Exponential functions and logarithms: Exponential functions
Hyperbolic functions
Some combinations of =powers are so common that they have been given special names. They are the so-called hyperbolic functions.
Hyperbolic functions From the definitions it immediately follows that is an odd function, is an even function, and that is an odd function. The graphs of and can be quickly sketched as the graph of the sum or difference of two exponential functions. It also follows from the definitions of and that the graphs of these functions are approaching each other for large values of .
graph of sinh
graph of cosh
The graph of is less easily sketched on the basis of the graphs of exponential functions and it is shown below.
The graphs above also illustrate the domain and range of the hyperpbolic functions: they are defined for all real numbers and the range of , and is , and , respectively.
Basic properties, sum formulas and doubling formulas Some properties of the trigonometric functions follow almost directly from the definitions.
Basic identity
Inverse hyperbolic functions Expressed in the natural logarithm, the inverse hyperbolic functions are defined as follows: The domain of , , and is , , and , respectively. The graphs of the inverse hyperbolic functions are shown below. They can be constructed by reflection of the graphs of the hyperbolic functions in the line with equation . This can be seen in the diagrams because the graph of the corresponding hyperbolic function is also shown as a dotted line..
graph of arsinh
graph of arcosh
graph of artanh