Hyperbolic sine and hyperbolic cosine

A straight line through the zero point intersects the hyperbola at the point , where stands for the area between the straight line, its mirror image in relation to the axis and the hyperbola. (See also the animated version with comparison to the trigonometric (circular) functions.) The hyperbola is also referred to as the unit hyperbola . ${\displaystyle x^{2}-y^{2}=1}$ ${\displaystyle (\cosh \,A,\sinh \,A)}$ ${\displaystyle A}$ ${\displaystyle x}$

Hyperbolic sine and hyperbolic cosine are mathematical hyperbolic functions , also called hyperbolic sine and hyperbolic cosine ; they carry the symbols or , in older sources also and . The hyperbolic cosine describes, among other things, the course of a rope suspended at two points. Its graph is therefore also called a catenoid (chain line). ${\displaystyle \sinh }$ ${\displaystyle \cosh }$ ${\displaystyle {\mathfrak {Sin}}}$ ${\displaystyle {\mathfrak {Cos}}}$

Definitions

• Hyperbolic sine

${\displaystyle \sinh x={\frac {1}{2}}\left(e^{x}-e^{-x}\right)=-i\,\sin(i\,x)}$

• Hyperbolic cosine

${\displaystyle \cosh x={\frac {1}{2}}\left(e^{x}+e^{-x}\right)=\cos(i\,x)}$

The functions sinh and cosh are the odd or even part of the exponential function ( ). ${\displaystyle \exp x=\cosh x+\sinh x}$

properties

Hyperbolic sine (red) and hyperbolic cosine (blue) for real x.

	Hyperbolic sine	Hyperbolic cosine
Domain of definition	${\displaystyle -\infty <x<+\infty }$	${\displaystyle -\infty <x<+\infty }$
Range of values	${\displaystyle -\infty <f(x)<+\infty }$	${\displaystyle 1\leq f(x)<+\infty }$
periodicity	no	no
monotony	strictly monotonously increasing	${\displaystyle -\infty <x\leq 0}$strictly monotonically decreasing strictly monotonically increasing ${\displaystyle 0\leq x<\infty }$
Symmetries	Point symmetry to the origin	Axial symmetry to the ordinate
Asymptotic functions	${\displaystyle a_{1}(x)={\frac {1}{2}}e^{x},\quad x\to \infty }$	${\displaystyle a_{1}(x)={\frac {1}{2}}e^{x},\quad x\to \infty }$
	${\displaystyle a_{2}(x)=-{\frac {1}{2}}e^{-x},\quad x\to -\infty }$	${\displaystyle a_{2}(x)={\frac {1}{2}}e^{-x},\quad x\to -\infty }$
zeropoint	${\displaystyle x=0}$	no
Jump points	no	no
Poles	no	no
Extremes	no	Minimum at ${\displaystyle x=0}$
Turning points	${\displaystyle x=0}$	no

Special values

${\displaystyle \sinh(\ln \Phi )={\tfrac {1}{2}}}$with the golden ratio ${\displaystyle \Phi }$

Improper integral

For the hyperbolic cosine the following applies in particular:

${\displaystyle \int \limits _{-\infty }^{\infty }{\frac {\mathrm {d} x}{\cosh x}}=\pi .}$

Inverse functions

The hyperbolic sine is bijective on , and therefore has an inverse function , which one area hyperbolic sine calls. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

The hyperbolic cosine forms the interval bijectively on the interval and can therefore be inverted to a limited extent. The inverse function of this is called the hyperbolic areakosine ${\displaystyle [0,+\infty [}$ ${\displaystyle [1,+\infty [}$ ${\displaystyle [0,+\infty [}$

Both inverse functions, hyperbolic area and hyperbolic area, can be calculated with the help of more elementary functions as follows:

${\displaystyle \operatorname {arsinh} x=\ln \left(x+{\sqrt {x^{2}+1}}\right)\ }$.

${\displaystyle \operatorname {arcosh} x=\ln \left(x+{\sqrt {x^{2}-1}}\right)\ }$.

Derivatives

The derivative of the hyperbolic sine is the hyperbolic cosine and the derivative of the hyperbolic cosine is the hyperbolic sine:

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\sinh x&=\cosh x\\{\frac {\mathrm {d} }{\mathrm {d} x}}\cosh x&=\sinh x\end{aligned}}}$

Antiderivatives

${\displaystyle {\begin{aligned}\int \sinh x\,\mathrm {d} x&=\cosh x+C\\\int \cosh x\,\mathrm {d} x&=\sinh x+C\end{aligned}}}$

Relationships (between the two functions and others)

${\displaystyle \cosh ^{2}x\!\;-\sinh ^{2}x=1}$

${\displaystyle \cosh x\,\;+\sinh x\,\,=e^{x}}$( Euler's identity )

${\displaystyle \cosh({\rm {arsinh}}(x))={\sqrt {x^{2}+1}}}$

${\displaystyle \sinh({\rm {arcosh}}(x))={\sqrt {x^{2}-1}}}$( Hyperbolic equation )

Addition theorems

${\displaystyle {\begin{aligned}\sinh(x\pm y)&=\sinh x\cosh y\pm \cosh x\sinh y\\\cosh(x\pm y)&=\cosh x\cosh y\pm \sinh x\sinh y\end{aligned}}}$

in particular applies to : ${\displaystyle y:=x}$

${\displaystyle {\begin{aligned}\sinh 2x&=2\cdot \sinh x\cosh x\ \\\cosh 2x&=\cosh ^{2}x+\sinh ^{2}x=2\cdot \cosh ^{2}x-1=2\cdot \sinh ^{2}x+1\end{aligned}}}$

and for : ${\displaystyle y:=2x}$

${\displaystyle {\begin{aligned}\sinh 3x&=4\cdot \sinh ^{3}x+3\sinh x\ \\\cosh 3x&=4\cdot \cosh ^{3}x-3\cosh x\end{aligned}}}$

Molecular formulas

${\displaystyle {\begin{aligned}\sinh x\pm \sinh y&=2\sinh {\frac {x\pm y}{2}}\cosh {\frac {x\mp y}{2}}\\\cosh x+\cosh y&=2\cosh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\\\cosh x-\cosh y&=2\sinh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\end{aligned}}}$

Series developments

The Taylor series of the hyperbolic sine or hyperbolic cosine with the development point is: ${\displaystyle x=0}$

${\displaystyle {\begin{aligned}\sinh x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+\dotsb \\\cosh x&=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+\dotsb \end{aligned}}}$

Product developments

${\displaystyle {\begin{aligned}&\sinh x=x\cdot \prod _{k=1}^{\infty }\left(1+{\frac {x^{2}}{(k\pi )^{2}}}\right)\qquad \qquad \quad \\&\sinh {(\pi x)}=\pi x\cdot \prod _{k=1}^{\infty }\left(1+{\frac {x^{2}}{k^{2}}}\right)\\&\cosh x=\prod _{k=1}^{\infty }\left(1+{\frac {4x^{2}}{(2k-1)^{2}\pi ^{2}}}\right)\end{aligned}}}$

Multiplication formulas

Be . Then for all complex : ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle z}$

${\displaystyle {\begin{aligned}&\sinh z={\left({\frac {2}{i}}\right)}^{\!\!n-1}\,\prod \limits _{k=0}^{n-1}\sinh {\frac {z+k\,\pi \,i}{n}}\\&\cosh z=2^{n-1}\prod \limits _{k=0}^{n-1}\cosh {\frac {z+\left(k-{\frac {n-1}{2}}\right)\,\pi \,i}{n}}\end{aligned}}}$

Complex arguments

With applies: ${\displaystyle x,y\in \mathbb {R} }$

${\displaystyle {\begin{aligned}\sinh(x+i\,y)&=\cos y\,\sinh x+i\sin y\,\cosh x\\\cosh(x+i\,y)&=\cos y\,\cosh x+i\sin y\,\sinh x\\\sin(x+i\,y)&=\sin x\,\cosh y+i\cos x\,\sinh y\\\cos(x+i\,y)&=\cos x\,\cosh y-i\sin x\,\sinh y\\\end{aligned}}}$

For example, the third and fourth equations follow in the following way:

With applies ${\displaystyle z=x+i\,y}$

${\displaystyle {\begin{aligned}\exp(iz)&=\cos(x+i\,y)+i\sin(x+i\,y)\\&=\exp(i\,(x+i\,y))\\&=\exp(i\,x)\,\exp(i\,(i\,y))\\&=(\cos x\,\cos(i\,y)-\sin x\,\sin(i\,y))+i\,(\cos x\,\sin(i\,y)+\sin x\,\cos(i\,y))\\&=(\cos x\,\cosh y-i\sin x\,\sinh y)+i\,(\sin x\,\cosh y+i\cos x\,\sinh y)\\\end{aligned}}}$

By comparison of coefficients it follows:

${\displaystyle {\begin{aligned}\cos(x+i\,y)&=\cos x\,\cosh y-i\sin x\,\sinh y\\\sin(x+i\,y)&=\sin x\,\cosh y+i\cos x\,\sinh y\\\end{aligned}}}$

Applications

Solution of a differential equation

The function

${\displaystyle f(x)=a\cdot \sinh x+b\cdot \cosh x}$ With ${\displaystyle a,b\in \mathbb {R} }$

solves the differential equation

${\displaystyle f''(x)-f(x)=0\ }$.

Chain line

A homogeneous rope that only sags due to its own weight can be described by a hyperbolic cosine function. Such a curve is also called a chain line, chain curve or catenoid .

Lorentz transformation

With the help of the rapidity , the transformation matrix for a special Lorentz transformation (also Lorentz boost ) in the x direction can be represented as follows (for transformations in other directions there are similar matrices): ${\displaystyle \lambda }$

${\displaystyle L={\begin{pmatrix}\cosh \lambda &-\sinh \lambda &0&0\\-\sinh \lambda &\cosh \lambda &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}$

You can see a great similarity to rotary dies ; So one can easily see the analogy between special Lorentz transformations in four-dimensional space - time and rotations in three-dimensional space.

cosmology

The hyperbolic sine also occurs in cosmology . The evolution of the scale factor over time in a flat universe containing essentially only matter and dark energy (which is a good model for our actual universe) is described by

${\displaystyle a(t)=\left({\sqrt {\frac {1-\Omega _{\Lambda ,0}}{\Omega _{\Lambda ,0}}}}\sinh \left({\frac {t}{t_{\mathrm {ch} }}}\right)\right)^{2/3}}$,

in which

${\displaystyle t_{\mathrm {ch} }={\frac {2}{3{\sqrt {\Omega _{\Lambda ,0}}}H_{0}}}}$

is a characteristic timescale. is the current value of the Hubble parameter, the density parameter for dark energy. The derivation of this result can be found in the Friedmann equations . In the case of the time dependence of the density parameter of matter, on the other hand, the hyperbolic cosine occurs: ${\displaystyle H_{0}}$ ${\displaystyle \Omega _{\Lambda ,0}}$

${\displaystyle \Omega _{M}(t)=\cosh ^{-2}\left({\frac {t}{t_{\mathrm {ch} }}}\right)}$.

See also

• Areasinus hyperbolicus and Areakosinus hyperbolicus
• Trigonometric functions
• Circle and hyperbolic functions .

Web links

• Eric W. Weisstein: Hyperbolic Sine and Hyperbolic Cosine on MathWorld (engl.)

Individual evidence

1. ↑ Dr. Franz Brzoska, Walter Bartsch: Mathematical formula collection . 2nd improved edition. Fachbuchverlag Leipzig, 1956.