Exponential Function

By definition… (Taylor Approximation)

$$e^x=\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^x=e$$

Or by definition, the family of functions that satisfy:

$$f(x+y)=f(x)\cdot f(y)$$

(See Euler’s formula with introductory group theory - YouTube)

Another Definition:

$$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}=1+x+\frac{x^2}{2}+\dots$$