Hyperbolic functions

From secondary school we know very well trigonometric functions, namely sine, cosine, and the derived from them tg, ctg and others. However, no all people know that there exist so-called hyperbolic functions, namely hyperbolic sine, hyperbolic cosine and others. Trigonometric and hyperbolic functions together with some other functions form a very important class of elementary functions, which are used in mathematics very often.

In this page of our Museum we will try to tell you about the hyperbolic functions and their role in natural sciences.

Let's start from a definition of the hyperbolic functions. In contrast to trigonometric functions, which have not definition in analytic form, the hyperbolic functions can be presented in the following form:

(1) Hyperbolic sine

(2) Hyperbolic cosine

By analogy with trigonometric functions one may define the following hyperbolic functions:

(3) Hyperbolic tangent

(4) Hyperbolic cotangent

Note that graphically hyperbolic functions shx, chx and thx can be presented as the following:

There exists a very deep mathematical analogy between trigonometric and hyperbolic functions. For example, there is well known the following formulas connecting trigonometric functions:

However, similar formulas exist for the hyperbolic functions:

Hyperbolic functions are used very widely for modeling of natural phenomena. Let us consider two important examples. Nikolay Lobatchevski used hyperbolic functions in your non-Euclidean geometry and therefore Lobatchevski's geometry is called hyperbolic geometry. The famous mathematician Minkovski used basic relationships of hyperbolic functions for highly interesting geometric interpretation of Einstein's theory of relativity (Minkovski's geometry).

You can know hyperbolic functions more in detail in the brochure "Hyperbolic Functions" (1958) written by the Russian mathematician V.G. Shervatov.

But why we tell about the hyperbolic functions in our Museum? Of what is connection of subject of our Museum with hyperbolic functions? Later we will show that this connection is very deep and fundamental! Follow us!