Hyperbolic functions

Certain combination of the exponential functions e^x and e^−x arise so frequently in mathematics and its applications that they deserve to be given special names. These are the hyperbolic functions, defined as follows

Hyperbolic sine

\sinh x = \frac{e^{x} - e^{x}}{2}

Hyperbolic cosine

\cosh x = \frac{e^{x} - e^{x}}{2}

Hyperbolic tangent

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

Note that sinh has domain ℝ and range ℝ, while cosh has domain ℝ and range [1, ∞]. From the definitions the following properties follow:

Sh (−x) = − Sh (x) (odd function);
Ch (−x) = Ch (−x) (even function);
Th (−x) = −Th (x) (odd function);

The variable

The reason for the names of these functions is that they are related to the hyperbola in much the same way that trigonometric functions are related to the circle; A point on the circle x² + y² = a² may be expressed parametrically as x = a cos θ, y = a sin θ, whereas a point on one branch of the rectangular hyperbola x² − y² = a² may be represented by x = a cosh φ, y = a sinh φ, satisfying the equation x² − y² = a².

hyperbolic functions — 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 most famous application is the use of hyperbolic cosine to describe the shape of a hanging wire (such as a telephone or power line). Galilei thought the chain curve was a parabola. catenary

But later it was proved that it takes the shape of a curve with equation

y = h + a \cdot \cosh \frac{x - x_{0}}{a}

called a catenary. (The Latin word catena means “chain.”). In which x₀ represent the value for which the catenary has either a minimum or maximum, h is the value of the catenary at x₀; The paramter a also characterizes the form of the catenary: for a > 0 it is a sagging chain, for a < 0 it is an arch. The actual value for a depends on the tension in the cable and the density of the cable.

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