Exponential

The definition of the exponential function is the solution of the equation $ f' = f $ with $ f(0) = 1 $, i.e. the function which is its own derivative and which has the value 1 at 0.

The exponential function is denoted by exp that is, by default, based on the number $ e \approx 2.71828\ldots $ (check also the decimals of the number e).

Example: $ \exp (7) = e^7 \approx 1096.633 $

The $ e^x $ notation is sometimes ambiguous, because $ e $ may be used as a variable, prefer using the $ \exp(x) $ notation.

The exponential has several remarkable properties

$ \exp(0) = 1 \\ \exp(1) = e \approx 2.71828\ldots \\ e^x + e^y = e^x \times e^y \\ (e^x)^b = e^{bx} \\ \ln(\exp(x)) = x \\ \exp(\ln(x)) = x $$

The derivative of the exponential function is the exponential function itself

$$ f(x) = \exp(x) \iff f'(x) = \exp(x) $$

The exponential is related to the exponentiation by the formula:

$$ a^b = e^{b\ln(a)} $$

In the complex plane, the exponential has several other properties (complex exponential form):

$$ \exp(i x) = \cos x + i \sin x \\ \exp(a + i b) = \exp(a) ( \cos b + i \sin b ) $$

The exponential function can be defined as a series expansion based on factorial and exponentiation:

$$ \exp(x)=\sum _{{n=0}}^{{\infty }}{x^{n} \over n!} $$