Exponential function

The exponential function is

\exp(x)=e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!},\qquad x\in\mathbb R\text{ or }\mathbb C.

This power series has infinite radius of convergence. Termwise differentiation gives $\exp'(x)=\exp(x)$ and $\exp(0)=1$ . Multiplication of absolutely convergent series gives

\exp(x+y)=\exp(x)\exp(y).

Real restriction

For real $x$ , $\exp(x)>0$ , $\exp(-x)=1/\exp(x)$ , and $\exp$ is strictly increasing from $\mathbb R$ onto $(0,\infty)$ .

Growth and decay

For every nonnegative integer $m$ , $x^m e^{-x}\to0$ as $x\to+\infty$ . Indeed, positivity of the series gives $e^x\ge x^{m+1}/(m+1)!$ for $x>0$ . This estimate is a basic source of functions vanishing to every order at an endpoint after replacing $x$ by a reciprocal distance.

Complex extension

The same power series is holomorphic everywhere in $\mathbb C$ . Separating even and odd powers gives

e^{x+iy}=e^x(\cos y+i\sin y),\qquad x,y\in\mathbb R.

It never vanishes, since $e^ze^{-z}=1$ , and has imaginary period $2\pi i$ . Its real restriction is one-to-one, but its complex extension is periodic and needs a branch choice for an inverse logarithm.

References

John K. Hunter, Introduction to Analysis, Chapter 10: Power Series.