Exponential function
The natural exponential function has as base e and is written as
however only, if x is a natural number.
Explanation
Exponential functions play a central role in mathematics. And that is precisely why this constant also is abbreviated with the letter e - of exponent. You can write each number as an exponential function with logarithms as exponent. Trigonometric functions and hyperbolic functions are defined with exponential functions.
Power series
The power series for the exponential function can be written as
Formulas
Euler's formula shows the connection with hyperbolic functions
Here x is specified in radians.