Exponential function

The exponential function $e x p : ℝ \to ℝ : 𝑥 \mapsto 𝑒 𝑥$ was originally defined and motivated exponentiation (repeated multiplication), but is usually defined by the following Power series #m/def/anal

e x p 𝑥 = \infty \sum 𝑘 = 0 𝑥 𝑘 𝑘! = 1 + 𝑥 + 𝑥 2 2 + 𝑥 3 6 + 𝑥 4 24 + \dots

or equivalently by the limit

e x p 𝑥 = l i m 𝑘 \to \infty (1 + 𝑥 𝑘) 𝑘

It solves the First-order linear differential equation

𝑑 𝑦 𝑑 𝑥 = 𝑦

Remarks

The power series definition is used to define the Matrix exponential

#state/develop | #lang/en | #SemBr