Hyperbolic function

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In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine sinh, and the hyperbolic cosine cosh, from which are derived the hyperbolic tangent tanh, etc., in analogy to the derived trigonometric functions. The inverse functions are the inverse hyperbolic sine arsinh (also called arcsinh in the US or asinh by programmers) and so on.

Just as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the right half of the equilateral hyperbola. Hyperbolic functions are also useful because they occur in the solutions of some simple linear differential equations, notably that defining the shape of a hanging cable, the catenary.

The hyperbolic functions take a real value for real argument called a hyperbolic angle. In complex analysis, they are simply algebraic functions of exponentials, and so are entire.

Contents 1 Standard algebraic expressions 2 Taylor series expressions 3 Relationship to regular trigonometric functions 4 Relationship to the exponential function 5 Inverse hyperbolic functions 6 Applications of inverse trigonometric functions and inverse hyperbolic functions to integrals 7 Hyperbolic functions for complex numbers 8 See also

Standard algebraic expressions

Image:Sinh cosh tanh.svg Image:Csch sech coth.svg

The hyperbolic functions are:

• Hyperbolic sine, pronounced "sinch" or "shine":

<math>\sinh(x) = \frac{e^x - e^{-x}}{2} = -i \sin(i x)</math>

• Hyperbolic cosine, pronounced "cosh":

<math>\cosh(x) = \frac{e^{x} + e^{-x}}{2} = \cos(i x) </math>

• Hyperbolic tangent, pronounced "than" or "tanch":

<math>\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = -i \tan(i x)</math>

• Hyperbolic cotangent, pronounced "coth" or "chot":

<math>\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = \frac{e^{2x} + 1} {e^{2x} - 1} = i \cot(i x)</math>

• Hyperbolic secant, pronounced "sheck" or "sech":

<math>\operatorname{sech}(x) = \frac{1}{\cosh(x)} = \frac {2} {e^x + e^{-x}} = \sec(i x)</math>

• Hyperbolic cosecant, pronounced "cosheck" or "cosech"

<math>\operatorname{csch}(x) = \frac{1}{\sinh(x)} = \frac {2} {e^x - e^{-x}} = i \csc(i x)</math>

where

<math>i \equiv \sqrt{-1}</math>

is the imaginary unit.

The complex forms in the definitions above derive from Euler's formula.

Taylor series expressions

It is possible to express the above functions as Taylor series:

<math>\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)!}</math>

<math>\cosh x = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}</math>

<math>\tanh x = x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=1}^\infty \frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_nx^{2n-1}}{(2n)!}, \left |x \right | < \frac {\pi} {2} </math>

<math>\coth x = \frac {1} {x} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = \frac {1} {x} + \sum_{n=1}^\infty \frac{(-1)^{n-1}2^{2n} B_n x^{2n-1}} {(2n)!}, 0 < \left |x \right | < \pi </math> (Laurent series)

<math>\operatorname {sech} x = 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = 1 + \sum_{n=1}^\infty \frac{(-1)^n E_n x^{2n}}{(2n)!} , \left |x \right | < \frac {\pi} {2} </math>

<math>\operatorname {csch} x = \frac {1} {x} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = \frac {1} {x} + \sum_{n=1}^\infty \frac{(-1)^n 2 (2^{2n}-1) B_n x^{2n-1}}{(2n)!} , 0 < \left |x \right | < \pi </math> (Laurent series)

where

<math>B_n \,</math> is the nth Bernoulli number

<math>E_n \,</math> is the nth Euler number

Relationship to regular trigonometric functions

Just as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the right half of the equilateral hyperbola x² - y² = 1. This is based on the easily verified identity

<math>\cosh^2(t) - \sinh^2(t) = 1 \,</math>

and the property that cosh t > 0 for all t.

The hyperbolic functions are periodic with complex period <math>2 \pi i</math>.

The parameter t is not a circular angle, but rather a hyperbolic angle which represents twice the area between the x-axis, the hyperbola and the straight line which links the origin with the point (cosh t, sinh t) on the hyperbola.

The function cosh x is an even function, that is symmetric with respect to the y-axis, and cosh 0 = 1.

The function sinh x is an odd function, that is symmetric with respect to the origin, and sinh 0 = 0.

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborne's rule states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of two sinh's. This yields for example the addition theorems

<math>\sinh(x+y) = \sinh(x) \cosh(y) + \cosh(x) \sinh(y) \,</math>

<math>\cosh(x+y) = \cosh(x) \cosh(y) + \sinh(x) \sinh(y) \,</math>

and the "half-angle formulas"

<math>\cosh^2\left(\frac{x}{2}\right) = \frac{1+\cosh(x)}{2}</math> Note: This corresponds to its circular counterpart.

<math>\sinh^2\left(\frac{x}{2}\right) = \frac{\cosh(x)-1}{2}</math> Note: This is equivalent to its circular counterpart <math>\times-1</math> .

The derivative of sinh x is given by cosh x and the derivative of cosh x is sinh x.

The graph of the function cosh x is the catenary curve.

Relationship to the exponential function

From the definitions of the hyperbolic sine and cosine, we can derive the following identities:

<math>e^x = \cosh(x) + \sinh(x) \!</math>

and

<math>e^{-x} = \cosh(x) - \sinh(x) \! </math>

These expressions are analogous to the expressions for sine and cosine, based on Euler's formula, as sums of complex exponentials.

Inverse hyperbolic functions

Image:Area tangent.svg The inverses of the hyperbolic functions are the area hyperbolic functions. The usual abbreviations for them in mathematical notation are, e.g., arsinh, arcsinh (US), or asinh in computer science.

<math>\operatorname{arsinh}(x) = \ln(x + \sqrt{x^2 + 1})</math>

<math>\operatorname{arcosh}(x) = \ln(x \pm \sqrt{x^2 - 1})</math>

<math>\operatorname{artanh}(x) = \ln\left(\frac{\sqrt{1 - x^2}}{1-x}\right) = \begin{matrix} \frac{1}{2} \end{matrix} \ln\left(\frac{1+x}{1-x}\right)</math>

<math>\operatorname{arcoth}(x) = \ln\left(\frac{\sqrt{x^2 - 1}}{x-1}\right) = \begin{matrix} \frac{1}{2} \end{matrix} \ln\left(\frac{x+1}{x-1}\right)</math>

<math>\operatorname{arsech}(x) = \ln\left(\frac{1 \pm \sqrt{1 - x^2}}{x}\right)</math>

<math>\operatorname{arcsch}(x) = \ln\left(\frac{1 \pm \sqrt{1 + x^2}}{x}\right)</math>

Expansion series can be obtained for the above functions:

<math>\operatorname{arsinh} (x) = x - \left( \frac {1} {2} \right) \frac {x^3} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {x^5} {5} - \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {x^7} {7} +\cdots = \sum_{n=0}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{2n+1}} {(2n+1)} , \left| x \right| < 1

</math>

<math>\operatorname{arcosh} (x) = \ln 2 - (\left( \frac {1} {2} \right) \frac {x^{-2}} {2} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {x^{-4}} {4} + \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {x^{-6}} {6} +\cdots ) = \ln 2 - \sum_{n=1}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{-2n}} {(2n)} , x > 1

</math>

<math>\operatorname{artanh} (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)} , \left| x \right| < 1 </math>

<math>\operatorname{arcsch} (x) = \operatorname{asinh} (x^{-1}) = x^{-1} - \left( \frac {1} {2} \right) \frac {x^{-3}} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {x^{-5}} {5} - \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {x^{-7}} {7} +\cdots = \sum_{n=0}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{-(2n+1)}} {(2n+1)} , \left| x \right| < 1 </math>

<math>\operatorname{arsech} (x) = \operatorname{acosh} (x^{-1}) = \ln 2 - (\left( \frac {1} {2} \right) \frac {x^{2}} {2} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {x^{4}} {4} + \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {x^{6}} {6} +\cdots ) = \ln 2 - \sum_{n=1}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{2n}} {(2n)} , 0 < x \le 1 </math>

<math>\operatorname{arcoth} (x) = \operatorname{atanh} (x^{-1}) = x^{-1} + \frac {x^{-3}} {3} + \frac {x^{-5}} {5} + \frac {x^{-7}} {7} +\cdots = \sum_{n=0}^\infty \frac {x^{-(2n+1)}} {(2n+1)} , \left| x \right| > 1 </math>

Applications of inverse trigonometric functions and inverse hyperbolic functions to integrals

<math>\int \frac {dx} {\sqrt{1 - x^2}} = \operatorname{arcsin}(x)+{C} = - \operatorname{arccos}(x) + \frac {\pi}{2}+{C}</math>

<math>\int \frac {dx} {\sqrt{x^2 + 1}} = \operatorname{arsinh}(x)+{C} = \ln(x + \sqrt{x^2 + 1})+{C}</math>

<math>\int \frac {dx} {\sqrt{x^2 - 1}} = \operatorname{arcosh}(x)+{C} = \ln(x + \sqrt{x^2 - 1})+{C}</math>

<math>\int \sqrt{1 - x^2} dx = \frac{\operatorname{arcsin}(x) + x\sqrt{1 - x^2}}{2}+{C}</math>

<math>\int \sqrt{x^2 + 1} dx = \frac{\operatorname{arsinh}(x) + x\sqrt{x^2 + 1}}{2}+{C} = \frac{\ln(x + \sqrt{x^2 + 1}) + x\sqrt{x^2 + 1}}{2}+{C}</math>

<math>\int \sqrt{x^2 - 1} dx = \frac{- \operatorname{arcosh}(x) + x\sqrt{x^2 - 1}}{2}+{C}

= \frac{- \ln(x + \sqrt{x^2 - 1}) + x\sqrt{x^2 - 1}}{2}+{C}</math>

<math>\int \frac {dx} {1 + x^2} = \operatorname{arctan}(x)+{C}</math>

<math>\int \frac {dx} {1 - x^2} = \operatorname{artanh}(x)+{C} = \begin{matrix} \frac{1}{2} \end{matrix} \ln\left(\frac{1+x}{1-x}\right)+{C}</math>

Hyperbolic functions for complex numbers

Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh z and cosh z are then holomorphic; their Taylor series expansions are given in the Taylor series article.

Relationships to regular trigonometric functions are given by Euler's formula for complex numbers:

<math>e^{i x} = \cos x + i \;\sin x</math>

<math>\cosh(i x) = \frac{(e^{i x} + e^{-i x})}{2} = \cos(x)</math>

<math>\sinh(i x) = \frac{(e^{i x} - e^{-i x})}{2} = i \sin(x)</math>

<math>\tanh(i x) = i \tan(x) \,</math>

<math>\sinh(x) = -i \sin(i x) \,</math>

<math>\cosh(x) = \cos(i x) \,</math>

<math>\tanh(x) = -i \tan(i x) \,</math>

<math>\operatorname{arsinh}(x) = i \arcsin(-i x)</math>

<math>\operatorname{arcosh}(x) = i \arccos(x)</math>

<math>\operatorname{artanh}(x) = i \arctan(-i x)</math>

<math>\ 2\sum_{j=n}^{kn-1} \operatorname{artanh}\left(\frac{1}{1 + 2\,j}\right)=\ln k </math>

See also

• Sigmoid functionde:Hyperbelfunktion

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