Hyperbolic function

Hyperbolic sine (red)
Hyperbolic cosine (blue)
Hyperbolic tangent (green)

Hyperbolic cotangent (red)
Hyperbolic secant (blue)
Hyperbolic cotangent (green)

The hyperbolic functions are the corresponding functions of the trigonometric functions (which are also referred to as angle or circle functions), but not on the unit circle , but on the unit hyperbola . ${\displaystyle x^{2}+y^{2}=1}$ ${\displaystyle x^{2}-y^{2}=1}$

How closely these functions are related to one another becomes even clearer in the complex number level. It is conveyed through the relation . So z. B. . ${\displaystyle (iy)^{2}=-y^{2}}$ ${\displaystyle \cos(ix)=\cosh x}$

The following functions belong to the hyperbolic functions :

• Hyperbolic or lat hyperbolic sine (symbol:. ) ${\displaystyle \sinh }$
• Hyperbolic cosine or Latin hyperbolic cosine ( ) ${\displaystyle \cosh }$
• Hyperbolic tangent or Latin hyperbolic tangent ( ) ${\displaystyle \tanh }$
• Hyperbolic cotangent or Latin hyperbolic cotangent ( ) ${\displaystyle \coth }$
• Hyperbolic secant or Latin hyperbolic secant ( ) ${\displaystyle \operatorname {sech} }$
• Hyperbolic kosekans or Latin hyperbolic kosekans ( ). ${\displaystyle \operatorname {csch} }$

In the German and Dutch languages, the Latin names are still used very often, sometimes with Germanized spelling.

Hyperbolic sines and hyperbolic cosines are defined for all complex numbers and holomorphic in the entire field of complex numbers . The other hyperbolic functions have poles on the imaginary axis.

definition

A straight line from the origin intersects the hyperbola at the point , with the area between the straight line, its mirror image on the -axis, and the hyperbola. ${\displaystyle x^{2}-y^{2}=1}$ ${\displaystyle (\cosh A,\sinh A)}$ ${\displaystyle A}$ ${\displaystyle x}$

Definition via the exponential function

Using the exponential function , and can be defined as follows: ${\displaystyle \sinh }$ ${\displaystyle \cosh }$

${\displaystyle \sinh(z):={\frac {e^{z}-e^{-z}}{2}}}$

${\displaystyle \cosh(z):={\frac {e^{z}+e^{-z}}{2}}}$

Therefore the hyperbolic functions are periodic (with a purely imaginary period). The power series of and are ${\displaystyle \cosh }$ ${\displaystyle \sinh }$

${\displaystyle {\begin{aligned}\sinh(z)&=z+{\frac {z^{3}}{3!}}+{\frac {z^{5}}{5!}}+{\frac {z^{7}}{7!}}+\dots =\sum _{n=0}^{\infty }{\frac {z^{2n+1}}{(2n+1)!}}\\\cosh(z)&=1+{\frac {z^{2}}{2!}}+{\frac {z^{4}}{4!}}+{\frac {z^{6}}{6!}}+\dots =\sum _{n=0}^{\infty }{\frac {z^{2n}}{(2n)!}}\,,\end{aligned}}}$

where the expression stands for the factorial of , the product of the first natural numbers . In contrast to the power series expansions of and , all terms have a positive sign. ${\displaystyle n!}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \cos }$ ${\displaystyle \sin }$

Geometric definition with the help of the hyperbola

Because of their use to parameterize the unit hyperbola : ${\displaystyle x^{2}-y^{2}=1}$

${\displaystyle x=\cosh(t),y=\sinh(t)}$

they are called hyperbolic functions, in analogy to the circular functions sine and cosine, which parameterize the unit circle : ${\displaystyle x^{2}+y^{2}=1}$

${\displaystyle x=\cos(t),y=\sin(t)}$

The functions create a connection between the area enclosed by a straight line starting from the zero point and its mirror image on the axis and the hyperbola, and the length of various lines. ${\displaystyle A}$ ${\displaystyle x}$

Here, the (positive) coordinate of the intersection point of the straight line with the hyperbola and the associated coordinate; is the coordinate of the straight line at , i.e. H. the slope of the straight line. ${\displaystyle \sinh(A)}$ ${\displaystyle y}$ ${\displaystyle \cosh(A)}$ ${\displaystyle x}$ ${\displaystyle \tanh(A)}$ ${\displaystyle y}$ ${\displaystyle x=1}$

If the area is calculated by integration , the representation is obtained using the exponential function.

Properties of the real hyperbolic functions

Graph of the real hyperbolic functions

• For all real numbers are and real. ${\displaystyle x}$ ${\displaystyle \sinh(x)}$ ${\displaystyle \cosh(x)}$
• The real function is strictly monotonically increasing and has only one inflection point. ${\displaystyle \sinh }$ ${\displaystyle x=0}$
• The real function is for strictly monotonically decreasing on the interval, strictly monotonically increasing on the interval and has a global minimum at. ${\displaystyle \cosh }$ ${\displaystyle (-\infty ,0]}$ ${\displaystyle [0,\infty )}$ ${\displaystyle x=0}$

Because of this , all the properties of the complex hyperbolic functions listed in the following paragraph also apply to the functions that are restricted to the real numbers. ${\displaystyle \sinh ,\cosh \colon \mathbb {R} \mapsto \mathbb {R} }$

Properties of the complex hyperbolic functions

The following applies to all complex numbers : ${\displaystyle z,z_{1},z_{2}}$

Symmetry and periodicity

• ${\displaystyle \sinh(z)=-\sinh(-z)}$, d. H. sinh is an odd function .
• ${\displaystyle \cosh(z)=\cosh(-z)}$, d. H. cosh is an even function.

• ${\displaystyle \sinh(z)=\sinh(z+2\pi i)\quad {\text{ und }}\quad \cosh(z)=\cosh(z+2\pi i)}$,

d. H. there is purely “imaginary periodicity” with a minimum period length . ${\displaystyle 2\pi }$

Addition theorems

• ${\displaystyle \sinh(z_{1}\pm z_{2})=\sinh(z_{1})\cdot \cosh(z_{2})\pm \sinh(z_{2})\cdot \cosh(z_{1})}$
• ${\displaystyle \cosh(z_{1}\pm z_{2})=\cosh(z_{1})\cdot \cosh(z_{2})\pm \sinh(z_{1})\cdot \sinh(z_{2})}$
• ${\displaystyle \tanh(z_{1}\pm z_{2})={\frac {\tanh(z_{1})\pm \tanh(z_{2})}{1\pm \tanh(z_{1})\tanh(z_{2})}}}$

Connections

${\displaystyle {\cosh }^{2}(z)-{\sinh }^{2}(z)=1}$

${\displaystyle \cosh z+\sinh z\ =e^{z}}$

${\displaystyle \cosh z-\sinh z\ =e^{-z}}$

Derivation

The derivative of the hyperbolic sine is:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\sinh }(z)=\cosh(z)}$.

The derivative of the hyperbolic cosine is:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\cosh }(z)=\sinh(z)}$.

The derivative of the hyperbolic tangent is:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\tanh }(z)=1-{\tanh }^{2}(z)={\frac {1}{\cosh ^{2}(z)}}}$.

Differential equation

The functions and form a solution basis ( fundamental system ) of the linear differential equation like and ${\displaystyle \sinh(z)}$ ${\displaystyle \cosh(z)}$ ${\displaystyle e^{z}}$ ${\displaystyle e^{-z}}$

${\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}f(z)=f(z)}$.

If one demands , and , for the two basic solutions of this differential equation of the second order , then they are already clearly defined by and . In other words, this property can also be used to define these two hyperbolic functions. ${\displaystyle f_{i}(z)}$ ${\displaystyle f_{1}(0)=0}$ ${\displaystyle f_{1}'(0)=1}$ ${\displaystyle f_{2}(0)=1}$ ${\displaystyle f_{2}'(0)=0}$ ${\displaystyle \sinh }$ ${\displaystyle \cosh }$

Bijectivity of the complex hyperbolic functions

sinh

The following subsets of the complex numbers are defined:

${\displaystyle A:=\{z\in \mathbb {C} \mid -\pi /2<\operatorname {Im} \,z<\pi /2\}}$

${\displaystyle B:=\{z\in \mathbb {C} \mid \operatorname {Re} \,z\neq 0\vee |\operatorname {Im} \,z|<1\}}$

Then the complex function is to "strip" bijective on from. ${\displaystyle \sinh }$ ${\displaystyle A}$ ${\displaystyle B}$

cosh

The following subsets of the complex numbers are defined:

${\displaystyle A:=\{z\in \mathbb {C} \mid 0<\operatorname {Im} \,z<\pi \}}$

${\displaystyle B:=\{z\in \mathbb {C} \mid \operatorname {Im} \,z\neq 0\vee |\operatorname {Re} \,z|<1\}}$

Then the complex function is to "strip" bijective on from. ${\displaystyle \cosh }$ ${\displaystyle A}$ ${\displaystyle B}$

Historical notation

In German literature the hyperbolic functions have long been in for the distinction of the trigonometric functions Frakturschrift shown - with initial capitalization and no final hour:

${\displaystyle {\mathfrak {Sin}}\,x{\widehat {=}}\sinh x}$

${\displaystyle {\mathfrak {Cos}}\,x{\widehat {=}}\cosh x}$

${\displaystyle {\mathfrak {Tan}}\,x\ /\ {\mathfrak {Tg}}\,x{\widehat {=}}\tanh x\ /\ \operatorname {tgh} x}$

${\displaystyle {\mathfrak {Cot}}\,x\ /\ {\mathfrak {Ctg}}\,x{\widehat {=}}\coth x\ /\ \operatorname {ctgh} x}$

${\displaystyle {\mathfrak {Sec}}\,x{\widehat {=}}\operatorname {sech} x}$

${\displaystyle {\mathfrak {Csc}}\,x{\widehat {=}}\operatorname {csch} x}$

Alternative names

• The name hyperbolic functions is also used for the hyperbolic functions .
• The names hsin , hyperbolic sine and hyperbolic sine are also used for . ${\displaystyle \sinh }$
• The names hcos , hyperbolic cosine and hyperbolic cosine are also used for . The graph corresponds to the chain line (catenoids). ${\displaystyle \cosh }$

Derived functions

• Hyperbolic tangent : ${\displaystyle \tanh(x):={\frac {\sinh(x)}{\cosh(x)}}}$
• Hyperbolic cotangent : ${\displaystyle \coth(x):={\frac {1}{\tanh(x)}}={\frac {\cosh(x)}{\sinh(x)}}}$
• Secans hyperbolicus : ${\displaystyle \operatorname {sech} (x):={\frac {1}{\cosh(x)}}}$
• Kosecans hyperbolicus : ${\displaystyle \operatorname {csch} (x):={\frac {1}{\sinh(x)}}}$

Conversion table

function	${\displaystyle \sinh }$	${\displaystyle \cosh }$	${\displaystyle \tanh }$	${\displaystyle \coth }$	${\displaystyle \operatorname {sech} }$	${\displaystyle \operatorname {csch} }$
${\displaystyle \sinh(x)=}$	${\displaystyle \sinh(x)\,}$	${\displaystyle \operatorname {sgn}(x){\sqrt {\cosh ^{2}(x)-1}}}$	${\displaystyle {\frac {\tanh(x)}{\sqrt {1-\tanh ^{2}(x)}}}}$	${\displaystyle {\frac {\operatorname {sgn}(x)}{\sqrt {\coth ^{2}(x)-1}}}}$	${\displaystyle \operatorname {sgn}(x){\frac {\sqrt {1-\operatorname {sech} ^{2}(x)}}{\operatorname {sech} (x)}}}$	${\displaystyle {\frac {1}{\operatorname {csch} (x)}}}$
${\displaystyle \cosh(x)=}$	${\displaystyle \,{\sqrt {1+\sinh ^{2}(x)}}}$	${\displaystyle \,\cosh(x)}$	${\displaystyle \,{\frac {1}{\sqrt {1-\tanh ^{2}(x)}}}}$	${\displaystyle \,{\frac {\left|\coth(x)\right|}{\sqrt {\coth ^{2}(x)-1}}}}$	${\displaystyle \,{\frac {1}{\operatorname {sech} (x)}}}$	${\displaystyle \,{\frac {\sqrt {1+\operatorname {csch} ^{2}(x)}}{\left|\operatorname {csch} (x)\right|}}}$
${\displaystyle \tanh(x)=}$	${\displaystyle \,{\frac {\sinh(x)}{\sqrt {1+\sinh ^{2}(x)}}}}$	${\displaystyle \,\operatorname {sgn}(x){\frac {\sqrt {\cosh ^{2}(x)-1}}{\cosh(x)}}}$	${\displaystyle \,\tanh(x)}$	${\displaystyle \,{\frac {1}{\coth(x)}}}$	${\displaystyle \,\operatorname {sgn}(x){\sqrt {1-\operatorname {sech} ^{2}(x)}}}$	${\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}$
${\displaystyle \coth(x)=}$	${\displaystyle \,{\frac {\sqrt {1+\sinh ^{2}(x)}}{\sinh(x)}}}$	${\displaystyle \,\operatorname {sgn}(x){\frac {\cosh(x)}{\sqrt {\cosh ^{2}(x)-1}}}}$	${\displaystyle \,{\frac {1}{\tanh(x)}}}$	${\displaystyle \,\coth(x)}$	${\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}$	${\displaystyle \,\operatorname {sgn}(x){\sqrt {1+\operatorname {csch} ^{2}(x)}}}$
${\displaystyle \operatorname {sech} (x)=}$	${\displaystyle \,{\frac {1}{\sqrt {1+\sinh ^{2}(x)}}}}$	${\displaystyle \,{\frac {1}{\cosh(x)}}}$	${\displaystyle \,{\sqrt {1-\tanh ^{2}(x)}}}$	${\displaystyle \,{\frac {\sqrt {\coth ^{2}(x)-1}}{\left|\coth(x)\right|}}}$	${\displaystyle \,\operatorname {sech} (x)}$	${\displaystyle \,{\frac {\left|\operatorname {csch} (x)\right|}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}$
${\displaystyle \operatorname {csch} (x)=}$	${\displaystyle \,{\frac {1}{\sinh(x)}}}$	${\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {\cosh ^{2}(x)-1}}}}$	${\displaystyle \,{\frac {\sqrt {1-\tanh ^{2}(x)}}{\tanh(x)}}}$	${\displaystyle \,\operatorname {sgn}(x){\sqrt {\coth ^{2}(x)-1}}}$	${\displaystyle \,\operatorname {sgn}(x){\frac {\operatorname {sech} (x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}$	${\displaystyle \,\operatorname {csch} (x)}$

Inverse functions

The inverse functions of the hyperbolic functions are called area functions .

See also: Connection with the circular functions

literature

• Ilja N. Bronstein: Paperback of mathematics . German (Harri).

Web links

Commons : Hyperbolic functions  - collection of images, videos and audio files

Individual evidence

1. ^ Stefan Hildebrandt : Analysis . Springer , 2002, ISBN 978-3-540-42838-1 , pp. 243 , doi : 10.1007 / 978-3-662-05694-3 ( limited preview in Google Book search).