Hyperbolic Functions
The hyperbolic functions are similar to the trigonometric functions, often called circular functions, in that they play an important role in problems in mathematics and the mathematics of physics. The hyperbolic functions are involved in mathematical problems in which integrals involving (1 + x2) arise whereas trigonometric functions involve functions with integrals of the type (1 - x2). Although the hyperbolic functions are similar to trigonometric functions, their definitions are much more straightforward. As the trigonometric functions yield parameters describing the unit circle, the hyperbolic functions yield parameters describing the standard hyperbola: x2 - y2 = 1 for x > 1. For every trigonometric function there is a corresponding hyperbolic function, though they are not necessarily identical.
Like the trigonometric functions the hyperbolic functions are periodic because they are all elliptic functions. Elliptic functions can be doubly periodic, singly periodic, or non-periodic, which is often referred to as trivially periodic. The exponential function, on which the hyperbolic functions are based, is a singly periodic elliptic function that means the modulus of periodicity is 2. This means that 2 is the smallest constant for which the identity is true. The two main hyperbolic functions are hyperbolic sine, sinh x, and hyperbolic cosine, cosh x. They are defined as: sinh x = (ex - e-x)/2 and cosh x = (ex + e-x)/2. Analogous to the trigonometric functions sinh x is an odd function with sinh 0 = 0, and cosh x is an even functions with cosh 0 = 1. Because of the properties of ex, it follows that sinh x < 0 for x < 0, sinh x > 0 for x > 0, and cosh x > 0 for all x. The four other hyperbolic functions, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant, can all be defined in terms of sinh x and cosh x: tanh x = sinh x / cosh x, coth x = cosh x / sinh x, sech x = 1 / cosh x, and csch x = 1 / sinh x. Just as there are trigonometric identities there are hyperbolic identities, the most important being the analogous version of the Pythagorean theorem: (cosh x)2 - (sinh x)2 = 1.
Although Johann Heinrich Lambert, a French mathematician, is often credited for the introduction of the hyperbolic functions, it was actually Vincenzo Riccati, an Italian mathematician, who did so in the mid 18th century. He studied these functions and employed them to obtain solutions of cubic equations. Riccati found the standard addition formulas, similar to trigonometric identities, for hyperbolic functions as well as their derivatives. He revealed the relationship between the hyperbolic functions and the exponential function. This work, some done jointly with Saladini, was published between 1757 and 1767. In these publications Riccati used Sh. and Ch. for hyperbolic sine and cosine similar to his abbreviations for the circular functions, Sc. and Cc. Later, in 1768, Lambert published further developments concerning the theory of hyperbolic functions. He used sin h and cos h to abbreviate hyperbolic sine and hyperbolic cosine at that time. In 1771 Lambert began using sinh, the Latin sinus hyperbolus, and cosh to represent these functions. Although this notation is widely used today to abbreviate the hyperbolic functions, in some instances hysin and hycos are used. These notations were introduced in 1902 by George Minchin and published in the journal Nature that year.
The hyperbolic functions are important to many mathematical and physical problems. Hyperbolic sine arises in the formulation of the gravitational potential of a cylinder and the calculation of the Roche limit. Hyperbolic cosine is connected with the catenary form, meaning chain. This shape is formed when a flexible inelastic chain of uniform density is suspended by its two ends and is subjected to only the influence of gravity. This shape imparts great strength to structures when built in the shape of a catenary. The hyperbolic tangent is involved in the calculation of the magnetic moment as well as the rapidity of special relativity. The rapidity of special relativity is defined as: Θ = tanh-1B, where B = /c, and is the speed of a particle and c is the speed of light in a vacuum. The hyperbolic secant is involved in the profile of a laminar jet and the hyperbolic cotangent is connected with the Langevin function for magnetic polarization. As can be seen the hyperbolic functions are not only important in mathematics but in physics and engineering.
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