Definition:Exponential Function/Real/Differential Equation

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\exp: \R \to \R_{>0}$ denote the (real) exponential function.

The exponential function can be defined as the unique solution $y = \map f x$ to the first order ODE:

$\dfrac {\d y} {\d x} = y$

satisfying the initial condition $\map f 0 = 1$.


That is, the defining property of $\exp$ is that it is its own derivative.


The number $\exp x$ is called the exponential of $x$.


Also see

  • Results about the exponential function can be found here.


Sources