### 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.