The hyperbolic sine, cosine and tangent functions, written sinh x , cosh x and

tanh x , are defined in terms of the exponential function as follows:

sinh x ex ex

2 (hyperbolic sine function),

cosh x ex ex

2 (hyperbolic cosine function),

tanh x sinh x

cosh x (hyperbolic tangent function):

By analogy with trigonometric functions the hyperbolic cosecant, secant

and cotangent functions written cosech x , sech x and coth x , are defined as

cosech x 1

sinh x (x"0) (hyperbolic cosecant function),

sech x 1

cosh x (hyperbolic secant function),

coth x

tanh x (x"0) (hyperbolic cotangent function):

Their graphs, which are not periodic, are shown in Fig. 64. It can be seen

from the definitions, and also from the graphs, that sinh x , tanh x , cosech x and coth x are odd functions, while cosh x and sech x are even functions.