Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined for the unit hyperbola rather than on the unit 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 hyperbola.

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations, cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics.

The basic hyperbolic functions are:

• hyperbolic sine "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),
• hyperbolic cosine "cosh" (/ˈkɒʃ, ˈkoʊʃ/),

from which are derived:

• hyperbolic tangent "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/),
• hyperbolic cosecant "csch" or "cosech" (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/)
• hyperbolic secant "sech" (/ˈsɛtʃ, ˈʃɛk/),
• hyperbolic cotangent "coth" (/ˈkɒθ, ˈkoʊθ/),

corresponding to the derived trigonometric functions.

The inverse hyperbolic functions are:

• area hyperbolic sine "arsinh" (also denoted "sinh⁻¹", "asinh" or sometimes "arcsinh")
• area hyperbolic cosine "arcosh" (also denoted "cosh⁻¹", "acosh" or sometimes "arccosh"
• and so on.

A ray through the unit hyperbola x² − y² = 1 in the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).

The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.

Definitions

sinh, cosh and tanh

csch, sech and coth

There are various equivalent ways to define the hyperbolic functions.

Exponential definitions

sinh x is half the difference of e^x and e^−x

cosh x is the average of e^x and e^−x

In terms of the exponential function:

• Hyperbolic sine: the odd part of the exponential function, that is

  $\sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.$

• Hyperbolic cosine: the even part of the exponential function, that is

  $\cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.$

• Hyperbolic tangent:

  $\tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}$

• Hyperbolic cotangent: for x ≠ 0,

  $\coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}$

• Hyperbolic secant:

  $\operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}$

• Hyperbolic cosecant: for x ≠ 0,

  $\operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}$

Useful relations

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities.

Odd and even functions:

${\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\end{aligned}}$

Hence:

${\begin{aligned}\tanh(-x)&=-\tanh x\\\coth(-x)&=-\coth x\\\operatorname {sech} (-x)&=\operatorname {sech} x\\\operatorname {csch} (-x)&=-\operatorname {csch} x\end{aligned}}$

Thus, cosh x and sech x are even functions; the others are odd functions.

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

Hyperbolic sine and cosine satisfy:

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

the last of which is similar to the Pythagorean trigonometric identity.

One also has

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

for the other functions.

Comparison with circular functions

Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of circular sector area u and hyperbolic functions depending on hyperbolic sector area u.

The hyperbolic functions represent an expansion of trigonometry beyond the functions defined on unit circle. Both types depend on one 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 sectors together depict an area and hyperbolic angle magnitude.

The legs of the two right triangles with hypotenuse on the ray defining the angles are of length ${\sqrt {2}}$ times the circular and hyperbolic functions.

Relationship to the exponential function

The decomposition of the exponential function in its even and odd parts gives the identities

$e^{x}=\cosh x+\sinh x,$

and

$e^{-x}=\cosh x-\sinh x.$

The first one is analogous to Euler's formula

$e^{ix}=\cos x+i\sin x.$

Additionally,

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

Related pages

• e (mathematical constant)
• Trigonometric functions

Other websites

• Hyperbolic functions on PlanetMath
• GonioLab: Visualization of the unit circle, trigonometric and hyperbolic functions (Java Web Start)
• Web-based calculator of hyperbolic functions