergence/4Un0ANfP">domain of convergence</a>, where the series is convergent and its sum equals the function. tanh x = x − x 3 3 + 2 x 5 15 − 17 x 7 315 + ⋯ = ∑ n = 1 ∞ 2 2 n ( 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! , | x | < π 2 coth x = x − 1 + x 3 − x 3 45 + 2 x 5 945 + ⋯ = ∑ n = 0 ∞ 2 2 n B 2 n x 2 n − 1 ( 2 n ) ! , 0 < | x | < π sech x = 1 − x 2 2 + 5 x 4 24 − 61 x 6 720 + ⋯ = ∑ n = 0 ∞ E 2 n x 2 n ( 2 n ) ! , | x | < π 2 csch x = x − 1 − x 6 + 7 x 3 360 − 31 x 5 15120 + ⋯ = ∑ n = 0 ∞ 2 ( 1 − 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! , 0 < | x | < π {\displaystyle {\begin{aligned}\tanh x&=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\qquad \left|x\right|<{\frac {\pi }{2}}\\\coth x&=x^{-1}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots =\sum _{n=0}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},\qquad 0<\left|x\right|<\pi \\\operatorname {sech} x&=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\qquad \left|x\right|<{\frac {\pi }{2}}\\\operatorname {csch} x&=x^{-1}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots =\sum _{n=0}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},\qquad 0<\left|x\right|<\pi \end{aligned}}} </p><p>where: </p> <ul><li> B n {\displaystyle B_{n}} is the <i>n</i>th <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli number</a></li> <li> E n {\displaystyle E_{n}} is the <i>n</i>th <a href="/facts/Euler_number/acpKy6uS">Euler number</a></li></ul> <h2 id="infinite-products-and-continued-fractions">Infinite products and continued fractions</h2> <p>The following expansions are valid in the whole complex plane: </p> sinh x = x ∏ n = 1 ∞ ( 1 + x 2 n 2 π 2 ) = x 1 − x 2 2 ⋅ 3 + x 2 − 2 ⋅ 3 x 2 4 ⋅ 5 + x 2 − 4 ⋅ 5 x 2 6 ⋅ 7 + x 2 − ⋱ {\displaystyle \sinh x=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{n^{2}\pi ^{2}}}\right)={\cfrac {x}{1-{\cfrac {x^{2}}{2\cdot 3+x^{2}-{\cfrac {2\cdot 3x^{2}}{4\cdot 5+x^{2}-{\cfrac {4\cdot 5x^{2}}{6\cdot 7+x^{2}-\ddots }}}}}}}}} cosh x = ∏ n = 1 ∞ ( 1 + x 2 ( n − 1 / 2 ) 2 π 2 ) = 1 1 − x 2 1 ⋅ 2 + x 2 − 1 ⋅ 2 x 2 3 ⋅ 4 + x 2 − 3 ⋅ 4 x 2 5 ⋅ 6 + x 2 − ⋱ {\displaystyle \cosh x=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{(n-1/2)^{2}\pi ^{2}}}\right)={\cfrac {1}{1-{\cfrac {x^{2}}{1\cdot 2+x^{2}-{\cfrac {1\cdot 2x^{2}}{3\cdot 4+x^{2}-{\cfrac {3\cdot 4x^{2}}{5\cdot 6+x^{2}-\ddots }}}}}}}}} tanh x = 1 1 x + 1 3 x + 1 5 x + 1 7 x + ⋱ {\displaystyle \tanh x={\cfrac {1}{{\cfrac {1}{x}}+{\cfrac {1}{{\cfrac {3}{x}}+{\cfrac {1}{{\cfrac {5}{x}}+{\cfrac {1}{{\cfrac {7}{x}}+\ddots }}}}}}}}} <h2 id="comparison-with-circular-functions">Comparison with circular functions</h2> <p>The hyperbolic functions represent an expansion of <a href="/facts/Trigonometry/K4hfaC6F">trigonometry</a> beyond the <a href="/facts/Circular_function/czej7ur0">circular functions</a>. Both types depend on an <a href="/facts/Argument_of_a_function/QkuPc28I">argument</a>, either <a href="/facts/Angle/gFUTEQYq">circular angle</a> or <a href="/facts/Hyperbolic_angle/m4IaIh0g">hyperbolic angle</a>. </p><p>Since the <a href="/facts/Circular_sector/TqwCaEA5">area of a circular sector</a> with radius r and angle u (in radians) is <i>r</i>2<i>u</i>/2, it will be equal to u when <i>r</i> = √2. In the diagram, such a circle is tangent to the hyperbola <i>xy</i> = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red regions together depict a <a href="/facts/Hyperbolic_sector/PCEFy8fd">hyperbolic sector</a> with area corresponding to hyperbolic angle magnitude. </p><p>The legs of the two <a href="/facts/Right_triangle/dD6M9GAn">right triangles</a> with hypotenuse on the ray defining the angles are of length √2 times the circular and hyperbolic functions. </p><p>The hyperbolic angle is an <a href="/facts/Invariant_measure/X75B3PcM">invariant measure</a> with respect to the <a href="/facts/Squeeze_mapping/CYxzaaKR">squeeze mapping</a>, just as the circular angle is invariant under rotation.<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a> </p><p>The <a href="/facts/Gudermannian_function/bA0B2KcC">Gudermannian function</a> gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. </p><p>The graph of the function <i>a</i> cosh(<i>x</i>/<i>a</i>) is the <a href="/facts/Catenary/fWVU770E">catenary</a>, the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity. </p> <h2 id="relationship-to-the-exponential-function">Relationship to the exponential function</h2> <p>The decomposition of the exponential function in its <a href="/facts/Even%25E2%2580%2593odd_decomposition/tGHWJqAa">even and odd parts</a> gives the identities e x = cosh x + sinh x , {\displaystyle e^{x}=\cosh x+\sinh x,} and e − x = cosh x − sinh x . {\displaystyle e^{-x}=\cosh x-\sinh x.} Combined with <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a> e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} this gives e x + i y = ( cosh x + sinh x ) ( cos y + i sin y ) {\displaystyle e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)} for the <a href="/facts/General_complex_exponential_function/ewlYxvR2">general complex exponential function</a>. </p><p>Additionally, e x = 1 + tanh x 1 − tanh x = 1 + tanh x 2 1 − tanh x 2 {\displaystyle e^{x}={\sqrt {\frac {1+\tanh x}{1-\tanh x}}}={\frac {1+\tanh {\frac {x}{2}}}{1-\tanh {\frac {x}{2}}}}} </p> <h2 id="hyperbolic-functions-for-complex-numbers">Hyperbolic functions for complex numbers</h2> Hyperbolic functions in the complex plane<table><tbody><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td> sinh ( z ) {\displaystyle \sinh(z)} </td><td> cosh ( z ) {\displaystyle \cosh(z)} </td><td> tanh ( z ) {\displaystyle \tanh(z)} </td><td> coth ( z ) {\displaystyle \coth(z)} </td><td> sech ( z ) {\displaystyle \operatorname {sech} (z)} </td><td> csch ( z ) {\displaystyle \operatorname {csch} (z)} </td></tr></tbody></table> <p>Since the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> can be defined for any <a href="/facts/Complex_number/2w7mqI7e">complex</a> argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions sinh <i>z</i> and cosh <i>z</i> are then <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a>. </p><p>Relationships to ordinary trigonometric functions are given by <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a> for complex numbers: e i x = cos x + i sin x e − i x = cos x − i sin x {\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\e^{-ix}&=\cos x-i\sin x\end{aligned}}} so: cosh ( i x ) = 1 2 ( e i x + e − i x ) = cos x sinh ( i x ) = 1 2 ( e i x − e − i x ) = i sin x cosh ( x + i y ) = cosh ( x ) cos ( y ) + i sinh ( x ) sin ( y ) sinh ( x + i y ) = sinh ( x ) cos ( y ) + i cosh ( x ) sin ( y ) tanh ( i x ) = i tan x cosh x = cos ( i x ) sinh x = − i sin ( i x ) tanh x = − i tan ( i x ) {\displaystyle {\begin{aligned}\cosh(ix)&={\frac {1}{2}}\left(e^{ix}+e^{-ix}\right)=\cos x\\\sinh(ix)&={\frac {1}{2}}\left(e^{ix}-e^{-ix}\right)=i\sin 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(ix)&=i\tan x\\\cosh x&=\cos(ix)\\\sinh x&=-i\sin(ix)\\\tanh x&=-i\tan(ix)\end{aligned}}} </p><p>Thus, hyperbolic functions are <a href="/facts/Periodic_function/Sm8lJQsF">periodic</a> with respect to the imaginary component, with period 2 π i {\displaystyle 2\pi i} ( π i {\displaystyle \pi i} for hyperbolic tangent and cotangent). </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/E_(mathematical_constant)/coEtSBk8">e (mathematical constant)</a></li> <li><a href="/facts/Equal_incircles_theorem/UYAqvTQw">Equal incircles theorem</a>, based on sinh</li> <li><a href="/facts/Hyperbolastic_functions/E9Isb9qW">Hyperbolastic functions</a></li> <li><a href="/facts/Hyperbolic_growth/S0Mbef3y">Hyperbolic growth</a></li> <li><a href="/facts/Inverse_hyperbolic_function/myk0wJty">Inverse hyperbolic functions</a></li> <li><a href="/facts/List_of_integrals_of_hyperbolic_functions/UIPA88e1">List of integrals of hyperbolic functions</a></li> <li><a href="/facts/Poinsot%2527s_spirals/uuX3PVRY">Poinsot's spirals</a></li> <li><a href="/facts/Sigmoid_function/lAXu3AwW">Sigmoid function</a></li> <li><a href="/facts/Soboleva_modified_hyperbolic_tangent/kCNXLpyb">Soboleva modified hyperbolic tangent</a></li> <li><a href="/facts/Trigonometric_functions/czej7ur0">Trigonometric functions</a></li></ul> <h2 id="external-links">External links</h2> Wikimedia Commons has media related to Hyperbolic functions. <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Hyperbolic_functions">"Hyperbolic functions"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><a href="https://planetmath.org/hyperbolicfunctions">Hyperbolic functions</a> on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a></li> <li><a href="https://web.archive.org/web/20071006172054/http://glab.trixon.se/">GonioLab</a>: Visualization of the unit circle, trigonometric and hyperbolic functions (<a href="/facts/Java_Web_Start/HPWVy5VS">Java Web Start</a>)</li> <li><a href="http://www.calctool.org/CALC/math/trigonometry/hyperbolic">Web-based calculator of hyperbolic functions</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Weisstein, Eric W. "Hyperbolic Functions". mathworld.wolfram.com. Retrieved 2020-08-29. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>(1999) Collins Concise Dictionary, 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p. 1386 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Collins Concise Dictionary, p. 328 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Hyperbolic Functions". www.mathsisfun.com. Retrieved 2020-08-29. <a href="https://www.mathsisfun.com/sets/function-hyperbolic.html" target="_blank">https://www.mathsisfun.com/sets/function-hyperbolic.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Collins Concise Dictionary, p. 1520 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Collins Concise Dictionary, p. 329 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>tanh <a href="http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/hyperbolicfunctions.pdf" target="_blank">http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/hyperbolicfunctions.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Collins Concise Dictionary, p. 1340 <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Collins Concise Dictionary, p. 328 <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Woodhouse, N. M. J. (2003), Special Relativity, London: Springer, p. 71, ISBN 978-1-85233-426-0 <a href="978-1-85233-426-0" target="_blank">978-1-85233-426-0</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>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 <a href="978-0-486-61272-0" target="_blank">978-0-486-61272-0</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Some examples of using arcsinh found in Google Books. <a href="https://www.google.com/books?q=arcsinh+-library" target="_blank">https://www.google.com/books?q=arcsinh+-library</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Niven, Ivan (1985). Irrational Numbers. Vol. 11. Mathematical Association of America. ISBN 9780883850381. JSTOR 10.4169/j.ctt5hh8zn. <a href="9780883850381" target="_blank">9780883850381</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>George F. Becker; C. E. Van Orstrand (1909). Hyperbolic Functions. Universal Digital Library. The Smithsonian Institution. <a href="https://archive.org/details/hyperbolicfuncti027682mbp/page/n49/mode/2up?q=mercator" target="_blank">https://archive.org/details/hyperbolicfuncti027682mbp/page/n49/mode/2up?q=mercator</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>McMahon, James (1896). Hyperbolic Functions. Osmania University, Digital Library Of India. John Wiley And Sons. <a href="https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up" target="_blank">https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>McMahon, James (1896). Hyperbolic Functions. Osmania University, Digital Library Of India. John Wiley And Sons. <a href="https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up" target="_blank">https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>McMahon, James (1896). Hyperbolic Functions. Osmania University, Digital Library Of India. John Wiley And Sons. <a href="https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up" target="_blank">https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>George F. Becker; C. E. Van Orstrand (1909). Hyperbolic Functions. Universal Digital Library. The Smithsonian Institution. <a href="https://archive.org/details/hyperbolicfuncti027682mbp/page/n49/mode/2up?q=mercator" target="_blank">https://archive.org/details/hyperbolicfuncti027682mbp/page/n49/mode/2up?q=mercator</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. Euler at 300: an appreciation. Mathematical Association of America, 2007. Page 100. <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>McMahon, James (1896). Hyperbolic Functions. Osmania University, Digital Library Of India. John Wiley And Sons. <a href="https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up" target="_blank">https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. Euler at 300: an appreciation. Mathematical Association of America, 2007. Page 100. <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>McMahon, James (1896). Hyperbolic Functions. Osmania University, Digital Library Of India. John Wiley And Sons. <a href="https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up" target="_blank">https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>McMahon, James (1896). Hyperbolic Functions. Osmania University, Digital Library Of India. John Wiley And Sons. <a href="https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up" target="_blank">https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. Euler at 300: an appreciation. Mathematical Association of America, 2007. Page 100. <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. Euler at 300: an appreciation. Mathematical Association of America, 2007. Page 100. <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Becker, Georg F. Hyperbolic functions. Read Books, 1931. Page xlviii. <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Weisstein, Eric W. "Hyperbolic Functions". mathworld.wolfram.com. Retrieved 2020-08-29. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>"Hyperbolic Functions". www.mathsisfun.com. Retrieved 2020-08-29. <a href="https://www.mathsisfun.com/sets/function-hyperbolic.html" target="_blank">https://www.mathsisfun.com/sets/function-hyperbolic.html</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Weisstein, Eric W. "Hyperbolic Functions". mathworld.wolfram.com. Retrieved 2020-08-29. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Weisstein, Eric W. "Hyperbolic Functions". mathworld.wolfram.com. Retrieved 2020-08-29. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>N.P., Bali (2005). Golden Integral Calculus. Firewall Media. p. 472. ISBN 81-7008-169-6. <a href="81-7008-169-6" target="_blank">81-7008-169-6</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>Steeb, Willi-Hans (2005). Nonlinear Workbook, The: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov Models, Fuzzy Logic With C++, Java And Symbolicc++ Programs (3rd ed.). World Scientific Publishing Company. p. 281. ISBN 978-981-310-648-2. Extract of page 281 (using lambda=1) <a href="978-981-310-648-2" target="_blank">978-981-310-648-2</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>Oldham, Keith B.; Myland, Jan; Spanier, Jerome (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator (2nd, illustrated ed.). Springer Science & Business Media. p. 290. ISBN 978-0-387-48807-3. Extract of page 290 <a href="978-0-387-48807-3" target="_blank">978-0-387-48807-3</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>Osborn, G. (July 1902). "Mnemonic for hyperbolic formulae". The Mathematical Gazette. 2 (34): 189. doi:10.2307/3602492. 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