Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how 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).

Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and fluid dynamics.

The basic hyperbolic functions are:^[1]

• hyperbolic sine "sinh" (/ˈsɪŋ,ˈsɪntʃ,ˈʃaɪn/),^[2]
• hyperbolic cosine "cosh" (/ˈkɒʃ,ˈkoʊʃ/),^[3]

from which are derived:^[4]

• الظل الزائدي " تانه " ( / ˈtæŋ , ˈtæntʃ , ˈθæn / ) ، ^{[ 5 ]}​​​​​​​
• الظل الزائدي " coth " ( / ˈ k ɒ θ , ˈ k oʊ θ / ), ^{[ 6 ] [ 7 ]}
• القاطع الزائدي " sech " ( / ˈ s ɛ tʃ , ˈ ʃ ɛ k / ) ، ^{[ 8 ]}
• قاطع التمام الزائدي " csch ^" أو " cosech " ( / ˈkoʊsɛtʃ , ˈkoʊʃɛk / ^{[ 3} ] )​​​​​

بما يتوافق مع الدوال المثلثية المشتقة.

الدوال الزائدية العكسية هي:

• الجيب الزائدي العكسي " arsinh " (يرمز له أيضًا بـ " sinh ⁻¹ " أو " asinh " أو أحيانًا " arcsinh ") ^{[ 9 ] [ 10 ] [ 11 ]}
• دالة جيب التمام الزائدية العكسية " arcosh " (يشار إليها أيضًا باسم " cosh ⁻¹ " أو " acosh " أو أحيانًا " arccosh ").
• الظل الزائدي العكسي " artanh " (يُشار إليه أيضًا بـ " tanh ⁻¹ " أو " atanh " أو أحيانًا " arctanh ")
• دالة الظل الزائدي العكسي " arcoth " (يشار إليها أيضًا باسم " coth ⁻¹ " أو " acoth " أو أحيانًا " arccoth ")
• القاطع الزائدي العكسي " arsech " (يُشار إليه أيضًا بـ " sech ⁻¹ " أو " asech " أو أحيانًا " arcsech ")
• inverse hyperbolic cosecant "arcsch" (also denoted "arcosech", "csch⁻¹", "cosech⁻¹","acsch", "acosech", or sometimes "arccsch" or "arccosech")

The hyperbolic functions take an argument called a hyperbolic angle. The magnitude of a hyperbolic angle is the area of its hyperbolic sector to xy = 1. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.

In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.

By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.^[12]

The first known calculation of a hyperbolic trigonometry problem is attributed to Gerardus Mercator when issuing the Mercator map projection circa 1566. It requires tabulating solutions to a transcendental equation involving hyperbolic functions.^[13]

The first to suggest a similarity between the sector of the circle and that of the hyperbola was Isaac Newton in his 1687 Principia Mathematica.^[14]

Roger Cotes suggested to modify the trigonometric functions using the imaginary unit${\displaystyle i={\sqrt {-1}}}$ to obtain an oblate spheroid from a prolate one.^[14]

Hyperbolic functions were formally introduced in 1757 by Vincenzo Riccati.^[14][13][15] Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions.^[14] As early as 1759, 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.^[15][14]

During the 1760s, Johann Heinrich Lambert systematized the use functions and provided exponential expressions in various publications.^[14][15] Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.^[15][16]

With hyperbolic angleu, the hyperbolic functions sinh and cosh can be defined with the exponential function e^u.^[1][4] In the figure ${\displaystyle A=(e^{-u},e^{u}),\ ​​B=(e^{u},\ e^{-u}),\ ​​OA+OB=OC}$ .

Exponential definitions

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

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

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule^[20] (named after George Osborn) states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for ${\displaystyle \theta }$, ${\displaystyle 2\theta }$, ${\displaystyle 3\theta }$ or ${\displaystyle \theta }$ and ${\displaystyle \varphi }$ into a hyperbolic identity, by:

1. expanding it completely in terms of integral powers of sines and cosines,
2. changing sine to sinh and cosine to cosh, and
3. switching the sign of every term containing a product of two sinhs.

Odd and even functions: $${\displaystyle {\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\\\tanh(-x)&=-\tanh x\\\coth(-x)&=-\coth x\\\operatorname {sech} (-x)&=\operatorname {sech} x\\\operatorname {csch} (-x)&=-\operatorname {csch} x\end{aligned}}}$$

Reciprocals:

$${\displaystyle {\begin{aligned}\operatorname {arsech} x&=\operatorname {arcosh} \left({\frac {1}{x}}\right)\\\operatorname {arcsch} x&=\operatorname {arsinh} \left({\frac {1}{x}}\right)\\\operatorname {arcoth} x&=\operatorname {artanh} \left({\frac {1}{x}}\right)\end{aligned}}}$$

Analogous to Euler's formula:

$${\displaystyle {\begin{aligned}\cosh x+\sinh x&=e^{x}\\\cosh x-\sinh x&=e^{-x}\end{aligned}}}$$

Analogous to the Pythagorean trigonometric identity:

$${\displaystyle {\begin{aligned}\cosh ^{2}x-\sinh ^{2}x&=1\\1-\tanh ^{2}x&=\operatorname {sech} ^{2}x\\\coth ^{2}x-1&=\operatorname {csch} ^{2}x\end{aligned}}}$$

$${\displaystyle {\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&&x\geq 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)&&|x|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)&&|x|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)&&0<x\leq 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)&&x\neq 0\end{aligned}}}$$

$${\displaystyle {\begin{aligned}{\frac {d}{dx}}\sinh x&=\cosh x\\{\frac {d}{dx}}\cosh x&=\sinh x\\{\frac {d}{dx}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {d}{dx}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}&&x\neq 0\\{\frac {d}{dx}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\\{\frac {d}{dx}}\operatorname {csch} x&=-\coth x\operatorname {csch} x&&x\neq 0\end{aligned}}}$$$${\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} x&={\frac {1}{\sqrt {x^{2}+1}}}\\{\frac {d}{dx}}\operatorname {arcosh} x&={\frac {1}{\sqrt {x^{2}-1}}}&&1<x\\{\frac {d}{dx}}\operatorname {artanh} x&={\frac {1}{1-x^{2}}}&&|x|<1\\{\frac {d}{dx}}\operatorname {arcoth} x&={\frac {1}{1-x^{2}}}&&1<|x|\\{\frac {d}{dx}}\operatorname {arsech} x&=-{\frac {1}{x{\sqrt {1-x^{2}}}}}&&0<x<1\\{\frac {d}{dx}}\operatorname {arcsch} x&=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}&&x\neq 0\end{aligned}}}$$

كل من الدالتين sinh و cosh تساوي مشتقتها الثانية ، أي: $${\displaystyle {\frac {d^{2}}{dx^{2}}}\sinh x=\sinh x}$$$${\displaystyle {\frac {d^{2}}{dx^{2}}}\cosh x=\cosh x\,.}$$

جميع الدوال التي تتمتع بهذه الخاصية هي تركيبات خطية من دالتي sinh و cosh ، وخاصة الدوال الأسية.${\displaystyle e^{x}}$و${\displaystyle e^{-x}}$[ ^{22 ]​}

$${\displaystyle {\begin{aligned}\int \sinh(ax)\,dx&=a^{-1}\cosh(ax)+C\\\int \cosh(ax)\,dx&=a^{-1}\sinh(ax)+C\\\int \tanh(ax)\,dx&=a^{-1}\ln(\cosh(ax))+C\\\int \coth(ax)\,dx&=a^{-1}\ln \left|\sinh(ax)\right|+C\\\int \operatorname {sech} (ax)\,dx&=a^{-1}\arctan(\sinh(ax))+C\\\int \operatorname {csch} (ax)\,dx&=a^{-1}\ln \left|\tanh \left({\frac {ax}{2}}\right)\right|+C=a^{-1}\ln \left|\coth \left(ax\right)-\operatorname {csch} \left(ax\right)\right|+C=-a^{-1}\operatorname {arcoth} \left(\cosh \left(ax\right)\right)+C\end{aligned}}}$$

يمكن إثبات التكاملات التالية باستخدام التعويض الزائدي : $${\displaystyle {\begin{aligned}\int {{\frac {1}{\sqrt {a^{2}+u^{2}}}}\,du}&=\operatorname {arsinh} \left({\frac {u}{a}}\right)+C\\\int {{\frac {1}{\sqrt {u^{2}-a^{2}}}}\,du}&=\operatorname {sgn} {u}\operatorname {arcosh} \left|{\frac {u}{a}}\right|+C\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {artanh} \left({\frac {u}{a}}\right)+C&&u^{2}<a^{2}\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {arcoth} \left({\frac {u}{a}}\right)+C&&u^{2}>a^{2}\\\int {{\frac {1}{u{\sqrt {a^{2}-u^{2}}}}}\,du}&=-a^{-1}\operatorname {arsech} \left|{\frac {u}{a}}\right|+C\\\int {{\frac {1}{u{\sqrt {a^{2}+u^{2}}}}}\,du}&=-a^{-1}\operatorname {arcsch} \left|{\frac {u}{a}}\right|+C\end{aligned}}}$$

حيث C هو ثابت التكامل .

من الممكن التعبير بشكل صريح عن متسلسلة تايلور عند الصفر (أو متسلسلة لوران ، إذا لم تكن الدالة معرفة عند الصفر) للدوال المذكورة أعلاه.

$${\displaystyle \sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}}$$ هذه المتسلسلة متقاربة لكل قيمة عقدية لـ x . وبما أن دالة sinh x فردية ، فإن الأسس الفردية فقط لـ x تظهر في متسلسلة تايلور الخاصة بها.

$${\displaystyle \cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}}$$ هذه المتسلسلة متقاربة لكل قيمة عقدية لـ x . وبما أن الدالة cosh x زوجية ، فإن الأسس الزوجية فقط لـ x تظهر في متسلسلة تايلور الخاصة بها.

مجموع متسلسلتي sinh و cosh هو تعبير المتسلسلة اللانهائية للدالة الأسية .

تلي السلسلة التالية وصف لمجموعة فرعية من مجال تقاربها ، حيث تكون السلسلة متقاربة ومجموعها يساوي الدالة. $${\displaystyle {\begin{aligned}\tanh x&=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\qquad \left|x\right|<{\frac {\pi }{2}}\\\coth x&=x^{-1}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots =\sum _{n=0}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},\qquad 0<\left|x\right|<\pi \\\operatorname {sech} x&=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\qquad \left|x\right|<{\frac {\pi }{2}}\\\operatorname {csch} x&=x^{-1}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots =\sum _{n=0}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},\qquad 0<\left|x\right|<\pi \end{aligned}}}$$

أين:

• ${\displaystyle B_{n}}$هو العدد النوني لبرنولي
• ${\displaystyle E_{n}}$هو العدد النوني لأويلر

تكون التوسعات التالية صالحة في المستوى المركب بأكمله:

${\displaystyle \sinh x=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{n^{2}\pi ^{2}}}\right)={\cfrac {x}{1-{\cfrac {x^{2}}{2\cdot 3+x^{2}-{\cfrac {2\cdot 3x^{2}}{4\cdot 5+x^{2}-{\cfrac {4\cdot 5x^{2}}{6\cdot 7+x^{2}-\ddots }}}}}}}}}$

${\displaystyle \cosh x=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{(n-1/2)^{2}\pi ^{2}}}\right)={\cfrac {1}{1-{\cfrac {x^{2}}{1\cdot 2+x^{2}-{\cfrac {1\cdot 2x^{2}}{3\cdot 4+x^{2}-{\cfrac {3\cdot 4x^{2}}{5\cdot 6+x^{2}-\ddots }}}}}}}}}$

${\displaystyle \tanh x={\cfrac {1}{{\cfrac {1}{x}}+{\cfrac {1}{{\cfrac {3}{x}}+{\cfrac {1}{{\cfrac {5}{x}}+{\cfrac {1}{{\cfrac {7}{x}}+\ddots }}}}}}}}}$

The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle.

Since the area of a circular sector with radius r and angle u (in radians) is r²u/2, it will be equal to u when r = √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 triangles with the hypotenuse on the ray defining the angles are of length √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.^[23]

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 ⁠${\displaystyle a\cosh(x/a)}$⁠ is the catenary, the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.

The decomposition of the exponential function in its even and odd parts gives the identities $${\displaystyle e^{x}=\cosh x+\sinh x,}$$ and $${\displaystyle e^{-x}=\cosh x-\sinh x.}$$ Combined with Euler's formula$${\displaystyle e^{ix}=\cos x+i\sin x,}$$ this gives $${\displaystyle e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)}$$ for the general complex exponential function.

Additionally, $${\displaystyle e^{x}={\sqrt {\frac {1+\tanh x}{1-\tanh x}}}={\frac {1+\tanh {\frac {x}{2}}}{1-\tanh {\frac {x}{2}}}}}$$

Hyperbolic functions in the complex plane

${\displaystyle \sinh(z)}$	${\displaystyle \cosh(z)}$	${\displaystyle \tanh(z)}$	${\displaystyle \coth(z)}$	${\displaystyle \operatorname {sech} (z)}$	${\displaystyle \operatorname {csch} (z)}$

Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions sinh z and cosh z are then holomorphic.

تُعطى العلاقات مع الدوال المثلثية العادية بواسطة صيغة أويلر للأعداد المركبة: $${\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\e^{-ix}&=\cos x-i\sin x\end{aligned}}}$$ لذا: $${\displaystyle {\begin{aligned}\cosh(ix)&={\frac {1}{2}}\left(e^{ix}+e^{-ix}\right)=\cos x\\\sinh(ix)&={\frac {1}{2}}\left(e^{ix}-e^{-ix}\right)=i\sin x\\\tanh(ix)&=i\tan x\\\cosh(x+iy)&=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\\\sinh(x+iy)&=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\\\tanh(x+iy)&={\frac {\tanh(x)+i\tan(y)}{1+i\tanh(x)\tan(y)}}\\\cosh x&=\cos(ix)\\\sinh x&=-i\sin(ix)\\\tanh x&=-i\tan(ix)\end{aligned}}}$$

وبالتالي، فإن الدوال الزائدية دورية بالنسبة للمكون التخيلي، ولها دورة.${\displaystyle 2\pi i}$(${\displaystyle \pi i}$(للدالة الظل الزائدي ودالة ظل التمام).

• e (ثابت رياضي)
• نظرية الدوائر الداخلية المتساوية ، بناءً على دالة الجيب الزائدي
• الدوال الزائدية
• النمو الزائدي
• الدوال الزائدية العكسية
• قائمة تكاملات الدوال الزائدية
• حلزونات بوينسو
• دالة سيجمويد
• الدوال المثلثية

• "Hyperbolic functions", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Hyperbolic functions on PlanetMath
• GonioLab: Visualization of the unit circle, trigonometric and hyperbolic functions (Java Web Start)
• Web-based calculator of hyperbolic functions