ABSTRACT
Hyperbolic functions There are two combinations of e x $ {{e}}^{\boldsymbol{x}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351232876/7c39b375-15cb-4b0b-ad64-16e4787c18a4/content/inline-math24_1.tif"/> and e - x $ {{e}}^{\boldsymbol{-\ x}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351232876/7c39b375-15cb-4b0b-ad64-16e4787c18a4/content/inline-math24_2.tif"/> which are used so often in engineering that they are given their own name. They are the hyperbolic sine, sinh, and the hyperbolic cosine, cosh. They are of interest because they have many properties analogous to those of trigonometric functions and because they arise in the study of falling bodies, hanging cables, ocean waves, and many other phenomena in science and engineering. The shape of a chain hanging under gravity is well described by cosh x and the deformation of uniform beams can be expressed in terms of hyperbolic tangents. Other applications of hyperbolic functions are found in fluid dynamics, optics, heat, mechanical engineering, and in astronomy when dealing with the curvature of light in the presence of black holes.
