Hyperbolic functions

 ( Exponentials ) 

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

• Hyperbolic cotangent: for , $$\backslash coth\; x\; =\; \backslash frac\{\backslash cosh\; x\}\{\backslash sinh\; x\}\; =\; \backslash frac\; \{e^x\; +\; e^\{-x\}\}\; \{e^x\; -\; e^\{-x\}\}$$

• Hyperbolic secant: $$\backslash operatorname\{sech\}\; x\; =\; \backslash frac\{1\}\{\backslash cosh\; x\}\; =\; \backslash frac\; \{2\}\; \{e^x\; +\; e^\{-x\}\}$$

• Hyperbolic cosecant: for , $$\backslash operatorname\{csch\}\; x\; =\; \backslash frac\{1\}\{\backslash sinh\; x\}\; =\; \backslash frac\; \{2\}\; \{e^x\; -\; e^\{-x\}\}$$

and  are also the unique solution of the equation ,
     

• Hyperbolic sine: $$\backslash sinh\; x\; =\; -i\; \backslash sin\; (i\; x).$$
• Hyperbolic cosine: $$\backslash cosh\; x\; =\; \backslash cos\; (i\; x).$$
• Hyperbolic tangent: $$\backslash tanh\; x\; =\; -i\; \backslash tan\; (i\; x).$$
• Hyperbolic cotangent: $$\backslash coth\; x\; =\; i\; \backslash cot\; (i\; x).$$
• Hyperbolic secant: $$\backslash operatorname\{sech\}\; x\; =\; \backslash sec\; (i\; x).$$
• Hyperbolic cosecant:$$\backslash operatorname\{csch\}\; x\; =\; i\; \backslash csc\; (i\; x).$$

The above definitions are related to the exponential definitions via Euler's formula (See below).

\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
     

\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)
     

\cosh x + \sinh x &= e^x \\
\cosh x - \sinh x &= e^{-x}
     

\cosh^2 x - \sinh^2 x = 1
     

\operatorname{sech} ^{2} x &= 1 - \tanh^{2} x \\
\operatorname{csch} ^{2} x &= \coth^{2} x - 1
     

\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 } \\
     

\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)\\
     

\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 } \\
     

\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)\\
     

\sinh\left(\frac{x}{2}\right) &= \frac{\sinh x}{\sqrt{2 (\cosh x + 1)} } &&= \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} &&= \sgn x \, \sqrt \frac{\cosh x-1}{\cosh x+1} = \frac{e^x - 1}{e^x + 1}
     

where is the sign function.

If , then

$$\backslash tanh\backslash left(\backslash frac\{x\}\{2\}\backslash right)\; =\; \backslash frac\{\backslash cosh\; x\; -\; 1\}\{\backslash sinh\; x\}\; =\; \backslash coth\; x\; -\; \backslash operatorname\{csch\}\; x$$

It can be proved by comparing the Taylor series of the two functions term by term.

\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 \ne 0
     

\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
     

\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}
\end{align}
     
     

 
 

 
 
  

  Second derivatives
 
 

 
Each of the functions  and  is equal to its second derivative, that is:
     $$\backslash frac\{d^2\}\{dx^2\}\backslash sinh\; x\; =\; \backslash sinh\; x$$
     $$\backslash frac\{d^2\}\{dx^2\}\backslash cosh\; x\; =\; \backslash cosh\; x\; \backslash ,\; .$$
     

All functions with this property are linear combinations of  and , in particular the exponential functions $e^x$ and $e^\{-x\}$.
     

 

 
 
  

  Standard integrals
 
 

 
 
 
     $$\backslash begin\{align\}$$
\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
+ C = -a^{-1}\operatorname{arcoth} \left(\cosh\left(ax\right)\right) +C
     
\end{align}
     

The following integrals can be proved using hyperbolic substitution:
     $$\backslash begin\{align\}$$
\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} &= \sgn{u} \operatorname{arcosh} \left
+ 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
+ C \\
\int {\frac{1}{u\sqrt{a^2 + u^2}}\,du} & = -a^{-1}\operatorname{arcsch}\left
+ C
     
\end{align}
     

where  C is the constant of integration.
     

 

 
 
  

  Taylor series expressions
 
 

 
It is possible to express explicitly the Taylor series at zero (or the Laurent series, if the function is not defined at zero) of the above functions.
     

     $$\backslash sinh\; x\; =\; x\; +\; \backslash frac\; \{x^3\}\; \{3!\}\; +\; \backslash frac\; \{x^5\}\; \{5!\}\; +\; \backslash frac\; \{x^7\}\; \{7!\}\; +\; \backslash cdots\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{x^\{2n+1\}\}\{(2n+1)!\}$$
This series is convergent for every complex number value of . Since the function  is odd function, only odd exponents for  occur in its Taylor series.
     

     $$\backslash cosh\; x\; =\; 1\; +\; \backslash frac\; \{x^2\}\; \{2!\}\; +\; \backslash frac\; \{x^4\}\; \{4!\}\; +\; \backslash frac\; \{x^6\}\; \{6!\}\; +\; \backslash cdots\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{x^\{2n\}\}\{(2n)!\}$$
This series is convergent for every complex number value of . Since the function  is even function, only even exponents for  occur in its Taylor series.
     

The sum of the sinh and cosh series is the infinite series expression of the exponential function.
     

The following series are followed by a description of a subset of their domain of convergence, where the series is convergent and its sum equals the function.
     $$\backslash begin\{align\}$$

     
              \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
< \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
< \pi \\
     
     

     
\operatorname{sech} x &= 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = \sum_{n=0}^\infty \frac{E_{2 n} x^{2n}}{(2n)!} , \qquad \left
< \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
< \pi
     
     

\end{align}
     

where:
 

  

   $B\_n$ is the   nth Bernoulli number
  
  

   $E\_n$ is the   nth Euler number
  
 

 

 
 
  

  Infinite products and continued fractions
 
 

 
The following expansions are valid in the whole complex plane:
     

     
$\backslash sinh\; x\; =\; x\backslash prod\_\{n=1\}^\backslash infty\backslash left(1+\backslash frac\{x^2\}\{n^2\backslash pi^2\}\backslash right)\; =$
     
\cfrac{x}{1 - \cfrac{x^2}{2\cdot3+x^2 -
\cfrac{2\cdot3 x^2}{4\cdot5+x^2 -
     
     

     

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

     

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

 

 
 
  

  Comparison with circular functions
 
 

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

Since the area of a circular sector with radius  and angle  (in radians) is , it will be equal to  when . 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  with hypotenuse on the ray defining the angles are of length  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.Haskell, Mellen W., "On the introduction of the notion of hyperbolic functions",  Bulletin of the American Mathematical Society  1:6:155–9,  full text
     

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  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
 
 

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

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

 

 
 
  

  Hyperbolic functions for complex numbers
 
 

 
{
style="text-align:center"
+ Hyperbolic functions in the complex plane
 
 
 
 
 
 
$\backslash sinh(z)$
$\backslash cosh(z)$
$\backslash tanh(z)$
$\backslash coth(z)$
$\backslash operatorname\{sech\}(z)$
$\backslash operatorname\{csch\}(z)$
Since the exponential function can be defined for any complex number argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions  and  are then holomorphic.
     

Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:
     $$\backslash begin\{align\}$$
 e^{i x} &= \cos x + i \sin x \\
e^{-i x} &= \cos x - i \sin x
     
\end{align}
so:
     $$\backslash begin\{align\}$$
  \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(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{align}
     

Thus, hyperbolic functions are periodic with respect to the imaginary component, with period $2\; \backslash pi\; i$ ($\backslash pi\; i$ for hyperbolic tangent and cotangent).
     

 

 
 
  

  See also
 
 

 
 

  

e (mathematical constant)
  
  

Equal incircles theorem, based on sinh
  
  

Hyperbolastic functions
  
  

Hyperbolic growth
  
  

Inverse hyperbolic functions
  
  

List of integrals of hyperbolic functions
  
  

Poinsot's spirals
  
  

     Sigmoid function
  
  

Soboleva modified hyperbolic tangent
  
  

Trigonometric functions
  
 

 
 
 

 
 
  

  External links
 
 

 
 
 

  

      Hyperbolic functions on PlanetMath
  
  

      GonioLab: Visualization of the unit circle, trigonometric and hyperbolic functions (Java Web Start)
  
  

      Web-based calculator of hyperbolic functions
  
 

 
 
 
 
 
 
Categories: Exponentials, Hyperbolic Geometry, Analytic Functions